( PDF ) Scaling Limits: d-dimensional models with conductances, velocity, reservatories and random environment

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Instituto Nacional de Matem´atica Pura e Aplicada

Scaling limits: d-dimensional models with

conductances, velocity, reservatories and random

environment

abio J

ulio da Silva Valentim

Tese apresentada para obten¸c˜ao do t´ıtulo de Doutor em Ciˆencias

Orientador: Claudio Landim

Rio de Janeiro

Junho, 2010

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Resumo

Nesta tese consideramos trˆes modelos de processo de exclus˜ao em dimens˜ao d ≥ 1: Processo de Ex-

clus˜ao com Condutˆancias, com Condutˆancias em Ambiente Aleat´orio e com Bordos e Velocidades. Para

o primeiro, obtemos o limite hidrodinˆamico, no segundo obtemos limite hidrodinˆamico e as ﬂutua¸c˜oes

no equil´ıbrio, e no ´ultimo provamos o princ´ıpio dos grandes desvios.

Keywords: Exclusion Processes, Boundary Driven Exclusion Processes, Hydrodynamic Limit, Equi-

librium Fluctuations, Large Deviations, Conductances, Random Environment, Homogenization.

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Aos amores de minha vida:

minha m˜ae, minha esposa Tatiana,

a meus ﬁlhos Arthur (Tuco) e Estela (Teia)

e em mem´oria do meu pai, meu grande her´oi.

Agradecimentos

Agrade¸co a Deus por ter me aben¸coado com a intensidade de virtudes superior aos meus defeitos de

modo que consegui ﬁnalizar com sucesso o programa de Doutorado. Isto tamb´em, somente foi poss´ıvel,

pela for¸ca, amizade, companheirismo e incentivo que in´umeras pessoas sempre depositaram em mim

durante toda a minha forma¸c˜ao acadˆemica. Em especial, destaco a minha amada e companheira esposa

Tatiana, minha querida m˜ae Maria da Penha e ao estimado amigo professor Carlos Roberto Alves dos

Santos, meu sincero muito obrigado.

Agrade¸co a meu orientador de Mestrado e Doutorado, professor Claudio Landim, pelas oportunidades,

os ensinamentos, paciˆencia, pela seguran¸ca e competˆencia com que conduziu toda minha orienta¸c˜ao.

Seguramente, tenho nele o exemplo de conduta a seguir em minha vida proﬁssional.

Tive a felicidade de usufruir de um programa de doutorado sanduiche no Courant - NYU, de forma

que tenho a agradecer a excelente recep¸c˜ao e suporte com que professores, alunos e funcion´arios deste

renomado instituto me acolheram. Em especial, sou muito grato ao meu orientador no exterior, professor

S.R.S. Varadhan, por ter me propiciado inesquec´ıveis e valiosos momentos de ensinamento de conduta

como matem´atico e pessoa e ao amigo Antonio Carlos Auﬃnger (Tuca) pelo apoio e companheirismo

desde os tempos de gradua¸c˜ao.

Agrade¸co aos parceiros de pesquisa Alexandre de Bustamante Simas e Jonathan Farfan que, efetiva-

mente, contribuiram para o sucesso desta etapa de minha vida, dividimos momentos de descontra¸c˜ao,

concentra¸c˜ao, tristeza e alegria. A matem´atica ﬁcou mais prazerosa com as in´umeras e intermin´aveis

discurss˜oes no IMPA, em casa, nos ˆonibus, na praia,...

Agrade¸co a todos os professores que contribuiram para minha forma¸c˜ao. Em especial, destaco no

CEFETES os professores Oscar Rezende. Na UFES os professores Jos´e Gilvan, Valmecir Bayer, Ademir

Sartim, Luzia Casati e Luiz Fernando Camargo (tamb´em orientador de Inicia¸c˜ao Cient´ıﬁca, que al´em do

rigor matem´atico, me instru´ıa em minha forma¸c˜ao pessoal), sou muito grato. No IMPA, os professores

Fernado Cod´a, Carlos Gustavo Moreira (Gugu), Manfredo do Carmo, Milton Jara que com sua brilhante

capacidade de conduzir as aulas tornavam a Matem´atica mais bela e inspiradora.

Agrade¸co aos funcion´arios e alunos do IMPA que, durante minha estada, ﬁzeram com que o ambiente

de trabalho fosse agrad´avel e prazeroso. De fato, um local que ﬁz muitas amizades. Em especial,

agrade¸co a Alexandre Oliveira (Xand˜ao), F´atima Russo, Jos´e Paulo Fahl (Paulinho), F´abio do Santos,

aos funcion´arios do ensino e a turma do futebol de quinta, obrigado pela acolhida. Um agradecimento

especial a Jean Silva, Jeﬀerson Melo, Fernando Marek, Etereldes, Thiago Fassarela (Mocha) e a turma

da Probabilidade, obrigado pelo companheirismo.

Agrade¸co aos professores Leandro Pimentel, Rob Morris, Serguei Popov e Vladas Sidoravicius por

terem aceitado participar de minha banca de defesa de tese contribuindo com sugest˜oes, cr´ıticas e elogios.

Agrade¸co ao CNPq pelo ﬁnanciamento de minha bolsa de doutorado no pa´ıs e no exterior.

Contents

Introduction 1

1 Exclusion process with conductances 3

1.1 Notation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 The hydrodynamic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 The operator L

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Some remarks on the one-dimensional case . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 The d-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Random walk with conductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Discrete approximation of the operator L

. . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Semigroups and resolvents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Scaling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.1 Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.2 Uniqueness of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.3 Replacement lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5 Energy estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 W -Sobolev spaces 25

2.1 W -Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1 The W -Sobolev space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.2 Approximation by smooth functions and the energetic space . . . . . . . . . . . . . 28

2.1.3 A Rellich-Kondrachov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1.4 The space H

−1

) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 W -Generalized elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 W -Generalized parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.1 Uniqueness of weak solutions of the parabolic equation . . . . . . . . . . . . . . . . 37

2.4 W -Generalized Sobolev spaces: Discrete version . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4.1 Connections between the discrete and continuous Sobolev spaces . . . . . . . . . . 42

2.5 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5.1 H-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5.2 Random environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5.3 Homogenization of random operators . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6 Hydrodynamic limit of processes with conductances in random environment . . . . . . . . 46

2.6.1 The exclusion processes with conductances in random environments . . . . . . . . 46

2.6.2 The hydrodynamic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.6.3 Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.6.4 Uniqueness of limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Equilibrium ﬂuctuations 53

3.1 Notation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 The space S

) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Equilibrium Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Martingale Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.2 Generalized Ornstein-Uhlenbeck Processes . . . . . . . . . . . . . . . . . . . . . . . 63

iii

3.4 Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Boltzmann-Gibbs Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6 Appendix: Stochastic diﬀerential equations on nuclear spaces . . . . . . . . . . . . . . . 70

3.6.1 Countably Hilbert nuclear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.6.2 Stochastic diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Dynamical Large Deviations 73

4.1 Notation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.1 The boundary driven exclusion process . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.2 Mass and momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.1.3 Dynamical large deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 The rate function I

(·|γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 I

(·|γ)-Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.5 Large deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5.1 Superexponential estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5.2 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.5.3 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.5.4 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Introduction

O limite hidrodinˆamico permite obter uma descri¸c˜ao das caracter´ısticas termodinˆamicas (por exem-

plo, temperatura, densidade, press˜ao) de sistemas inﬁnitos assumindo que a dinˆamica das part´ıculas

´e estoc´astica. Seguindo a abordagem da mecˆanica estat´ıstica introduzida por Boltzmann, deduzimos

o comportamento macrosc´opico de um sistema a partir da itera¸c˜ao microsc´opica entre as part´ıculas.

Considera-se a dinˆamica microsc´opica consistindo de caminhos aleat´orios sobre um grafo submetida a

alguma itera¸c˜ao local, denominado sistema de part´ıculas interagentes introduzido por Spitzer [36], veja

tamb´em [24]. Ademais, esta abordagem justiﬁca rigorosamente um m´etodo algumas vezes utilizado pe-

los f´ısicos para estabelecer equa¸c˜oes diferenciais parciais que descrevem a evolu¸c˜ao de caracter´ısticas

termodinˆamicas de um ﬂuido. Assim, a existˆencia de solu¸c˜oes fracas de tais EDPs podem ser vistas

como um dos objetivos do limite hidrodinˆamico.

Um conhecido sistema de part´ıculas interagentes ´e o processo de exclus˜ao simples. Informalmente

´e um processo onde apenas uma part´ıcula por sitio ´e permitida (dai o nome exclus˜ao), e o salto das

part´ıculas somente ocorrem para os vizinhos pr´oximos. Nesta tese consideramos o processo de exclus˜ao

simples sobre o toro discreto d-dimensional, T

, e obtemos o comportamento hidrodinˆamico nos seguintes

modelos:

No cap´ıtulo 1, consideramos o processo de exclus˜ao com condutˆancias induzida por uma classe de

fun¸c˜oes W e obtemos que, sobre uma escala difusiva, a evolu¸c˜ao das densidades emp´ıricas do processo

de exclus˜ao sobre o toro d-dimensional, T

, ´e descrita pela ´unica solu¸c˜ao fraca da equa¸c˜ao diferencial

parcial generalizada n˜ao-linear

∂

ρ =



k=1

∂

Φ(ρ), (0.0.1)

Onde a fun¸c˜ao Φ : [l, r] → R ´e ﬁxada e suave, deﬁnida sobre um intervalo [l, r] de R. Esta fun¸c˜ao est´a

associada a um fator na taxa de salto das part´ıculas no processo microsc´opico e depende das conﬁgura¸c˜oes

do sistema. O adjetivo generalizada decorre do termo ∂

cuja deﬁni¸c˜ao e referˆencias s˜ao dadas na Se¸c˜ao

1.2. Em Particular, se considerarmos W

(x) = x

, obtemos que (0.0.1) ´e a equa¸c˜ao do calor. Para a

prova do limite hidrodinˆamico, n´os tamb´em obtemos algumas propriedades do operador el´ıptico do lado

direito de (0.0.1).

Ultimamente, a evolu¸c˜ao de processos de exclus˜ao uni-dimensional com condutˆancias tem atra´ıdo

aten¸c˜ao [13, 14, 18, 21]. Um dos prop´ositos desta tese ´e estender esta an´alise para dimens˜oes maiores.

Este processo pode, por exemplo, modelar difus˜oes de part´ıculas em um meio com membranas perme´aveis,

nos pontos de descontinuidade de W , tendendo a reﬂetir part´ıculas, criando espa¸cos de descontinuidade

nos perﬁs de densidade. Nas primeiras linhas do cap´ıtulo 1, encontra-se uma detalhamento maior desta

aplica¸c˜ao e da real conex˜ao deste operador com os famosos operadores diferenciais de F´eller.

No cap´ıtulo 2, consideramos um processo de exclus˜ao com condutˆancias em ambiente aleat´orio e

obtemos o limite hidrodinˆamico. A condutˆancia ´e a mesma considerada no cap´ıtulo 1, no entanto a

novidade neste cap´ıtulo n˜ao se resume ao ambiente aleat´orio. Isto porque a prova do comportamento

hidrodinˆamico no cap´ıtulo 1 ´e baseada em estimativas do semigrupo e resolventes entre o processo original

e um corrigido. O elo entre o casos d = 1 [18, 14] e d ≥ 1 ´e ent˜ao estabelecido via uma classe especial de

fun¸c˜oes W , a saber:

W (x

, . . . , x

) =



k=1

) x ∈ R,

onde cada W

´e da forma considerada em [18]. Enquanto no cap´ıtulo 2, usando as propriedades obtidas

do operador el´ıptico em (0.0.1), constru´ımos o espa¸co W -Sobolev, o qual consiste das fun¸c˜oes f tendo

gradiente generalizado fraco ∇

f = (∂

f, . . . , ∂

f). Obtemos varias propriedades para este espa¸co,

que s˜ao an´alogas aos cl´assicos resultados para espa¸cos de Sobolev. Equa¸c˜oes W -generalizada el´ıptica

e parab´olica s˜ao consideradas, alcan¸cando resultados de existˆencia e unicidade de solu¸c˜oes fracas para

estas equa¸c˜oes. Resultados de homogeniza¸c˜ao para uma classe de operadores aleat´orios s˜ao investigados,

ﬁnalmente, como primeira aplica¸c˜ao desta teoria desenvolvida, nos provamos o limite hidrodinˆamico

para o processo em quest˜ao. Em particular, substituindo a an´alise de semigrupos e resolventes feita no

cap´ıtulo 1, por homogeniza¸c˜ao.

A motiva¸c˜ao para este enfoque foi o artigo [20]. Nele os autores consideram um processo de exclus˜ao

gradiente em ambiente aleat´orio e usam a teoria de homogeniza¸c˜ao, desenvolvida em [31], para obterem

o limite hidrodinˆamico e ﬂutua¸c˜oes.

No cap´ıtulo 3, nos obtemos as ﬂutua¸c˜oes do equil´ıbrio para o processo considerado no cap´ıtulo 2. Esta

foi a segunda aplica¸c˜ao da teoria previamente desenvolvida. Nos obtemos que a distribui¸c˜ao emp´ırica

´e governada pela ´unica solu¸c˜ao de uma equa¸c˜ao diferencial estoc´astica, tomando valores em um certo

espa¸co Frechet Nuclear.

No cap´ıtulo 4, nos provamos os grandes desvios dinˆamicos para um processo boundary driven, i.e.

um sistema que possui dois reservat´orios inﬁnitos de part´ıculas na fronteira com part´ıculas que podem

ter diferentes velocidades. Este resultado baseia-se na recente abordagem introduzida em [15].

Cada cap´ıtulo desta tese resultou em um artigo, os quais salvo alguns cortes para evitar excessivas

repeti¸c˜oes, s˜ao os pr´oprios artigos. Em particular cada in´ıcio de cap´ıtulo tem uma pequena introdu¸c˜ao

que complementa esta. Ressalto que o cap´ıtulo 2 ´e um trabalho conjunto com Alexandre Bustamante de

Simas e os cap´ıtulos 3 e 4 s˜ao em parceria com Jonathan Farfan e Alexandre Bustamante de Simas.

Chapter 1

Hydrodynamic limit of a

d-dimensional exclusion process with

conductances

The evolution of one-dimensional exclusion processes with random conductances has attracted some

attention recently [21, 13, 14, 18]. The purpose of this chapter is to extend this analysis to higher

dimension.

Let W : R

→ R be a function such that W (x

, . . . , x

) =



k=1

), where d ≥ 1 and each

function W

: R → R is strictly increasing, right continuous with left limits (c`adl`ag) , and periodic in

the sense that W

(u + 1) − W

(u) = W

(1) − W

(0) for all u ∈ R. Informally, the exclusion process

with conductances associated to W is an interacting particle systems on the d-dimensional discrete torus

−1

, in which at most one particle per site is allowed, and only nearest-neighbor jumps are permitted.

Moreover, the jump rate in the direction e

is given by the reciprocal of the increments of W with respect

to the jth coordinate.

We show that, on the diﬀusive scale, the macroscopic evolution of the empirical density of exclusion

processes with conductances W is described by the nonlinear diﬀerential equation

∂

ρ =



k=1

∂

Φ(ρ) , (1.0.1)

where Φ is a smooth function, strictly increasing in the range of ρ, and such that 0 < b ≤ Φ



≤ b

−1

Furthermore, we denote by ∂

the generalized derivative with respect to W

, see [8, 18] and a revision

in Section 1.2. The partial diﬀerential equation (1.0.1) appears naturally as, for instance, scaling limits

of interacting particle systems in inhomogeneous media. It may model diﬀusions in which permeable

membranes, at the points of discontinuities of W , tend to reﬂect particles, creating space discontinuities

in the density proﬁles.

The proof of hydrodynamic limit relies strongly on some properties of the diﬀerential operator



k=1

∂

presented in Theorem 1.1.2. We prove, among other properties: that the operator



k=1

∂

, deﬁned on an appropriate domain, is non-positive, self-adjoint and dissipative; that its

eigenvalues are countable and have ﬁnite multiplicity; and that the associated eigenvectors form a com-

plete orthonormal system.

There is a wide literature on the so-called Feller’s generalized diﬀusion operator (d/du)(d/dv). Where,

typically, u and v are strictly increasing functions with v (but not necessarily u) being continuous. It

provides general diﬀusions operators and an appreciable simpliﬁcation of the theory of second-order

diﬀerential operators (see, for instance, [16, 17, 26]). The operator (d/dx)(d/du), considered in [18], is

the formal adjoint of (d/du)(d/dv) in the particular case v(x) = x (as in [17]). The goal of this work is

to extend this adjoint operator to higher dimensions and provide some results regarding this extension.

This chapter is organized as follows: in Section 1.1 we state the main results of the chapter; in

Section 1.2 we prove the main properties of the operator L



k=1

∂

; in Section 1.3 we prove

the convergence of random walks with random conductances to Markov processes with generator given

by L

; in Section 1.4 we prove the scaling limit of the exclusion process with conductances given by W ;

and, ﬁnally, in Section 1.5 we show that the unique solution of (1.0.1) has ﬁnite energy.

1.1 Notation and Results

We examine the hydrodynamic behavior of a d-dimensional exclusion process, with d ≥ 1, with conduc-

tances induced by a special class of functions W : R

→ R such that:

W (x

, . . . , x

) =



k=1

) (1.1.1)

where W

: R → R are strictly increasing right continuous functions with left limits (c`adl`ag), and periodic

in the sense that

(u + 1) − W

(u) = W

(1) − W

(0)

for all u ∈ R and k = 1, . . . , d. To keep notation simple, we assume that W

vanishes at the origin, that

is, W

(0) = 0.

Denote by T

= [0, 1)

the d-dimensional torus and by e

, . . . , e

the canonical basis of R

. For this

class of functions we have:

• W (0) = 0;

• W is strictly increasing on each coordinate:

W (x + ae

) > W (x)

for all 1 ≤ j ≤ d, a > 0, x ∈ R

;

• W is continuous from above:

W (x) = lim

y→x, y≥x

W (y),

where we say that y ≥ x if y

≥ x

for all 1 ≤ j ≤ d;

• W is deﬁned on the torus T

W (x

, . . . , x

j−1

, 0, x

j+1

, . . . , x

) = W (x

, . . . , x

j−1

, 1, x

j+1

, . . . , x

) − W (e

for all 1 ≤ j ≤ d, (x

, . . . , x

j−1

, x

j+1

, ..., x

) ∈ T

d−1

Unless explicitly stated W belongs to this class. Let T

= (Z/NZ)

= {0, . . . , N − 1}

be the

d-dimensional discrete torus with N

points. Distribute particles throughout T

in such a way that

each site of T

is occupied at most by one particle. Denote by η the conﬁgurations of the state space

{0, 1}

, so that η(x) = 0 if site x is vacant and η(x) = 1 if site x is occupied.

Fix b > −1/2 and W . For x = (x

, . . . , x

) ∈ T

let

x,x+e

(η) = 1 + b{η(x − e

) + η(x + 2 e

)} ,

where all sums are modulo N, and let

x,x+e

N[W ((x + e

)/N ) − W (x/N )]

N[W

((x

+ 1)/N ) − W

/N )]

We now describe the stochastic evolution of the process. Let x = (x

, . . . , x

) ∈ T

. At rate

x,x+e

(η) the occupation variables η(x), η(x + e

) are exchanged. If W is diﬀerentiable at x/N ∈

[0, 1)

, the rate at which particles are exchanged is of order 1 for each direction, but if some W

discontinuous at x

/N , it no longer holds. In fact, assume, to ﬁx ideas, that W

is discontinuous at

/N , and smooth on the segments (x

/N, x

/N + εe

) and (x

/N −εe

, x

/N ). Assume, also, that W

is diﬀerentiable in a neighborhood of x

/N for k = j. In this case, the rate at which particles jump over

the bonds {y −e

, y}, with y

= x

, is of order 1/N , whereas in a neighborhood of size N of these bonds,

particles jump at rate 1. Thus, note that a particle at site y − e

jumps to y at rate 1/N and jumps at

rate 1 to each one of the 2d − 1 other options. Particles, therefore, tend to avoid the bonds {y − e

, y}.

However, since time will be scaled diﬀusively, and since on a time interval of length N

a particle spends

a time of order N at each site y, particles will be able to cross the slower bond {y −e

, y}.

Then, this process models membranes that obstruct passages of particles. Note that these membranes

are (d −1)-dimensional hyperplanes embedded in a d-dimensional environment. Moreover, if we consider

having more than one discontinuity point for more than one j, these membranes will be more

sophisticated manifolds, for instance, unions of (d − 1)-dimensional boxes.

The eﬀect of the factor c

x,x+e

(η) is the following: if the parameter b is positive, the presence of

particles in the neighboring sites of the bond {x, x + e

} speeds up the exchange rate by a factor of order

one, and if the parameter b is negative, the presence of particles in the neighboring sites slows down the

exchange rate also by a factor of order one.

The dynamics informally presented describes a Markov evolution. The generator L

of this Markov

process acts on functions f : {0, 1}

→ R as

f(η) =



j=1



x∈T

x,x+e

(η) {f(σ

x,x+e

η) − f(η)} , (1.1.2)

where σ

x,x+e

η is the conﬁguration obtained from η by exchanging the variables η(x) and η(x + e

(σ

x,x+e

η)(y) =











η(x + e

) if y = x,

η(x) if y = x + e

η(y) otherwise.

(1.1.3)

A straightforward computation shows that the Bernoulli product measures {ν

: 0 ≤ α ≤ 1} are

invariant, and in fact reversible, for the dynamics. The measure ν

is obtained by placing a particle at

each site, independently from the other sites, with probability α. Thus, ν

is a product measure over

{0, 1}

with marginals given by

{η : η(x) = 1} = α,

for x in T

. For more details see [23, chapter 2]. We will often omit the index N on ν

Denote by {η

: t ≥ 0} the Markov process on {0, 1}

associated to the generator L

speeded up by

. Let D(R

, {0, 1}

) be the path space of c`adl`ag trajectories with values in {0, 1}

. For a measure

on {0, 1}

, denote by P

the probability measure on D(R

, {0, 1}

) induced by the initial state

, and the Markov process {η

: t ≥ 0}. Expectation with respect to P

is denoted by E

1.1.1 The hydrodynamic equation

Fix W =



k=1

as in (1.1.1). In [18] it was shown that there exist self-adjoint operators L

: D

⊂

(T) → L

(T). The domain D

is completely characterized in the following proposition:

Proposition 1.1.1. The domain D

consists of all functions f in L

(T) such that

f(x) = a + bW

(x) +



(0,x]

(dy)



f(z) dz

for some function f in L

(T) that satisﬁes



f(z) dz = 0 and



(0,1]

(dy)



b +



f(z) dz



= 0 .

The proof and further details can be found in [18]. Further, the set A

of the eigenvectors of L

forms a complete orthonormal system in L

(T). Let

= {f : T

→ R; f(x

, . . . , x

) =



k=1

), f

∈ A

, k = 1, . . . , d}, (1.1.4)

and denote by span(A) the space of ﬁnite linear combinations of the set A, and let D

:= span(A

Deﬁne the operator L

: D

→ L

) as follows: for f =



k=1

∈ A

, we have

(f)(x

, . . . x

) =



k=1



j=1,j=k

), (1.1.5)

and then extend to D

by linearity.

Lemma 1.2.2, in Section 1.2, shows that: L

is symmetric and non-positive; D

is dense in L

);

and the set A

forms a complete, orthonormal, countable system of eigenvectors for the operator L

Let A

= {h

}

k≥0

, {α

}

k≥0

be the corresponding eigenvalues of −L

, and consider

= {v =

∞



k=1

∈ L

);

∞



k=1

< +∞}. (1.1.6)

Deﬁne the operator L

: D

→ L

) by

−L

v =

+∞



k=1

(1.1.7)

The operator L

is clearly an extension of the operator L

, and we present in Theorem 1.1.2 some

properties of this operator.

Theorem 1.1.2. The operator L

: D

→ L

) enjoys the following properties:

(a) The domain D

is dense in L

). In particular, the set of eigenvectors A

= {h

}

k≥0

forms a

complete orthonormal system;

(b) The eigenvalues of the operator −L

form a countable set {α

}

k≥0

. All eigenvalues have ﬁnite

multiplicity, and it is possible to obtain a re-enumeration {α

}

k≥0

such that

0 = α

≤ α

≤ ··· and lim

n→∞

= ∞;

: D

→ L

) is bijective;

(d) L

: D

→ L

) is self-adjoint and non-positive:

−L

f, f  ≥ 0;

(e) L

is dissipative.

In view of (a), (b) and (d), we may use Hille-Yosida theorem to conclude that L

is the generator

of a strongly continuous contraction semigroup {P

: L

) → L

) }

t≥0

Denote by {G

: L

) → L

) }

λ>0

the semigroup of resolvents associated to the operator L

= (λ − L

)

−1

. G

can also be written in terms of the semigroup {P

; t ≥ 0}:



∞

−λt

dt.

In Section 1.3 we derive some properties and obtain some results for these operators.

The hydrodynamic equation is, roughly, a PDE that describes the time evolution of the thermody-

namical quantities of the model in a ﬂuid. A sequence of probability measures {µ

: N ≥ 1} on {0, 1}

is said to be associated to a proﬁle ρ

: T

→ [0, 1] if

lim

N→∞











x∈T

H(x/N )η(x) −



H(u)ρ

(u)du



> δ







= 0 (1.1.8)

for every δ > 0, and every continuous function H : T

→ R. For details, see [23, chapter 3].

For a positive integer m ≥ 1, denote by C

) the space of continuous functions H : T

→ R with

m continuous derivatives. Fix l < r, and a smooth function Φ : [l, r] → R, whose derivative is bounded

below by a strictly positive constant and bounded above by a ﬁnite constant, that is,

0 < B

−1

≤ Φ



(x) ≤ B,

for all x ∈ [l, r]. Let γ : T

→ [l, r] be a bounded density proﬁle, and consider the parabolic diﬀerential

equation



∂

ρ = L

Φ(ρ)

ρ(0, ·) = γ(·)

. (1.1.9)

A bounded function ρ : R

× T

→ [l, r] is said to be a weak solution of the parabolic diﬀerential

equation (1.1.9) if

ρ

, G

H − γ, G

H =



Φ(ρ

), L

Hds

for every continuous function H : T

→ R, all t > 0 and all λ > 0.

Existence of these weak solutions follows from tightness of the sequence of probability measures Q

W,N

introduced in Section 1.4. The proof of uniquenesses of weak solutions is analogous to [18].

Theorem 1.1.3. Fix a continuous initial proﬁle ρ

: T

→ [0, 1], and consider a sequence of probability

measures µ

on {0, 1}

associated to ρ

, in the sense of (1.1.8). Then, for any t ≥ 0,

lim

N→∞











x∈T

H(x/N )η

(x) −



H(u)ρ(t, u) du



> δ







= 0

for every δ > 0 and every continuous function H. Here, ρ is the unique weak solution of the non-linear

equation (1.1.9) with l = 0, r = 1, γ = ρ

, and Φ(α) = α + aα

Remark 1.1.4. As noted in [18, remark 2.3], the speciﬁc form of the rates c

x,x+e

is not important, but

two conditions must be fulﬁlled: the rates must be strictly positive, although they may not depend on the

occupation variables η(x), η(x + e

); but they have to be chosen in such a way that the resulting process

is gradient. (cf. Chapter 7 in [23] for the deﬁnition of gradient processes).

We may deﬁne rates c

x,x+e

to obtain any polynomial Φ of the form Φ(α) = α +



2≤j≤m

m ≥ 1, with 1 +



2≤j≤m

> 0. Let, for instance, m = 3. Then the rates

ˆc

x,x+e

(η) = c

x,x+e

(η) +

{η(x − 2e

)η(x − e

) + η(x − e

)η(x + 2e

) + η(x + 2e

)η(x + 3e

)},

satisfy the above three conditions, where c

x,x+e

is the rate deﬁned at the beginning of Section 2 and b,

are such that 1 + 2b + 3b

> 0. An elementary computation shows that Φ(α) = α + bα

+ b

In Section 1.5 we prove that any limit point Q

∗

of the sequence Q

W,N

is concentrated on trajectories

ρ(t, u)du, with ﬁnite energy in the following sense: for each 1 ≤ j ≤ d, there is a Hilbert space L

⊗W

associated to W

, such that



ds 

Φ(ρ(s, .))

⊗W

< ∞ ,

where .

⊗W

is the norm in L

⊗W

, and d/dW

is the derivative, which must be understood in the

generalized sense.

1.2 The operator L

The operator L

: D

⊂ L

) → L

) is a natural extension, for the d-dimensional case, of the

self-adjoint operator obtained for the one-dimensional case in [18]. We begin by presenting one of the

main results obtained in [18], and we then present the necessary modiﬁcations to conclude similar results

for the d-dimensional case.

1.2.1 Some remarks on the one-dimensional case

Let T ⊂ R be the one-dimensional torus. Denote by ·, · the inner product of L

(T):

f, g =



f(u) g(u) du .

Let W

: R → R be a strictly increasing right continuous function with left limits (c`adl`ag), and

periodic in the sense that W

(u + 1) − W

(u) = W

(1) − W

(0) for all u in R.

Let D

be the set of functions f in L

(T) such that

f(x) = a + bW

(x) +



(0,x]

(dy)



f(z) dz,

for a, b ∈ R and some function f in L

(T) that satisﬁes:



f(z) dz = 0 ,



(0,1]

(dy)



b +



f(z) dz



= 0.

Deﬁne the operator L

: D

→ L

(T) by L

f = f. Formally

f =

f , (1.2.1)

where the generalized derivative d/dW

is deﬁned as

(x) = lim

→0

f(x + ) − f(x)

(x + ) − W

(x)

, (1.2.2)

if the above limit exists and is ﬁnite.

Theorem 1.2.1. Denote by I the identity operator in L

(T). The operator L

: D

→ L

(T) enjoys

the following properties:

(a) D

is dense in L

(T);

(b) The operator I − L

: D

→ L

(T) is bijective;

: D

→ L

(T) is self-adjoint and non-positive:

−L

f, f  ≥ 0;

(d) L

is dissipative i.e., for all g ∈ D

and λ > 0, we have

λg ≤ (λI −L

)g;

(e) The eigenvalues of the operator −L

form a countable set {λ

: n ≥ 0}. All eigenvalues have

ﬁnite multiplicity, 0 = λ

≤ λ

≤ ···, and lim

n→∞

= ∞;

(f) The eigenvectors {f

}

n≥0

of the operator L

form a complete orthonormal system.

The proof can be found in [18].

1.2.2 The d-dimensional case

Consider W as in (1.1.1). Let A

be the countable complete orthonormal system of eigenvectors of the

operator L

: D

⊂ L

(T) → R given in Theorem 1.2.1.

Let A

be as in (1.1.4), and let the operator L

: D

:= span(A

) → L

) be as in (1.1.5). By

Fubini’s theorem, the set A

is orthonormal in L

), and the constant functions are eigenvectors of

the operator L

. Moreover, A

⊂ A

, in the sense that f

, . . . , x

) = f

), f

∈ A

By (1.2.1), the operators L

can be formally extended to functions deﬁned on T

as follows: given

a function f : T

→ R, we deﬁne L

f as

f = ∂

∂

f, (1.2.3)

where the generalized derivative ∂

is deﬁned by

∂

f(x

, . . . , x

) = lim

→0

f(x

, . . . , x

+ , . . . , x

) − f(x

, . . . , x

)

+ ) − W

)

, (1.2.4)

if the above limit exists and is ﬁnite. Hence, by (1.1.5), if f ∈ D

f =



k=1

f. (1.2.5)

Note that if f =



k=1

, where f

∈ A

is an eigenvector of L

associated to the eigenvalue λ

then f is an eigenvector of L

, with eigenvalue



k=1

Lemma 1.2.2. The following statements hold:

(a) The set D

is dense in L

);

(b) The operator L

: D

→ L

) is symmetric and non-positive:

−L

f, f  ≥ 0.

Proof. The strategy to prove the above Lemma is the following. We begin by showing that the set

S = span({f ∈ L

); f(x

, . . . , x

) =



k=1

), f

∈ D

})

is dense in

S = span({f ∈ L

); f(x

, . . . , x

) =



k=1

), f

∈ L

(T)}).

We then show that D

is dense in S. Since S is dense in L

), item (a) follows.

We now prove item (a) rigorously. Since S is a vector space, we only have to show that we can

approximate the functions



k=1

∈ L

), where f

∈ D

, by functions of D

. By Theorem 1.2.1,

the set D

is dense in L

(T), thus, there exists a sequence (f

)

n∈N

converging to f

in L

(T). Thus,

let

, . . . , x

) =



k=1

By the triangle inequality and Fubini’s theorem, the sequence (f

) converges to



k=1

. Fix  > 0, and

let

h(x

, . . . , x

) =



k=1

), h

∈ D

Since, for each k = 1 . . . , d, A

⊂ D

is a complete orthonormal set, there exist sequences

∈ A

, and α

∈ R, such that

h

−

n(k)



j=1



(T)

< δ ,

where δ = /dM

d−1

and M := 1 + sup

k=1:n

h

. Let

g(x

, . . . , x

) =



k=1

n(k)



j=1

) ∈ D

An application of the triangle inequality, and Fubini’s theorem, yields h − g < . This proves (a).

To prove (b), let

f(x

, . . . , x

) =



k=1

) and g(x

, . . . , x

) =



k=1

)

be functions belonging to A

. We have that

f, L

g = 



k=1



k=1



j=1,j=k

 =



k=1





j=1,j=k

, f

,

where ·, · denotes the inner product in L

). Since, by Theorem 1.2.1, L

is self-adjoint, we have



k=1





j=1,j=k

, g

 = L

f, g.

In particular, the operator L

is non-positive, and, therefore,

f, L

f =



k=1





j=1,j=k

, f

 ≤ 0.

Item (b) follows by linearity.

Lemma 1.2.2 implies that the set A

forms a complete, orthonormal, countable, system of eigenvec-

tors for the operator L

Let L

: D

→ L

) be the operator deﬁned in (1.1.7). The operator L

is clearly an extension

of the operator L

. Formally, by (1.2.5),

f =



k=1

f, (1.2.6)

where

f = ∂

∂

We are now in conditions to prove Theorem 1.1.2.

Proof of Theorem 1.1.2. By Lemma 1.2.2, D

is dense in L

). Since D

⊂ D

, we conclude that

is dense in L

If α

are eigenvalues of −L

, we may ﬁnd eigenvalues λ

, associated to some f

∈ A

, such that



j=1

. By item (e) of Theorem 1.2.1, (b) follows.

Let {α

}

k≥0

be the set of eigenvalues of −L

. Then, the set of eigenvalues of I − L

is {γ

}

k≥0

where γ

= α

+ 1, and the eigenvectors are the same as the ones of L

. By item (b), we have

1 = γ

≤ γ

≤ ··· and lim

n→∞

= ∞ .

Thus, I − L

is injective. For

v =

+∞



k=1

∈ L

) , such that

∞



k=1

< +∞ ,

let

u =

+∞



k=1

Then u ∈ D

and (I − L

)u = v. Hence, item (c) follows.

Let L

∗

: D

∗

⊂ L

) → L

) be the adjoint of L

. Since L

is symmetric, we have D

⊂

∗

. So, to show the equality of the operators it suﬃces to show that D

∗

⊂ D

. Given

ϕ =

+∞



k=1

∈ D

∗

let L

W ∗

ϕ = ψ ∈ L

). Therefore, for all v =



+∞

k=1

∈ D

v, ψ = v, L

W ∗

ϕ = L

v, ϕ =

+∞



k=1

−α

Hence

ψ =

+∞



k=1

−α

In particular,

+∞



k=1

< +∞ and ϕ ∈ D

Thus, L

is self-adjoint. Let v =



+∞

k=1

∈ D

. From item (b), α

≥ 0, and

−L

v, v =

+∞



k=1

≥ 0.

Therefore L

is non-positive, and item (d) follows.

Fix a function g in D

, λ > 0, and let f = (λI −L

)g. Taking inner product, with respect to g, on

both sides of this equation, we obtain

λg, g + −L

g, g = g, f  ≤ g, g

1/2

f, f 

1/2

Since g belongs to D

, by (d), the second term on the left hand side is non-negative. Thus, λg ≤

f = (λI −L

)g.

1.3 Random walk with conductances

Recall the decomposition obtained in (1.2.6) for the operator L

. In next subsection, we present the

discrete version L

of L

and we describe, informally, the Markovian dynamics generated by L

1.3.1 Discrete approximation of the operator L

Consider the random walk {X

}

t≥0

, which jumps from x/N (resp. (x + e

)/N ) to (x + e

)/N

(resp. x/N) with rate

x,x+e

= N/{W

((x

+ 1)/N ) − W

/N )}.

The generator L

of this Markov process acts on local functions f :

→ R as

f(x/N ) =



j=1

f(x/N ), (1.3.1)

where

f(x/N ) = N



x,x+e

[f((x + e

)/N ) − f(x/N )]

+ ξ

x−e

[f((x − e

)/N ) − f(x/N )]



Note that L

f(x/N ) is, in fact, a discrete version of the operator L

. The counting measure m

on T

is reversible for this process. The following estimate is a key ingredient for proving the results in

Section 1.4:

Lemma 1.3.1. Let f be a function on

. Then, for each j = 1, . . . , d:



x∈T



f(x/N )



≤



x∈T



f(x/N )



Proof. Let X

be the linear space of functions f on

over the ﬁeld R. Note that the dimension

of X

is N

. Denote by ·, ·

the following inner product in X

f, g



x∈T

f(x/N )g(x/N).

For each j = 1, . . . , d, consider the linear operators L

on X

(i.e., d = 1) given by

f = ∂

∂

where ∂

and ∂

are the diﬀerence operators:

∂

f(x/N ) = N[f((x + 1)/N ) − f(x/N )] and

∂

f(x/N ) =

f((x + 1)/N ) − f(x/N )

((x + 1)/N ) − W

(x/N )

The operators L

are symmetric and non-positive. In fact, a simple computation shows that

L

f, g

= −



x∈T



((x + 1)/N ) − W

(x/N )



∂

f(x/N )∂

g(x/N).

Using the spectral theorem, we obtain an orthonormal basis A

= {h

, . . . , h

} of X

formed by

the eigenvectors of L

, i.e.,

= α

and h

, h



= δ

i,k

where δ

i,k

is the Kronecker’s delta, which equals 0 if i = k, and equals 1 if i = k. Since L

is non-positive,

we have that the eigenvalues α

are non-positive: α

≤ 0, j = 1, . . . , d and i = 1, . . . , N.

Let A

= {φ

, . . . , φ

} ⊂ X

be set of functions of the form φ

, . . . , x

) =



j=1

), with

∈ A

Let α

be the eigenvalue of h

, i.e., L

= α

. The linear operator L

on X

, deﬁned in (1.3.1),

is such that L

= α

and L



j=1

. Furthermore, if φ

, . . . , x

) =



j=1

) and

, . . . , x

) =



j=1

), φ

, φ

∈ A

, we have that

φ

, φ





j=1

h

, g



= δ

i,k

for i, k = 1, . . . , N

. So, the set A

is an orthonormal basis of X

formed by the eigenvectors of L

and L

. In particular, for each f ∈ X

, there exist β

∈ R such that f =



i=1

. Thus,



x∈T



f(x/N )



= L

f

= L



i=1





i=1

(α

)

≤



i=1

(



j=1

)

(β

)

= L

f



x∈T



f(x/N)



where α

≤ 0 is the eigenvalue of the operator L

associated to the eigenvector φ

. This concludes the

proof of the lemma.

1.3.2 Semigroups and resolvents.

In this subsection we introduce families of semigroups and resolvents associated to the generators L

and L

. We present some properties and results regarding the convergence of these operators.

Denote by {P

: t ≥ 0} (resp. {G

: λ > 0}) the semigroup (resp. the resolvent) associated to the

generator L

, by {P

N,j

: t ≥ 0} the semigroup associated to the generator L

, by {P

: t ≥ 0} the

semigroup associated to the generator L

and by {P

: t ≥ 0} (resp. {G

: λ > 0}) the semigroup

(resp. the resolvent) associated to the generator L

Since the jump rates from x/N (resp. (x + e

)/N ) to (x + e

)/N (resp. x/N) are equal, P

symmetric: P

(x, y) = P

(y, x).

Using the decompositions (1.3.1) and (1.2.6), we obtain

(x, y) =



j=1

N,j

, y

) and P

(x, y) =



j=1

, y

By deﬁnition, for every H : N

−1

→ R,

H =



∞

dt e

−λt

H = (λI − L

)

−1

where I is the identity operator.

Lemma 1.3.2. Let H : T

→ R be a continuous function. Then

lim

N→+∞



x∈T

H(x/N ) − P

H(x/N )| = 0. (1.3.2)

Proof. If H : T

→ R has the form H(x

, . . . , x

) =



j=1

), we have

H(x) =



j=1

N,j

) and P

H(x) =



j=1

). (1.3.3)

Now, for any continuous function H : T

→ R, and any  > 0, we can ﬁnd continuous functions

j,k

: T → R, such that H



: T

→ R, which is given by



(x) =



j=1



k=1

j,k

satisﬁes H



− H

∞

≤ . Thus,



x∈T

H(x/N ) − P

H(x/N )| ≤ 2 +



x∈T



(x/N) − P



(x/N )|.

By (1.3.3) and similar identities for P



and P

N,j



, the sum on the right hand side in the previous

inequality is less than or equal to



x∈T



j=1



k=1

N,k

j,k

/N ) −



k=1

j,k

/N )| ≤



x∈T



j=1



k=1

N,k

j,k

/N ) − P

j,k

/N )|,

where C

is a constant that depends on the product



k=1

j,k

. The previous expressions can be rewritten



j=1



k=1



x∈T

d−1



i=1

N,k

j,k

(i/N ) − P

j,k

(i/N )| =



j=1



k=1



i=1

N,k

j,k

(i/N ) − P

j,k

(i/N )|.

Moreover, by [14, Lemma 4.5 item iii], when N → ∞, the last expression converges to 0.

Corollary 1.3.3. Let H : T

→ R be a continuous function. Then

lim

N→+∞



x∈T

H(x/N ) − G

H(x/N )| = 0. (1.3.4)

Proof. By the deﬁnition of resolvent, for each N, the previous expression is less than or equal to



∞

dt e

−λt



x∈T

H(x/N ) − P

H(x/N )|.

Corollary now follows from the previous lemma.

Let f

→ R be any function. Then, whenever needed, we consider f : T

→ R as the

extension of f

to T

given by:

f(y) = f

(x), if x ∈ T

, y ≥ x and y − x

∞

Let H : T

→ R be a continuous function. Then the extension of P

H : T

→ R to T

belongs to

), and by symmetry of the transition probability P

(x, y) we have



duP

H(u) =



x∈T

H(x/N ). (1.3.5)

The next Lemma shows that H can be approximated by P

H. As an immediate consequence, we

obtain an approximation result involving the resolvent.

Lemma 1.3.4. Let H : T

→ R be a continuous function. Then,

lim

t→0

lim

N→+∞



x∈T

H(x/N ) − H(x/N )| = 0, (1.3.6)

and

lim

λ→+∞

lim

N→+∞



x∈T

|λG

H(x/N ) − H(x/N )| = 0. (1.3.7)

Proof. Fix  > 0, and consider H



as in the proof of Lemma 1.3.2. Thus,



x∈T

H(x/N ) − H(x/N )| ≤ 2 +



x∈T



(x/N) − H



(x/N)|,

where the second term on the right hand side is less than or equal to

sup

j,k



x∈T

N,k

j,k

/N ) − H

j,k

/N )|,

with C

being a constant that depends on H



. By [14, Lemma 4.6], the last expression converges to 0,

when N → ∞, and then t → 0. This proves the ﬁrst equality.

To obtain the second limit, note that, by deﬁnition of the resolvent, the second expression is less than

or equal to



∞

dtλe

−λt



x∈T

H(x/N ) − H(x/N )|.

By (1.3.5) the sum is uniformly bounded in t and N . Furthermore, it vanishes as N → ∞ and t → 0.

This proves the second part.

Fix a function H : T

→ R. For λ > 0, let H

= G

H be the solution of the resolvent equation

λH

− L

= H. (1.3.8)

Taking inner product on both sides of this equation with respect to H

,we obtain



x∈T

(x/N))

−



x∈T

(x/N )L



x∈T

(x/N )H(x/N ).

A simple computation shows that the second term on the left hand side is equal to



j=1



x∈T

x,x+e

[∇

N,j

(x/N )]

where ∇

N,j

H(x/N ) = N[H((x + e

)/N ) − H(x/N)] is the discrete derivative of the function H in the

direction of the vector e

. In particular, by Schwarz inequality,



x∈T

(x/N )

≤



x∈T

H(x/N )

and



j=1



x∈T

x,x+e

[∇

N,j

(x/N )]

≤



x∈T

H(x/N )

(1.3.9)

1.4 Scaling limit

Let M be the space of positive measures on T

with total mass bounded by one, and endowed with the

weak topology. Recall that π

∈ M stands for the empirical measure at time t. This is the measure on

obtained by rescaling space by N, and by assigning mass 1/N

to each particle:



x∈T

(x) δ

x/N

, (1.4.1)

where δ

is the Dirac measure concentrated in u.

For a continuous function H : T

→ R, π

, H stands for the integral of H with respect to π

π

, H =



x∈T

H(x/N )η

(x) .

This notation is not to be mistaken with the inner product in L

) introduced earlier. Also, when π

has a density ρ, π(t, du) = ρ(t, u)du, we sometimes write ρ

, H for π

, H.

For a local function g : {0, 1}

→ R, let ˜g : [0, 1] → R be the expected value of g under the stationary

states:

˜g(α) = E

[g(η)] .

For  ≥ 1 and d-dimensional integer x = (x

, . . . , x

), denote by η



(x) the empirical density of

particles in the box B



(x) = {(y

, . . . , y

) ∈ Z

; 0 ≤ y

− x

< }:



(x) =





y∈B



(x)

η(y) .

Fix T > 0, and let D([0, T ], M) be the space of M-valued c`adl`ag trajectories π : [0, T ] → M endowed

with the uniform topology. For each probability measure µ

on {0, 1}

, denote by Q

W,N

the measure

on the path space D([0, T ], M) induced by the measure µ

and the process π

introduced in (1.4.1).

Fix a continuous proﬁle ρ

: T

→ [0, 1], and consider a sequence {µ

: N ≥ 1} of measures on

{0, 1}

associated to ρ

in the sense (1.1.8). Further, we denote by Q

be the probability measure

on D([0, T ], M) concentrated on the deterministic path π(t, du) = ρ(t, u)du, where ρ is the unique weak

solution of (1.1.9) with γ = ρ

, l

= 0, r

= 1, k = 1, . . . , d, and Φ(α) = α + aα

In subsection 1.4.1 we show that the sequence {Q

W,N

: N ≥ 1} is tight, and in subsection 1.4.2 we

characterize the limit points of this sequence.

1.4.1 Tightness

The proof of tightness of sequence {Q

W,N

: N ≥ 1} is motivated by [21, 18]. We consider, initially, the

auxiliary M-valued Markov process {Π

λ,N

: t ≥ 0}, λ > 0, deﬁned by

λ,N

(H) = π

, G

H =



x∈Z





(x/N)η

(x),

for H in C(T

), where {G

: λ > 0} is the resolvent associated to the random walk {X

: t ≥ 0}

introduced in Section 1.3.

We ﬁrst prove tightness of the process {Π

λ,N

: 0 ≤ t ≤ T } for every λ > 0, and we then show that

{λΠ

λ,N

: 0 ≤ t ≤ T } and {π

: 0 ≤ t ≤ T } are not far apart if λ is large.

It is well-known [23, proposition 4.1.7] that to prove tightness of {Π

λ,N

: 0 ≤ t ≤ T } it is enough

to show tightness of the real-valued processes {Π

λ,N

(H) : 0 ≤ t ≤ T } for a set of smooth functions

H : T

→ R dense in C(T

) for the uniform topology.

Fix a smooth function H : T

→ R. Denote by the same symbol the restriction of H to N

−1

. Let

= G

H, and keep in mind that Π

λ,N

(H) = π

, H

. Denote by M

N,λ

the martingale deﬁned by

N,λ

= Π

λ,N

(H) − Π

λ,N

(H) −



ds N

π

, H

 . (1.4.2)

Clearly, tightness of Π

λ,N

(H) follows from tightness of the martingale M

N,λ

and tightness of the additive

functional



ds N

π

, H

.

A simple computation shows that the quadratic variation M

N,λ



of the martingale M

N,λ

is given

by:



j=1



x∈T

x,x+e

[∇

N,j

(x/N )]



x,x+e

(η

) [η

(x + e

) − η

(x)]

ds .

In particular, by (1.3.9),

M

N,λ



≤



j=1



x∈T

x,x+e

[(∇

N,j

)(x/N )]

≤

C(H)t

λN

for some ﬁnite constant C(H) which depends only on H. Thus, by Doob inequality, for every λ > 0,

δ > 0,

lim

N→∞



sup

0≤t≤T



N,λ



> δ



= 0 . (1.4.3)

In particular, the sequence of martingales {M

N,λ

: N ≥ 1} is tight for the uniform topology.

It remains to be examined the additive functional of the decomposition (1.4.2). The generator of the

exclusion process L

can be decomposed in terms of generators of the random walks L

. By (1.3.1)

and a long but simple computation, we obtain that N

π

, H

 is equal to



j=1





x∈T

)(x/N ) η(x)



x∈T



)((x + e

)/N ) + (L

)(x/N)



(τ

1,j

)(η)

−



x∈T

)(x/N)(τ

2,j

)(η)



where {τ

: x ∈ Z

} is the group of translations, so that (τ

η)(y) = η(x + y) for x, y in Z

, and the sum

is understood modulo N. Also, h

1,j

, h

2,j

are the cylinder functions

1,j

(η) = η(0)η(e

) , h

2,j

(η) = η(−e

)η(e

) .

For all 0 ≤ s < t ≤ T , we have





dr N

π

, H





≤

(1 + 3|b|)(t − s)



j=1



x∈T

(x/N)| ,

from Schwarz inequality and Lemma 1.3.1, the right hand side of the previous expression is bounded

above by

(1 + 3|b|)(t − s)d









x∈T



(x/N )



Since H

is the solution of the resolvent equation (1.3.8), we may replace L

by U

= λH

−H

in the previous formula. In particular, It follows from the ﬁrst estimate in (1.3.9), that the right hand

side of the previous expression is bounded above by dC(H, b)(t − s) uniformly in N, where C(H, b) is

a ﬁnite constant depending only on b and H. This proves that the additive part of the decomposition

(1.4.2) is tight for the uniform topology and therefore that the sequence of processes {Π

λ,N

: N ≥ 1} is

tight.

Lemma 1.4.1. The sequence of measures {Q

W,N

: N ≥ 1} is tight for the uniform topology.

Proof. It is enough to show that for every smooth function H : T → R, and every  > 0, there exists

λ > 0 such that

lim

N→∞



sup

0≤t≤T

|Π

λ,N

(λH) − π

, H| > 



= 0,

since, in this case, tightness of π

follows from tightness of Π

λ,N

. Since there is at most one particle per

site, the expression inside the absolute value is less than or equal to



x∈T



λH

(x/N ) − H(x/N )



By Lemma 1.3.4, this expression vanishes as N ↑ ∞ and then λ ↑ ∞.

1.4.2 Uniqueness of limit points

We prove in this subsection that all limit points Q

∗

of the sequence Q

W,N

are concentrated on absolutely

continuous trajectories π(t, du) = ρ(t, u)du, whose density ρ(t, u) is a weak solution of the hydrodynamic

equation (1.1.9) with l = 0 < r = 1 and Φ(α) = α + aα

Let Q

∗

be a limit point of the sequence Q

W,N

and assume, without loss of generality, that Q

W,N

converges to Q

∗

Since there is at most one particle per site, it is clear that Q

∗

is concentrated on trajectories π

(du)

which are absolutely continuous with respect to the Lebesgue measure, π

(du) = ρ(t, u)du, and whose

density ρ is non-negative and bounded by 1.

Fix a continuously diﬀerentiable function H : T

→ R, and λ > 0. Recall the deﬁnition of the

martingale M

N,λ

introduced in the previous section. By (1.4.2) and (1.4.3), for ﬁxed 0 < t ≤ T and

δ > 0,

lim

N→∞

W,N





π

, G

H − π

, G

H −



ds N

π

, G

H



> δ



= 0.

Since there is at most one particle per site, we may use Corollary 1.3.3 to replace G

H by G

H in

the expressions π

, G

H, π

, G

H above. On the other hand, the expression N

π

, G

H

has been computed in the previous subsection. Since E

i,j

] = α

, i = 1, 2 and j = 1, . . . , d, Lemma

1.3.1 and the estimate (1.3.9), permit us use Corollary 1.4.4 to obtain, for every t > 0, λ > 0, δ > 0,

i = 1, 2,

lim

ε→0

lim

N→∞











x∈T

(x/N )



i,j

(η

) −



εN

(x)







> δ





= 0.

Recall that L

H = λG

H −H. As before, we may replace G

H by G

H. Let U

= λG

H −H.

Since η

εN

(x) = ε

−d

(



j=1

/N, x

/N + εe

]), we obtain, from the previous considerations, that

lim

ε→0

lim

N→∞

W,N





π

, G

H −

− π

, G

H −







−d

(



j=1

[·, · + εe

])



, U





> δ





= 0 .

Since H is a smooth function, G

H and U

can be approximated, in L

), by continuous functions.

Since we assumed that Q

W,N

converges in the uniform topology to Q

∗

, we have that

lim

ε→0

Q ∗





π

, G

H − π

, G

H −

−







−d

(



j=1

[·, · + εe

])



, U





> δ





= 0 .

Using the fact that Q

∗

is concentrated on absolutely continuous paths π

(du) = ρ(t, u)du, with

positive density bounded by 1, ε

−d

(



j=1

[·, · + εe

]) converges in L

) to ρ(s, .) as ε ↓ 0. Thus,

∗





π

, G

H − π

, G

H −



ds Φ(ρ

) , L

H



> δ



= 0,

because U

= L

H. Letting δ ↓ 0, we see that, Q

∗

a.s.,

π

, G

H − π

, G

H =



ds Φ(ρ

) , L

H .

This identity can be extended to a countable set of times t. Taking this set to be dense, by continuity of

the trajectories π

, we obtain that it holds for all 0 ≤ t ≤ T . In the same way, it holds for any countable

family of continuous functions H. Taking a countable set of continuous functions, dense for the uniform

topology, we extend this identity to all continuous functions H, because G

converges to G

H in

), if H

converges to H in the uniform topology. Similarly, we can show that it holds for all λ > 0,

since, for any continuous function H, G

H converges to G

H in L

), as λ

→ λ.

Proposition 1.4.2. As N ↑ ∞, the sequence of probability measures Q

W,N

converges in the uniform

topology to Q

Proof. In the previous subsection we showed that the sequence of probability measures Q

W,N

is tight for

the uniform topology. Moreover, we just proved that all limit points of this sequence are concentrated

on weak solutions of the parabolic equation (1.1.9). The proposition now follows from a straightforward

adaptation of the uniquenesses of weak solutions proved in [18] for the d-dimensional case.

Proof of Theorem 1.1.3. Since Q

W,N

converges in the uniform topology to Q

, a measure which is

concentrated on a deterministic path. For each 0 ≤ t ≤ T and each continuous function H : T

→ R,

π

, H converges in probability to



du ρ(t, u) H(u), where ρ is the unique weak solution of (1.1.9)

with l

= 0, r

= 1, γ = ρ

and Φ(α) = α + aα

1.4.3 Replacement lemma

We will use some results from [23, Appendix A1]. Denote by H

(µ

|ν

) the relative entropy of a

probability measure µ

with respect to a stationary state ν

, see [23, Section A1.8] for a precise deﬁnition.

By the explicit formula given in [23, Theorem A1.8.3], we see that there exists a ﬁnite constant K

depending only on α, such that

(µ

|ν

) ≤ K

, (1.4.4)

for all measures µ

Denote by ·, ·

the inner product of L

(ν

) and denote by I

the convex and lower semicontinuous

[23, Corollary A1.10.3] functional deﬁned by

(f) = −L



f ,



f

for all probability densities f with respect to ν

(i.e., f ≥ 0 and



fdν

= 1). By [23, proposition

A1.10.1], an elementary computation shows that

(f) =



j=1



x∈T

x,x+e

(f) , where

x,x+e

(f) = (1/2) ξ

x,x+e



x,x+e

(η)





f(σ

x,x+e

η) −



f(η)



dν

By [23, Theorem A1.9.2], if {S

: t ≥ 0} stands for the semigroup associated to the generator N

(µ

|ν

) + 2 N



) ds ≤ H

(µ

|ν

) ,

where f

stands for the Radon-Nikodym derivative of µ

with respect to ν

Recall the deﬁnition of B



(x) in begin of this section. For each y ∈ B



(x), such that y

> x

, let



x+e

= (z

)

0≤k≤M (y)

(1.4.5)

be a path from x + e

to y such that:

1. Λ



x+e

begins at x + e

and ends at y, i.e.:

= x + e

and z

M(y)

= y;

2. The distance between two consecutive sites of the Λ



x+e

= (z

)

0≤k≤M (y)

is equal to 1, i.e.:

k+1

= z

+ e

; for some j = 1 . . . , d and for all k = 0, . . . , M(y) − 1;

3. Λ



x+e

is injective:

= z

for all 0 ≤ i < j ≤ M(y);

4. The path begins by jumping in the direction of e

. Furthermore, the jump in the direction of e

j+1

is only allowed when it is not possible to jump in the direction of e

, for j = 1, . . . , d − 1.

Lemma 1.4.3. Fix a function F : N

−1

→ R. There exists a ﬁnite constant C

= C

(a, g, W ),

depending only on a, g and W , such that



x∈T

F (x/N )



{τ

g(η) − ˜g(η

εN

(x))}f(η)ν

(dη)

≤

εN

d+1



x∈T



F (x/N)



δN



x∈T

F (x/N)

d−2

(f),

for all δ > 0, ε > 0 and all probability densities f with respect to ν

Proof. Any local function can be written as a linear combination of functions in the form



x∈A

η(x),

for ﬁnite sets A



s. It is therefore enough to prove the Lemma for such functions. We will only prove the

result for g(η) = η(0)η(e

). The general case can be handled in a similar way.

We begin by estimating



x∈T

F (x/N)



η(x){η(x + e

) −

(εN)



y∈B

Nε

(x)

η(y)}f(η)ν

(dη) (1.4.6)

in terms of the functional I

(f). The integral in (1.4.6) can be rewritten as:

(Nε)



y∈B

Nε

(x)



η(x)[η(x + e

) − η(y)]f(η)ν

(dη).

For each y ∈ B

Nε

(x), such that y

> x

, let Λ



x+e

= (z

)

0≤k≤M (y)

be a path like the one in (1.4.5).

Then, by property (1) of Λ



x+e

and using telescopic sum we have the following:

η(x + e

) − η(y) =

M(y)−1



k=0

[η(z

) − η(z

k+1

)].

We can, therefore, bound (1.4.6) above by

(Nε)



x∈T



y∈B

Nε

(x)

M(y)−1



k=0



F (x/N )η(x)[η(z

) − η(z

k+1

)]f(η)ν

(dη) +

εN

d+1



x∈T



F (x/N)



where the last term in the previous expression comes from the contribution of the points y ∈ B

Nε

(x),

such that y

= x

. Recall that, by property (2) of Λ



x+e

, we have that z

k+1

= z

+ e

, for some

j = 1, . . . , d.

For each term of the form



F (x/N)η(x){η(z) − η(z + e

)}f(η)ν

(dη)

we can use the change of variables η



= σ

z,z+e

η to write the previous integral as

(1/2)



F (x/N )η(x){η(z) − η(z + e

)}



f(η) − f(σ

z,z+e

η)



(dη) .

Since a − b = (

√

a −

√

b)(

√

a +

√

b) and

√

ab ≤ a + b, by Schwarz inequality the previous expression is

less than or equal to

4(1 − 2a

−

)ξ

z,z+e



F (x/N )

η(x){η(z) − η(z + e

)}





f(η) +



f(σ

z,z+e

η)



(dη) +

z,z+e



z,z+e

(η)





f(η) −



f(σ

z,z+e

η)



(dη)

for every A > 0. In this formula we used the fact that c

z,z+e

(η) is bounded below by 1 − 2a

−

, where

−

= max{−a, 0}. Since f is a density with respect to ν

, the ﬁrst expression is bounded above by

A/(1 − 2a

−

)ξ

z,z+e

, whereas the second one is equal to 2A

−1

z,z+e

(f).

So, using all the previous calculations together with properties (3) and (4) of the path Λ



x+e

, we

obtain that (1.4.6) is less than or equal to

εN

d+1



x∈T



F (x/N )



(1 − 2a

−



x∈T

F (x/N )



j=1

εN



k=1

−1

x+(k−1)e

,x+ke

2ε

d−1



j=1



x∈T

x,x+e

(f) .

By deﬁnition of the sequence {ξ

x,x+e



εN

k=1

−1

x+ke

≤ N[W

(1) − W

(0)]. Thus, choosing A =

2εN

−1

, for some δ > 0, we obtain that the previous sum is bounded above by

εN

d+1



x∈T



F (x/N )



δN



x∈T

F (x/N )

d−2

(f) .

Up to this point we have succeeded to replace η(x)η(x + e

) by η(x)η

εN

(x). The same arguments

permit to replace this latter expression by [η

εN

(x)]

, which concludes the proof of the Lemma.

Corollary 1.4.4. Fix a cylinder function g, and a sequence of functions {F

: N ≥ 1}, F

: N

−1

→

R such that

lim

N→∞



x∈T

(x/N )

< ∞ .

Then, for any t > 0 and any sequence of probability measures {µ

: N ≥ 1} on {0, 1}

lim

ε→0

lim

N→∞









x∈T

(x/N)



g(η

) − ˜g(η

εN

(x)) ds







= 0 .

Proof. Fix 0 < α < 1. By the entropy and Jensen inequalities, the expectation appearing in the statement

of the Lemma is bounded above by

γN

log E



exp









x∈T

(x/N )



g(η

) − ˜g(η

εN

(x))







(µ

|ν

)

γN

for all γ > 0. In view of (1.4.4), in order to prove the corollary it is enough to show that the ﬁrst

term vanishes as N ↑ ∞, and then ε ↓ 0, for every γ > 0. We may remove the absolute value inside

the exponential by using the elementary inequalities e

|x|

≤ e

+ e

−x

and lim

N→∞

−1

log{a

+ b

} ≤

max{lim

N→∞

−1

log a

, lim

N→∞

−1

log b

}. Thus, to prove the corollary, it is enough to show that

lim

ε→0

lim

N→∞

log E



exp







x∈T

(x/N ){τ

g(η

) − ˜g(η

εN

(x))}



= 0,

for every γ > 0.

By Feynman-Kac formula, for each ﬁxed N the previous expression is bounded above by

tγ sup











x∈T

(x/N ){τ

g(η) − ˜g(η

εN

(x))}f(η) dν

−

d−2

(f)







where the supremum is carried over all density functions f with respect to ν

. Letting δ = 1 in Lemma

1.4.3, we obtain that the previous expression is less than or equal to

γt

εN

d+1



x∈T



(x/N )



γεt



x∈T

(x/N)

for some ﬁnite constant C

which depends on a, g and W . By assumption on the sequence {F

for every γ > 0, this expression vanishes as N ↑ ∞ and then ε ↓ 0. This concludes the proof of the

Lemma.

1.5 Energy estimate

We prove in this section that any limit point Q

∗

of the sequence Q

W,N

is concentrated on trajectories

ρ(t, u)du having ﬁnite energy. A more comprehensive treatment of energies can be found in [34].

Denote by ∂

the partial derivative of a function with respect to the j-th coordinate, and by

0,1

([0, T ] × T

) the set of continuous functions with continuous partial derivative in the j-th coor-

dinate. Let L

⊗W

([0, T ] × T

) be the Hilbert space of measurable functions H : [0, T ] × T

→ R such

that



d(x

⊗ W

) H(s, u)

< ∞ ,

where d(x

⊗W

) represents the product measure in T

obtained from Lesbegue’s measure in T

d−1

and

the measure induced by W

d(x

⊗ W

) = dx

. . . dx

j−1

j+1

. . . dx

endowed with the inner product H, G

⊗W

deﬁned by

H, G

⊗W



d(x

⊗ W

) H(s, u) G(s, u) .

Let Q

∗

be a limit point of the sequence Q

W,N

, and assume, without loss of generality, that the

sequence Q

W,N

converges to Q

∗

Proposition 1.5.1. The measure Q

∗

is concentrated on paths ρ(t, x)dx with the property that for all

j = 1, . . . , d there exists a function in L

⊗W

([0, T ] × T

), denoted by dΦ/dW

, such that



dx (∂

H)(s, x) Φ(ρ(s, x)) =

−



d(x

⊗ W

(x)) (dΦ/dW

)(s, x) H(s, x),

for all functions H in C

0,1

([0, T ] × T

The previous proposition follows from the next Lemma. Recall the deﬁnition of the constant K

given in (1.4.4).

Lemma 1.5.2. There exists a ﬁnite constant K

, depending only on a, such that

∗



sup





dx (∂

H)(s, x) Φ(ρ(s, x))

− K



H(s, x)

d(x

⊗ W

(x))



≤ K

where the supremum is carried over all functions H ∈ C

0,1

([0, T ] × T

Proof of Proposition 1.5.1. Denote by  : C

0,1

([0, T ] × T

) → R the linear functional deﬁned by

(H) =



dx (∂

H)(s, x) Φ(ρ(s, x)) .

Since C

0,1

([0, T ] × T

) is dense in L

⊗W

([0, T ] × T

), by Lemma 1.5.2,  is Q

∗

-almost surely ﬁnite

in L

⊗W

([0, T ] × T

). In particular, by Riesz representation theorem, there exists a function G in

⊗W

([0, T ] × T

) such that

(H) = −



d(x

⊗ W

(x)) H(s, x) G(s, x) .

This concludes the proof of the proposition.

For a smooth function H : T

→ R, δ > 0, ε > 0 and a positive integer N, deﬁne W

(ε, δ, H, η) by

(ε, δ, H, η) =



x∈T

H(x/N )

εN



Φ(η

δN

(x)) − Φ(η

δN

(x + εNe

))



−

εN



x∈T

H(x/N )

([x

+ εN + 1]/N ) −W

/N )} .

The proof of Lemma 1.5.2 relies on the following result:

Lemma 1.5.3. Consider a sequence {H



,  ≥ 1} dense in C

0,1

([0, T ] ×T

). For every k ≥ 1, and every

ε > 0,

lim

δ→0

lim

N→∞



max

1≤i≤k





(ε, δ, H

(s, ·), η

) ds



≤ K

Proof. It follows from the replacement lemma that in order to prove the Lemma we just need to show

that

lim

N→∞



max

1≤i≤k





(ε, H

(s, ·), η

) ds



≤ K

where

(ε, H, η) =

εN



x∈T

H(x/N )



g(η) − τ

x+εNe

g(η)



−

εN



x∈T

H(x/N )

([x

+ εN + 1]/N ) −W

/N )} ,

and g(η) = η(0) + aη(0)η(e

By the entropy and Jensen’s inequalities, for each ﬁxed N, the previous expectation is bounded above

H(µ

|ν

)

log E



exp



max

1≤i≤k





ds W

(ε, H

(s, ·), η

)



By (1.4.4), the ﬁrst term is bounded by K

. Since exp{max

1≤j≤k

} is bounded above by



1≤j≤k

exp{a

and since lim

−d

log{a

} is less than or equal to the maximum of lim

−d

log a

and lim

−d

log b

the limit, as N ↑ ∞, of the second term in the previous expression is less than or equal to

max

1≤i≤k

lim

N→∞

log E



exp





ds W

(ε, H

(s, ·), η

)



We now prove that, for each ﬁxed i, the above limit is non-positive for a convenient choice of the constant

Fix 1 ≤ i ≤ k. By Feynman–Kac formula and the variational formula for the largest eigenvalue of a

symmetric operator, the previous expression is bounded above by



ds sup





(ε, H

(s, ·), η)f(η)ν

(dη) −

d−2

(f)



for each ﬁxed N . In this formula the supremum is taken over all probability densities f with respect to

To conclude the proof, rewrite

η(x)η(x + e

) − η(x + εNe

)η(x + (εN + 1)e

)

η(x){η(x + e

) − η(x + (εN + 1)e

)} + η(x + (εN + 1)e

){η(x) − η(x + εN e

)},

and repeat the arguments presented in the proof of Lemma 1.4.3.

Proof of Lemma 1.5.2. Assume without loss of generality that Q

W,N

converges to Q

∗

. Consider a

sequence {H



,  ≥ 1} dense in C

0,1

([0, T ] × T

). By Lemma 1.5.3, for every k ≥ 1

lim

δ→0

∗



max

1≤i≤k





dx H

(s, x)



Φ(ρ

(x)) − Φ(ρ

(x + εe

))



−



dx H

(s, x)

+ ε) − W

)]



≤ K

where ρ

(x) = (ρ

∗ ι

)(x) and ι

is the approximation of the identity ι

(·) = (δ)

−d

1{[0, δ]

}(·).

Letting δ ↓ 0, changing variables, and then letting ε ↓ 0, we obtain that

∗



max

1≤i≤k





(∂

)(s, x)Φ(ρ(s, x)) dx

− K



(s, x)

d(x

⊗ W

(x))



≤ K

To conclude the proof, we apply the monotone convergence theorem, and recall that {H



,  ≥ 1} is a

dense sequence in C

0,1

([0, T ] × T

) for the norm H

∞

+ (∂

H)

∞

Chapter 2

W -Sobolev spaces: Theory,

Homogenization and Applications

The space of functions that admit diﬀerentiation in a weak sense has been widely studied in the mathe-

matical literature. The usage of such spaces provides a wide application to the theory of partial diﬀerential

equations (PDE), and to many other areas of pure and applied mathematics. These spaces have become

associated with the name of the late Russian mathematician S. L. Sobolev, although their origins predate

his major contributions to their development in the late 1930s. In theory of PDEs, the idea of Sobolev

space allows one to introduce the notion of weak solutions whose existence, uniqueness, regularities, and

well-posedness are based on tools of functional analysis.

In classical theory of PDEs, two important classes of equations are: elliptic and parabolic PDEs.

They are second-order PDEs, with some constraints (coerciveness) in the higher-order terms. The

elliptic equations typically model the ﬂow of some chemical quantity within some region, whereas the

parabolic equations model the time evolution of such quantities. Consider the following particular classes

of elliptic and parabolic equations:



i=1

∂

u(x) = g(x), and



∂

u(t, x) =



i=1

∂

u(t, x),

u(0, x) = g(x),

(2.0.1)

for t ∈ (0, T ] and x ∈ D, where D is some suitable domain, and g is a function. Sobolev spaces are the

natural environment to treat equations like (2.0.1) - an elegant exposition of this fact can be found in

[11].

Consider the following generalization of the above equations:



i=1

∂

u(x) = g(x), and



∂

u(t, x) =



i=1

∂

u(t, x),

u(0, x) = g(x),

(2.0.2)

where ∂

stands for the generalized derivative operator, and for each i, W

is a one-dimensional strictly

increasing (not necessarily continuous) function, as in Chapter 1. Note that if W

) = x

, we obtain

the equations in (2.0.1). This notion of generalized derivative has been studied by several authors in the

literature, see for instance, [8, 16, 25, 26, 27]. We also call attention to [8] since it provides a detailed

study of such notion. The equations in (2.0.2) have the same physical interpretation as the equations in

(2.0.1). However, the latter covers more general situations. For instance, [18] and chapter 1 argue that

these equations may be used to model a diﬀusion of particles within a region with membranes induced

by the discontinuities of the functions W

. Unfortunately, the standard Sobolev spaces are not suitable

for being used as the space of weak solutions of equations in the form of (2.0.2).

One of our goals in this work is to deﬁne and obtain some properties of a space, which we call W -

Sobolev space. This space lets us formalize a notion of weak generalized derivative in such a way that,

if a function is W -diﬀerentiable in the strong sense, it will also be diﬀerentiable in the weak sense, with

their derivatives coinciding. Moreover, the W -Sobolev space will coincide with the standard Sobolev

space if W

) = x

for all i. With this in mind, we will be able to deﬁne weak solutions of equations in

(2.0.2). We will prove that there exist weak solutions for such equations, and also, for some cases, the

uniqueness of such weak solutions. Some analogous to classical results of Sobolev spaces are obtained,

such as Poincar´e’s inequality and Rellich-Kondrachov’s compactness theorem.

Besides the treatment of elliptic and parabolic equations in terms of these W -Sobolev spaces, we are

also interested in studying Homogenization and Hydrodynamic Limits. The study of homogenization is

motivated by several applications in mechanics, physics, chemistry and engineering. For example, when

one studies the thermal or electric conductivity in heterogeneous materials, the macroscopic properties

of crystals or the structure of polymers, are typically described in terms of linear or non-linear PDEs for

medium with periodic or quasi-periodic structure, or, more generally, stochastic.

We will consider stochastic homogenization. In the stochastic context, several works on homogeniza-

tion of operators with random coeﬃcients have been published (see, for instance, [30, 31] and references

therein). In homogenization theory, only the stationarity of such random ﬁeld is used. The notion of

stationary random ﬁeld is formulated in such a manner that it covers many objects of non-probabilistic

nature, e.g., operators with periodic or quasi-periodic coeﬃcients.

The focus of our approach is to study the asymptotic behavior of eﬀective coeﬃcients for a family

of random diﬀerence schemes, whose coeﬃcients can be obtained by the discretization of random high-

contrast lattice structures. In this sense, we want to extend the theory of homogenization of random

operators developed in [31], as well as to prove its main Theorem (Theorem 2.16) to the context in which

we have weak generalized derivatives.

Lastly, as an application of all the theory developed for W -Sobolev spaces, elliptic operators, parabolic

equations and homogenization, we prove a hydrodynamic limit for a process with conductances in random

environments. Hydrodynamic limit for process with conductances have been obtained in [18] for the one-

dimensional setup and in Chapter 1 for the d-dimensional setup. However, with the tools developed in our

present Chapter, the proof of the hydrodynamic limit on a more general setup (in random environments)

turns out to be simpler and much more natural. Furthermore, the proof of this hydrodynamic limit also

provides an existence theorem for the generalized parabolic equations such as the one in (2.0.2).

The random environment we considered is governed by the coeﬃcients of the discrete formulation of

the model (the process on the lattice). It is possible to obtain other formulations of random environments,

for instance, in [14] they proved a hydrodynamic limit for a gradient process with conductances in

a random environment whose randomness consists of the random choice of the conductances. The

hydrodynamic limit for a gradient process without conductances on the random environment we are

considering was proved in [20]. We would like to mention that in [13] a process evolving on a percolation

cluster (a lattice with some bonds removed randomly) was considered and the resulting process turned

out to be non-gradient. However, the homogenization tools facilitated the proof of the hydrodynamic

limit, which made the proof much simpler than the usual proof of hydrodynamic limit for non-gradient

processes (see for instance [23, Chapter 7]).

We now describe the organization of the Chapter. In Section 2.1 we deﬁne the W -Sobolev spaces

and obtain some results, namely, approximation by smooth functions, Poincar´e’s inequality, Rellich-

Kondrachov theorem (compact embedding), and a characterization of the dual of the W -Sobolev spaces.

In Section 2.2 we deﬁne the W -generalized elliptic equations, and what we call by weak solutions. We

then obtain some energy estimates and use them together with Lax-Milgram’s theorem to conclude

results regarding existence, uniqueness and boundedness of such weak solutions. In Section 2.3 we

deﬁne the W -generalized parabolic equations, their weak solutions, and prove uniquenesses of these

weak solutions. Moreover, a notion of energy is also introduced in this Section. Section 2.4 consists in

obtaining discrete analogous results to the ones of the previous sections. This Section serves as preamble

for the subsequent sections. In Section 2.5 we deﬁne the random operators we are interested and obtain

homogenization results for them. Finally, Section 2.6 concludes the Chapter with an application that is

interesting for both probability and theoretical physics, which is the hydrodynamic limit for a process

in random environments with conductances. This application uses results from all the previous sections

and provides a proof for existence of weak solutions of W -generalized parabolic equations.

2.1 W -Sobolev spaces

This Section is devoted to the deﬁnition and derivation of properties of the W -Sobolev spaces. We

begin by introducing some notation, stating some known results, and giving a precise deﬁnition of these

spaces. Poincar´e’s inequality, Rellich-Kondrachov theorem and a characterization of the dual space of

these Sobolev spaces are also obtained.

Fix a function W : R

→ R as in Chapter 1:

W (x

, . . . , x

) =



k=1

where each W

: R → R is a strictly increasing right continuous function with left limits (c`adl`ag),

periodic in the sense that for all u ∈ R

(u + 1) − W

(u) = W

(1) − W

(0).

Let L

⊗W

) be the Hilbert space of measurable functions H : T

→ R such that



d(x

⊗ W

) H(x)

< ∞,

where d(x

⊗W

) represents the product measure in T

obtained from Lesbegue’s measure in T

d−1

and

the measure induced by W

in T:

d(x

⊗ W

) = dx

···dx

k−1

k+1

···dx

Denote by H, G

⊗W

the inner product of L

⊗W

H, G

⊗W



d(x

⊗ W

) H(x) G(x) ,

and by  · 

⊗W

the norm induced by this inner product.

Recall the deﬁnition of the operator L

: D

→ L

) given in (1.1.5).

Lemma 2.1.1. Let f, g ∈ D

, then for i = 1, . . . , d,





∂

f(x)



g(x) dx = −



(∂

f)(∂

g)d(x

⊗ W

In particular,



f(x)g(x) dx = −



i=1



(∂

f)(∂

g)d(x

⊗ W

Proof. Let f, g ∈ D

. By Fubini’s theorem



f(x)g(x)dx =



d−1





f(x)g(x)dx



where dx

is the Lebesgue product measure in T

d−1

on the coordinates x

, . . . , x

i−1

, x

i+1

, . . . , x

An application of [18, Lemma 3.1 (b)] and again Fubini’s theorem concludes the proof of this Lemma.

Let L

⊗W

) be the closed subspace of L

⊗W

) consisting of the functions that have zero

mean with respect to the measure d(x

⊗ W



fd(x

⊗ W

) = 0.

Finally, using the characterization of the functions in D

given in Proposition 1.1.1, and the deﬁni-

tion of D

, we have that the set {∂

h; h ∈ D

} is dense in L

⊗W

2.1.1 The W -Sobolev space

We deﬁne the Sobolev space of W -generalized derivatives as the space of functions g ∈ L

) such that

for each i = 1, . . . , d there exist fuctions G

∈ L

⊗W

) satisfying the following integral by parts

identity.





∂



g dx = −



(∂

f) G

d(x

⊗ W

), (2.1.1)

for every function f ∈ D

. We denote this space by

1,W

). A standard measure-theoretic argument

allows one to prove that for each function g ∈

1,W

) and i = 1, . . . , d, we have a unique function

that satisﬁes (2.1.1). Note that D

⊂

1,W

). Moreover, if g ∈ D

then G

= ∂

g. For this

reason for each function g ∈

1,W

we denote G

simply by ∂

g, and we call it the ith generalized weak

derivative of the function g with respect to W .

Lemma 2.1.2. The set

1,W

) is a Hilbert space with respect to the inner product

f, g

1,W

= f, g +



i=1



(∂

f)(∂

g) d(x

⊗ W

) (2.1.2)

Proof. Let (g

)

n∈N

be a Cauchy sequence in

1,W

), and denote by  · 

1,W

the norm induced by

the inner product (2.1.2). By the deﬁnition of the norm  · 

1,W

, we obtain that (g

)

n∈N

is a Cauchy

sequence in L

) and that (∂

)

n∈N

is a Cauchy sequence in L

⊗W

) for each i = 1, . . . , d.

Therefore, there exist functions g ∈ L

) and G

∈ L

⊗W

) such that g = lim

n→∞

, and

= lim

n→∞

∂

. It remains to be proved that G

is, in fact, the ith generalized weak derivative of

g with respect to W . But this follows from a simple calculation: for each f ∈ D

we have





∂



gdx = lim

n→∞





∂



= − lim

n→∞



(∂

f)(∂

g)d(x

⊗ W

)

= −



(∂

f)G

d(x

⊗ W

where we used H¨older’s inequality to pass the limit through the integral sign.

2.1.2 Approximation by smooth functions and the energetic space

We will now obtain approximation of functions in the Sobolev space

1,W

) by functions in D

. Note

that the functions in D

can be seen as smooth, in the sense that one may apply the operator L

these functions in the strong sense.

Let us introduce ·, ·

1,W

the inner product on D

deﬁned by

f, g

1,W

= f, g + −L

f, g, (2.1.3)

and note that by Lemma 2.1.1,

f, g

1,W

= f, g +



i=1



(∂

f)(∂

g)d(x

⊗ W

Let H

1,W

(T) be the set of all functions f in L

) for which there exists a sequence (f

)

n∈N

in D

such that f

converges to f in L

) and f

is a Cauchy sequence for the inner product ·, ·

1,W

. Such

sequence (f

)

n∈N

is called admissible for f.

For f , g in H

1,W

), deﬁne

f, g

1,W

= lim

n→∞

f

, g



1,W

, (2.1.4)

where (f

)

n∈N

, (g

)

n∈N

are admissible sequences for f, and g, respectively. By [40, Proposition 5.3.3],

this limit exists and does not depend on the admissible sequence chosen; the set D

is dense in H

1,W

;

and the embedding H

1,W

⊂ L

) is continuous. Moreover, H

1,W

) endowed with the inner product

·, ·

1,W

just deﬁned is a Hilbert space. Denote ·

1,W

the norm in H

1,W

induced by ·, ·

1,W

. The space

1,W

) is called energetic space. For more details on the theory of energetic spaces see [40, Chapter

5].

Note that H

1,W

is the space of functions that can be approximated by functions in D

with respect

to the norm  · 

1,W

. The following Proposition shows that this space is, in fact, the Sobolev space

1,W

Proposition 2.1.3 (Approximation by smooth functions). We have the equality of the sets

1,W

) = H

1,W

In particular, we can approximate any function f in the Sobolev space

1,W

) by functions in D

Proof. Fix g ∈ H

1,W

). By deﬁnition, there exists a sequence g

in D

such that g

converges to g in

) and g

is Cauchy for the inner product ·, ·

1,W

. So, for each i = 1, . . . , d there exists functions

∈ L

⊗W

) such that ∂

converges to G

in L

⊗W

). Applying the H¨older’s inequality,

we deduce that for every f ∈ D





∂



g dx = lim

n→∞





∂



dx.

By Lemma 2.1.1, we obtain

lim

n→∞





∂



dx = lim

n→∞



(∂

f)(∂

) d(x

⊗ W

)

= −



(∂

f)G

d(x

⊗ W

Then, g ∈

1,W

) and therefore H

1,W

) ⊂

1,W

We will now prove that H

1,W

) is dense in

1,W

), and since both of them are complete, they

are equal. Note that since D

is dense in L

) and D

⊂ H

1,W

), we have that H

1,W

) is also

dense in L

Therefore, given a function g ∈

1,W

), we can approximate g by a sequence of functions (f

)

n∈N

in H

1,W

) with respect to the L

) norm. Let F

i,n

be the ith generalized weak derivative of f

with

respect to W . We have, therefore, for each h ∈ D

lim

n→∞



(∂

h)(F

i,n

− G

)d(x

⊗ W

) = − lim

n→∞





∂



− g)dx = 0.

Denote by F

i,n

: L

⊗W

) → R the sequence of bounded linear functionals induced by F

i,n

−G

i,n

(h) :=



h[F

i,n

− G

]d(x

⊗ W

for h ∈ L

⊗W

). We then note that, since the set {∂

h; h ∈ D

} is dense in L

⊗W

), F

i,n

converges to 0 pointwisely. By Banach-Steinhaus’ Theorem, F

i,n

converges strongly to 0, and, thus, F

i,n

converges to G

in L

⊗W

), for each i = 1, . . . , d. Therefore, f

converges to g in L

) and ∂

converges to G

in L

⊗W

) for each i, i.e., f

converges to g with the norm ·

1,W

, and the density

of H

1,W

) in

1,W

) follows.

The next Corollary shows an analogous of the classic result for Sobolev spaces with dimension d = 1,

which states that every function in the one-dimensional Sobolev space is absolutely continuous.

Corollary 2.1.4. A function f in L

(T) belongs to the Sobolev space

1,W

(T) if and only if there exists

F in L

(T) and a ﬁnite constant c such that



(0,1]

F (y) dW (y) = 0 and f(x) = c +



(0,x]

F (y) dW (y)

Lebesgue almost surely.

Proof. In [18] the energetic extension H

1,W

(T) has the characterization given in Corollary 2.1.4. By

Proposition 2.1.3 we have that these spaces coincide, and hence the proof follows.

From Proposition 2.1.3, we may use the notation H

1,W

) for the Sobolev space

1,W

). Another

interesting feature we have on this space, which is very useful in the study of elliptic equations, is the

Poincar´e inequality:

Corollary 2.1.5 (Poincar´e Inequality). For all f ∈ H

1,W

) there exists a ﬁnite constant C such that



f −



f dx



)

≤ C



i=1



(∂

d(x

⊗ W

)

:= C∇

f

)

Proof. We begin by introducing some notations. For x, y ∈ T

, i = 0, . . . , d and t ∈ T, denote

z(x, y, i) = (x

, . . . , x

d−i

, y

d−i+1

, . . . , y

) ∈ T

and

z(x, y, t, i) = (x

, . . . , x

d−i

, t, y

d−i+2

, . . . , y

) ∈ T

With this notation, we may write f(x) − f(y) as the telescopic sum

f(x) − f(y) =



i=1

f(z(x, y, i − 1)) − f(z(x, y, i)).

We are now in conditions to prove this Lemma. Let f ∈ D

, then



f −



fdx



)







f(x) − f(y)dy











i=1



∂

f(z(x, y, t, i))dW

(t)dy



≤









i=1





∂

f(z(x, y, t, i))



(t)dy



≤







i=1



d−i+1



∂

f(z(x, y, t, i))



d−i

(t)⊗ y

d−i+1

⊗ ···⊗ y



≤ C





i=1



d−i+1



∂

f(z(x, y, t, i))



d−i

(t)⊗ dy

d−i+1

⊗ ···⊗ dy

= C



i=1





∂



d(x

⊗ W

where in the next-to-last inequality, we used Jensen’s inequality and the elementary inequality (



)

≤



for some positive constant C. To conclude the proof, one uses Proposition 2.1.3 to approximate

functions in H

1,W

) by functions in D

2.1.3 A Rellich-Kondrachov theorem

In this subsection we prove an analogous of the Rellich-Kondrachov theorem for the W -Sobolev spaces.

We begin by stating this result in dimension 1, whose proof can be found in [18, Lemma 3.3].

Lemma 2.1.6. Fix some k ∈ {1, . . . , d}. The embedding H

1,W

(T) ⊂ L

(T) is compact.

Recall that they proved this result for the energetic extension, but in view of Proposition 2.1.3, this

result holds for our Sobolev space H

1,W

(T).

Proposition 2.1.7 (Rellich-Kondrachov). The embedding H

1,W

) ⊂ L

) is compact.

Proof. We will outline the strategy of the proof. Using the deﬁnition of the set D

and the fact that it

is dense in H

1,W

), it is enough to show this fact for sequences in D

. From this point, the main tool

is Lemma 2.1.6 and Cantor’s diagonal method to obtain converging subsequences.

We begin by noting that by Proposition 2.1.3, it is enough to prove that the embed D

⊂ L

) is

compact.

Let C > 0 and consider a sequence (v

)

n∈N

in D

, with v



1,W

≤ C for all n ∈ N. We have, by

deﬁnition of D

(see the deﬁnition at the beginning of Section 2.1), that each v

can be expressed as a

ﬁnite linear combination of elements in A

. Furthermore, each element in A

is a product of elements

in A

for k = 1, . . . , d. Therefore, we can write v

N(n)



j=1



k=1

k,j

N(n)



j=1

where g

k,j

∈ A

, α

∈ R, g



k=1

j,k

, and N(n) is chosen such that N(n) ≥ n (we can complete

with zeros if necessary). Recall that these functions g

k,j

have g

k,j



(T)

= 1, and hence, g



)

= 1.

Moreover, the set {g

, . . . , g

N(n)

} is orthogonal in L

From orthogonality, we obtain that

N(n)



j=1

(α

)

≤ C

, uniformly in n ∈ N.

Note that the uniform boundedness of v

in H

1,W

) implies the uniform boundedness of g

k,j



1,W

for all k = 1, . . . , d, j = 1, . . . , N(n) and n ∈ N. Our goal now is to apply Lemma 2.1.6 to our current

setup.

Consider the sequence of functions α

1,1

in H

1,W

(T). By Lemma 2.1.6, this sequence has a con-

verging subsequence, and we call the limit point α

1,1

. Repeat this step d − 1 times for the sequences

k,1

in H

1,W

(T), for k = 2, . . . , d, considering in each step a subsequence of the previous step, to obtain

converging subsequences, and call their limit points g

k,1

. At the end of this procedure, we obtain a

converging subsequence of



k=1

1,k

, with limit point



k=1

1,k

∈ L

), which we will denote by

In the jth step, in which we want to obtain the limit point α

, we repeat the previous idea, with the

sequences α

j,1

and g

j,k

, with n ≤ j and k = 2, . . . , d. We note that it is always necessary to consider

a subsequence of all the previous steps.

This procedure provides limiting functions α

, for all j ∈ N. From now on, we use the notation v

to mean the diagonal sequence obtained to ensure the convergence of the functions α

to α

. We

claim that the function

v =

∞



j=1

is well-deﬁned and belongs to L

). To prove this claim, note that the set {g

}

k∈N

is orthonormal by

the continuity of the inner product. Suppose that there exists N ∈ N such that



j=1

(α

)

> C

We have that the sequence of functions



j=1

converges to



j=1

Since v

 ≤ C uniformly in n ∈ N, this yields a contradiction. Therefore v ∈ L

) with the bound

v ≤ C.

It remains to be proved that v

has a subsequence that converges to v. Choose N so large that

v − v

 < /3, v

−v

 < /3 and v

−v

 < /3, and use the triangle inequality to conclude the

proof.

2.1.4 The space H

−1

)

Let H

−1

) be the dual space to H

1,W

), that is, H

−1

) is the set of bounded linear functionals

on H

1,W

). Our objective in this subsection is to characterize the elements of this space. This proof

is based on the characterization of the dual of the standard Sobolev space in R

(see [11]).

We will write (·, ·) to denote the pairing between H

−1

) and H

1,W

Lemma 2.1.8. f ∈ H

−1

) if and only if there exist functions f

∈ L

), and f

∈ L

⊗W

such that

f = f

−



i=1

∂

, (2.1.5)

in the sense that for v ∈ H

1,W

)

(f, v) =



vdx +



i=1



(∂

v)d(x

⊗ W

Furthermore,

f

−1

= inf













i=0



1/2

; f satisﬁes (2.1.5)







Proof. Let f ∈ H

−1

). Applying the Riesz Representation Theorem, we deduce the existence of a

unique function u ∈ H

1,W

) satisfying (f, v) = u, v

1,W

, for all v ∈ H

1,W

), that is



uvdx +



j=1



(∂

u)(∂

v)d(x

⊗ W

) = (f, v), for all v ∈ H

1,W

). (2.1.6)

This establishes the ﬁrst claim of the Lemma for f

= u and f

= ∂

u, for i = 1, . . . , d.

Assume now that f ∈ H

−1

(f, v) =



vdx +



i=1



(∂

v)d(x

⊗ W

), (2.1.7)

for g

, g

, . . . , g

∈ L

⊗W

). Setting v = u in (2.1.6), using (2.1.7), and applying the Cauchy-

Schwartz inequality twice, we deduce

u

1,W

≤



dx +



i=1



∂

d(x

⊗ W

). (2.1.8)

From (2.1.6) it follows that

|(f, v)| ≤ u

1,W

if v

1,W

≤ 1. Consequently

f

−1

≤ u

1,W

Setting v = u/u

1,W

in (2.1.6), we deduce that, in fact,

f

−1

= u

1,W

The result now follows from the above expression and equation (2.1.8).

2.2 W -Generalized elliptic equations

This subsection investigates the solvability of uniformly elliptic generalized partial diﬀerential equations

deﬁned below. Energy methods within Sobolev spaces are, essentially, the techniques exploited.

Let A = (a

(x))

d×d

, x ∈ T

, be a diagonal matrix function such that there exists a constant θ > 0

satisfying

−1

≤ a

(x) ≤ θ, (2.2.1)

for every x ∈ T

and i = 1, . . . , d. To keep notation simple, we write a

(x) to mean a

(x).

Our interest lies on the study of the problem

u = f, (2.2.2)

where u : T

→ R is the unknown function and f : T

→ R is given. Here T

denotes the generalized

elliptic operator

u := λu − ∇A∇

u := λu −



i=1

∂



(x)∂



. (2.2.3)

The bilinear form B[·, ·] associated with the elliptic operator T

is given by

B[u, v] = λu, v +



i=1



(x)(∂

u)(∂

v) d(W

⊗ x

), (2.2.4)

where u, v ∈ H

1,W

Let f ∈ H

−1

). A function u ∈ H

1,W

) is said to be a weak solution of the equation T

u = f if

B[u, v] = (f, v) for all v ∈ H

1,W

Recall a classic result from linear functional analysis, which provides in certain circumstances the

existence and uniqueness of weak solutions of our problem, and whose proof can be found, for instance,

in [11]. Let H be a Hilbert space endowed with inner product < ·, ·> and norm |·|. Also, (·, ·) denotes

the pairing of H with its dual space.

Theorem 2.2.1 (Lax-Milgram Theorem). Assume that B : H×H → R is a bilinear mapping on Hilbert

space H, for which there exist constants α > 0 and β > 0 such that for all u, v ∈ H,

|B[u, v]| ≤ α|u| · |v| and B[u, u] ≥ β|u|

Let f : H → R be a bounded linear functional on H. Then there exists a unique element u ∈ H such

that

B[u, v] = (f, v),

for all v ∈ H.

Return now to the speciﬁc bilinear form B[·, ·] deﬁned in (2.2.4). Our goal now is to verify the

hypothesis of Lax-Milgram Theorem for our setup. We consider the cases λ = 0 and λ > 0 separately.

We begin by analyzing the case in which λ = 0.

Let H

⊥

1,W

) be the set of functions in H

1,W

) which are orthogonal to the constant functions:

⊥

1,W

) = {f ∈ H

1,W

);



f dx = 0}.

The space H

⊥

1,W

) is the natural environment to treat elliptic operators with Neumann condition.

Proposition 2.2.2 (Energy estimates for λ = 0). Let B be the bilinear form on H

1,W

) deﬁned in

(2.2.4) with λ = 0. There exist constants α > 0 and β > 0 such that for all u, v ∈ H

1,W

|B[u, v]| ≤ αu

1,W

v

1,W

and for all u ∈ H

⊥

1,W

B[u, u] ≥ βu

1,W

Proof. By (2.2.1), the computation of the upper bound α easily follows. For the lower bound β, we have

for u ∈ H

⊥

1,W

u

1,W



dx +



i=1





∂



d(x

⊗ W

Using Poincar´e’s inequality and (2.2.1), we obtain a constant C > 0 such that the previous expression is

bounded above by





∂



d(x

⊗ W

) ≤ CB[u, u].

The lemma follows from the previous estimates.

Corollary 2.2.3. Let f ∈ L

). There exists a weak solution u ∈ H

1,W

) for the equation

∇A∇

u = f (2.2.5)

if and only if



fdx = 0.

In this case, we have uniquenesses of the weak solutions if we disregard addition by constant functions.

Also, let u be the unique weak solution of (2.2.5) in H

⊥

1,W

). Then

u

1,W

≤ Cf

)

for some constant C independent of f.

Proof. Suppose that there exists a weak solution u ∈ H

1,W

) of (2.2.5). Since the function v ≡ 1 ∈

1,W

), we have by deﬁnition of weak solution that



fdx = B[u, v] = 0.

Now, let f ∈ L

) with



fdx = 0. Consider the bilinear form B, deﬁned in (2.2.4) with λ = 0,

on the Hilbert space H

⊥

1,W

). By Proposition 2.2.2, B satisﬁes the hypothesis of the Lax-Milgram’s

Theorem. Further, f deﬁnes the bounded linear functional in H

⊥

1,W

) given by (f, g) = f, g for

every g ∈ H

⊥

1,W

). Then, an application of Lax-Milgram’s Theorem yields that there exists a unique

u ∈ H

⊥

1,W

) such that

B[u, v] = f, v for all v ∈ H

⊥

1,W

Moreover, by Proposition 2.2.2, there is a β > 0 such that

βu

1,W

≤ B[u, u] = f, u ≤ f 

)

u

)

≤ f

)

u

1,W

The existence of weak solutions and the bound C in the statement of the Corollary follows from the

previous expression.

We now analyze the case in which λ > 0.

Proposition 2.2.4 (Energy estimates for λ > 0). Let f ∈ L

). There exists a unique weak solution

u ∈ H

1,W

) for the equation

λu − ∇A∇

u = f, λ > 0. (2.2.6)

This solution enjoys the following bounds

u

1,W

≤ Cf

)

for some constant C > 0 independent of f, and

u ≤ λ

−1

f

)

Proof. Let β = min{λ, θ

−1

} > 0 and α = max{λ, θ} < ∞, where θ is given in (2.2.1). An elementary

computation shows that

B[u, v]| ≤ αu

1,W

v

1,W

and B[u, u] ≥ βu

1,W

By Lax-Milgram’s Theorem, there exists a unique solution u ∈ H

1,W

) of (2.2.6). Note that

βu

1,W

≤ B[u, u] = f, u ≤ f 

)

u

)

≤ f

)

u

1,W

and therefore u

1,W

≤ Cf 

)

for some constant C > 0 independent of f. The computation to

obtain the other bound is analogous.

Remark 2.2.5. Let L

: D

→ L

) be given by L

= ∇A∇

. This operator has the properties

stated in Theorem 1.1.2. We now outline the main steps to prove it. We may prove an analogous of

Lemma 1.2.2 for the operator L

. Using the bounds on the diagonal matrix A and Proposition 2.1.7

(Rellich-Kondrachov), we conclude that the energetic extension of the space induced by this operator has

compact embedding in L

). The previous results together with [39, Theorems 5.5.a and 5.5.c] implies

that L

has a self-adjoint extension L

, which is dissipative and non-positive, and its eigenvectors form

a complete orthonormal set in L

). Furthermore, the set of eigenvalues of this extension is countable

and its elements can be ordered resulting in a non-increasing sequence that tends to −∞.

Remark 2.2.6. Let L

be the self-adjoint extension given in Remark 2.2.5, and D

its domain. For

λ > 0 the operator λI − L

: D

→ L

) is bijective. Therefore, the equation

λu − ∇A∇

u = f,

has strong solution in D

if and only if f ∈ (λI − L

)(D

), where I is the identity operator and

(λI −L

)(D

) stands for the range of D

under the operator λI −L

. Moreover, this strong solution

coincides with the weak solution obtained in Proposition 2.2.4.

2.3 W -Generalized parabolic equations

In this Section, we study a class of W-generalized PDEs that involves time: the parabolic equations. The

parabolic equations are often used to describe in physical applications the time-evolution of the density

of some quantity, say a chemical concentration within a region. The motivation of this generalization

is to enlarge the possibility of such applications, for instance, these equations may be used to model a

diﬀusion of particles within a region with membranes (see Chapter 1 and [18]).

We begin by introducing the class of W -generalized parabolic equations we are interested. Then,

we deﬁne what is meant by weak solution of such equations, using the W -Sobolev spaces, and prove

uniquenesses of these weak solutions. In Section 2.6, we obtain existence of weak solutions of these

equations.

Fix T > 0 and let (B,  · 

) be a Banach space. We denote by L

([0, T ], B) the Banach space of

measurable functions U : [0, T ] → B for which

U

([0,T ],B)



U



dt < ∞.

Let A = A(t, x) be a diagonal matrix satisfying the ellipticity condition (2.2.1) for all t ∈ [0, T ],

Φ : [l, r] → R be a continuously diﬀerentiable function such that

−1

< Φ



(x) < B,

for all x, where B > 0, l, r ∈ R are constants. We will consider the equation



∂

u = ∇A∇

Φ(u) in (0, T ] × T

u = γ in {0}×T

(2.3.1)

where u : [0, T ] × T

→ R is the unknown function and γ : T

→ R is given.

We say that a function ρ = ρ(t, x) is a weak solution of the problem (2.3.1) if:

• For every H ∈ D

the following integral identity holds



ρ(t, x)H(x)dx −



γ(x)H(x)dx =



Φ(ρ(s, x))∇A∇

H(x)dx ds

• Φ(ρ(·, ·)) and ρ(·, ·) belong to L

([0, T ], H

1,W

)):



Φ(ρ(s, x))

)

+ ∇

Φ(ρ(s, x))

)

ds < ∞,

and



ρ(s, x)

)

+ ∇

ρ(s, x)

)

ds < ∞.

Consider the energy in jth direction of a function u(s, x) as

(u) = sup

H∈D





(∂

∂

H)(s, x) u(s, x)dx ds

−



[∂

H(s, x)]

d(x

⊗ W

)



and the total energy of a function u(s, x) as

Q(u) =



j=1

(u).

The notion of energy is important in probability theory and is often used in large deviations of Markov

processes. We also use this notion to prove the hydrodynamic limit in Section 2.6. The following lemma

shows the connection between the functions of ﬁnite energy and functions in the Sobolev space.

Lemma 2.3.1. A function u ∈ L

([0, T ], L

)) has ﬁnite energy if and only if u belongs to L

([0, T ], H

1,W

)).

In the case the energy is ﬁnite, we have

Q(u) =



∇

u

)

dt.

Proof. Consider functions U ∈ L

([0, T ], L

⊗W

)) as trajectories in L

⊗W

), that is, consider

a trajectory U : [0, T ] → L

⊗W

) and deﬁne U(s, x) as U(s, x) := [U(s)](x).

Let u ∈ L

([0, T ], L

)) and recall that the set {∂

H; H ∈ D

} is dense in L

⊗W

). Then

the set {∂

H(s, x); H ∈ L

([0, T ], D

)} is dense in L

([0, T ], L

⊗W

)). Suppose that u has ﬁnite

energy, and let H ∈ L

([0, T ], D

), then

(∂

H) =



(∂

∂

H)(s, x) u(s, x)dx ds

is a bounded linear functional in L

([0, T ], L

⊗W

)). Consequently, by Riesz’s representation theo-

rem, there exists a function G

∈ L

([0, T ], L

⊗W

)) such that

(∂

H) =



(∂

H)(x) G

(s, x)dx ds,

for all H ∈ L

([0, T ], D

From the uniqueness of the generalized weak derivative, we have that G

(s, x) = −∂

u(s, x).

Now, suppose u belongs to L

([0, T ], H

1,W

)) and let H ∈ L

([0, T ], D

). Then, we have



(∂

∂

H)(s, x) u(s, x)dx ds −





∂

H(s, x)



d(x

⊗ W

) =

−2



∂

H(s, x)∂

u(s, x)d(x

⊗ W

) −





∂

H(s, x)



d(x

⊗ W

)

We can rewrite the right-hand side of the above expression as

−2∂

H, 2∂

u + ∂

H

⊗W

. (2.3.2)

A simple calculation shows that, for a Hilbert space H with inner product < ·, ·>, the following

inequality holds:

− <v, u + v> ≤

<u, u>,

for all u, v ∈ H, and we have equality only when v = −1/2u.

Therefore, by the previous estimates and (2.3.2)



(∂

∂

H)(s, x) u(s, x)dx ds −





∂

H(s, x)



d(x

⊗ W

) ≤





∂

u(s, x)



d(x

⊗ W

By the deﬁnition of energy, we have for each j = 1, . . . , d,

(u) ≤





∂

u(s, x)



d(x

⊗ W

Hence, the total energy is ﬁnite. Using the fact that L

([0, T ], D

) is dense in L

([0, T ], H

1,W

)), we

have that

Q(u) =



j=1



∂

u

⊗W



∇

u

)

dt.

2.3.1 Uniqueness of weak solutions of the parabolic equation

Recall that we denote by ·, · the inner product of the Hilbert space L

). Fix H, G ∈ L

), λ > 0,

and denote by H

and G

in H

1,W

) the unique weak solutions of the elliptic equations

λH

− ∇A∇

= H,

and

λG

− ∇A∇

= G,

respectively. Then, we have the following symmetry property

G

, H = G, H

.

In fact, both terms in the previous equality are equal to





j=1



(∂

)(∂

)d(x

⊗ W

Let ρ : R

× T → [l, r] be a weak solution of the parabolic equation (2.3.1). Since ρ, Φ(ρ) ∈

([0, T ], H

1,W

)), and the set D

is dense in H

1,W

), we have for every H in H

1,W

ρ

, H − γ, H = −



j=1



∂

Φ(ρ

), ∂

H

⊗W

ds (2.3.3)

for all t > 0.

Denote by ρ

∈ H

1,W

) the unique weak solution of the elliptic equation

λρ

− ∇A∇

= ρ(s, ·). (2.3.4)

We claim that

ρ

, ρ

 − ρ

, ρ

 = −2



j=1



∂

Φ(ρ

) , ∂



⊗W

ds (2.3.5)

for all t > 0.

To prove this claim, ﬁx t > 0 and consider a partition 0 = t

< t

< ··· < t

= t of the interval [0, t].

Using the telescopic sum, we obtain

ρ

, ρ

 − ρ

, ρ

 =

n−1



k=0

ρ

k+1

, ρ

k+1

 − ρ

k+1

, ρ



n−1



k=0

ρ

k+1

, ρ

 − ρ

, ρ

 .

We handle the ﬁrst term, the second one being similar. From the symmetric property of the weak

solutions, ρ

k+1

belongs to H

1,W

) and since ρ is a weak solution of (2.3.1),

ρ

k+1

, ρ

k+1

 − ρ

k+1

, ρ

 = −



j=1



k+1

∂

Φ(ρ

) , ∂

k+1

ds .

Add and subtract ∂

Φ(ρ

) , ∂

 inside the integral on the right hand side of the above expression.

The time integral of this term is exactly the expression announced in (2.3.5) and the remainder is given



j=1



k+1



∂

Φ(ρ

) , ∂

 − ∂

Φ(ρ

) , ∂

k+1





ds .

Since ρ

is the unique weak solution of the elliptic equation (2.3.4), and the weak solution has the

symmetric property, we may rewrite the previous diﬀerence as



Φ(ρ

) , ρ

k+1

 − Φ(ρ

) , ρ





− λ



Φ(ρ

)

, ρ

k+1

 − Φ(ρ

)

, ρ





The time integral between t

and t

k+1

of the second term is equal to

−λ



k+1



k+1

∂

Φ(ρ

)

, ∂

Φ(ρ

) dr

because ρ is a weak solution of (2.3.1) and Φ(ρ

) belongs to H

1,W

). It follows from the boundedness

of the weak solution given in Proposition 2.2.4 and from the boundedness of the L

⊗W

) norm of

∂

Φ(ρ) obtained in expression (2.3.3), that this expression is of order (t

k+1

− t

)

To conclude the proof of claim (2.3.5) it remains to be shown that

n−1



k=0



k+1



Φ(ρ

) , ρ

k+1

 − Φ(ρ

) , ρ





vanishes as the mesh of the partition tends to 0. Using, again, the fact that ρ is a weak solution, we may

rewrite the sum as

−

n−1



k=0



k+1



k+1

∂

Φ(ρ

) , ∂

Φ(ρ

) dr .

We have that this expression vanishes as the mesh of the partition tends to 0 from the boundedness

of the L

⊗W

) norm of ∂

Φ(ρ). This proves (2.3.5).

Recall the deﬁnition of the constant B given at the beginning of this Section.

Lemma 2.3.2. Fix λ > 0, two density proﬁles γ

, γ

: T → [l, r] and denote by ρ

, ρ

weak solutions of

(2.3.1) with initial value γ

, γ

, respectively. Then,



− ρ

, ρ

1,λ

− ρ

2,λ



≤



− γ

, γ

1,λ

− γ

2,λ



Bλt/2

for all t > 0. In particular, there exists at most one weak solution of (2.3.1).

Proof. We begin by showing that if there exists λ > 0 such that

H, H

 = 0,

then H = 0. In fact, we would have the following



λ(H

)

dx +



j=1





∂



d(x

⊗ W

) =



dx = 0,

which implies that H



1,W

)

= 0, and hence H

= 0, which yields H = 0.

Fix two density proﬁles γ

, γ

: T

→ [l, r]. Let ρ

, ρ

be two weak solutions with initial values γ

, respectively. By (2.3.5), for any λ > 0,



− ρ

, ρ

1,λ

− ρ

2,λ



−



− γ

, γ

1,λ

− γ

2,λ



−2



Φ(ρ

) − Φ(ρ

) , ρ

− ρ

ds + 2λ





Φ(ρ

) − Φ(ρ

) , ρ

1,λ

− ρ

2,λ



ds .

(2.3.6)

Deﬁne the inner product in H

1,W

)

u, v

= u, v

.

This is, in fact, an inner product, since u, v

= v, u

by the symmetric property, and if u = 0, then

u, u

> 0:



dx = λ



dx +



j=1





∂



d(x

⊗ W

The linearity of this inner product can be easily veriﬁed.

Then, we have

2λ





Φ(ρ

) − Φ(ρ

) , ρ

1,λ

− ρ

2,λ



ds = 2λ





Φ(ρ

) − Φ(ρ

) , ρ

− ρ



ds.

By using the Cauchy-Schwartz inequality twice, the term on the right hand side of the above formula

is bounded above by





Φ(ρ

) − Φ(ρ

) , Φ(ρ

)

− Φ(ρ

)



ds + Aλ





− ρ

, ρ

1,λ

− ρ

2,λ



for every A > 0. From Proposition 2.2.4, we have that u

 ≤ λ

−1

u, and since Φ



is bounded by B,

the ﬁrst term of the previous expression is less than or equal to

Aλ





− ρ

, Φ(ρ

) − Φ(ρ

)



ds .

Choosing A = B/2λ, this expression cancels with the ﬁrst term on the right hand side of (2.3.6). In

particular, the left hand side of this formula is bounded by

Bλ





− ρ

, ρ

1,λ

− ρ

2,λ



ds .

To conclude, recall Gronwall’s inequality.

Remark 2.3.3. Let L

: D

→ L

) be the self-adjoint extension given in Remark 2.2.5. For λ > 0,

deﬁne the resolvent operator G

= (λI − L

)

−1

. Following the Chapter 1 and [18], another possible

deﬁnition of weak solution of equation (2.3.1) is given as follows: a bounded function ρ : [0, T ]×T

→ [l, r]

is said to be a weak solution of the parabolic diﬀerential equation (2.3.1) if

ρ

, G

h − γ, G

h =



Φ(ρ

), L

hds (2.3.7)

for every continuous function h : T

→ R, t ∈ [0, T ], and all λ > 0. We claim that this deﬁnition of weak

solution coincides with our deﬁnition introduced at the beginning of Section 2.3. Indeed, for continuous

h : T

→ R, G

h belongs to D

. Since D

is dense in D

with respect to the H

1,W

)-norm, it follows

that our deﬁnition implies the current deﬁnition. Conversely, since the set of continuous functions is

dense in L

), the identity (2.3.7) is valid for all h ∈ L

). Therefore, for each H ∈ D

we have

ρ

, H − γ, H =



Φ(ρ

), L

Hds.

In particular, the above identity holds for every H ∈ D

, and therefore the integral identity in our

deﬁnition of weak solutions holds.

It remains to be checked that the weak solution of the current deﬁnition belongs to L

([0, T ], H

1,W

)).

This follows from the fact that there exists at most one weak solution satisfying (2.3.7), that this unique

solution has ﬁnite energy, and from Lemma 2.3.1. A proof of the fact that there exists at most one

solution satisfying (2.3.7), and that this unique solution has ﬁnite energy, can be found in [18].

Finally, the integral identity of our deﬁnition of weak solution has an advantage regarding the integral

identity (2.3.7), due to the fact that we do not need the resolvent operator G

for any λ. Moreover, we

have an explicit characterization of our test functions.

2.4 W -Generalized Sobolev spaces: Discrete version

We will now establish some of the results obtained in the above sections to the discrete version of the

W -Sobolev space. Our motivation to obtain these results is that they will be useful when studying

homogenization in Section 2.5. We begin by introducing some deﬁnitions and notations.

Fix W as in (1.1.1) and functions f, g deﬁned on N

−1

. Consider the following diﬀerence operators:

∂

, which is the standard diﬀerence operator,

∂





= N





x + e



− f







and ∂

, which is the W

-diﬀerence operator:

∂







x+e



− f







x+e



− W





for x ∈ T

. We introduce the following scalar product

f, g



x∈T

f(x)g(x),

f, g

d−1



x∈T

f(x)g(x)



W ((x + e

)/N ) − W (x/N )



f, g

1,W,N

:= f, g



j=1

∂

f, ∂

g

and its induced norms

f

)

= f, f 

, f 

)

= f, f 

and f

1,W

)

= f, f 

1,W,N

These norms are natural discretizations of the norms introduced in the previous sections. Note

that the properties of the Lebesgue’s measure used in the proof of Corollary 2.1.5, also holds for the

normalized counting measure. Therefore, we may use the same arguments of this Corollary to prove its

discrete version.

Lemma 2.4.1 (Discrete Poincar´e Inequality). There exists a ﬁnite constant C such that



f −



x∈T



)

≤ C∇

f

)

where

∇

f

)



j=1

∂

f

)

for all f : N

−1

→ R.

Let A be a diagonal matrix satisfying (2.2.1). We are interested in studying the problem

u = f, (2.4.1)

where u : N

−1

→ R is the unknown function, f : N

−1

→ R is given, and T

denotes the discrete

generalized elliptic operator

u := λu − ∇

A∇

u, (2.4.2)

with

∇

A∇

u :=



i=1

∂



(x/N )∂



The bilinear form B

[·, ·] associated with the elliptic operator T

is given by

[u, v] = λu, v

d−1



i=1



x∈T

(x/N )(∂

u)(∂

v)[W

((x

+ 1)/N ) − W

/N )],

(2.4.3)

where u, v : N

−1

→ R.

A function u : N

−1

→ R is said to be a weak solution of the equation T

u = f if

[u, v] = f, v

for all v : N

−1

→ R.

We say that a function f : N

−1

→ R belongs to the discrete space of functions orthogonal to the

constant functions H

⊥

) if



x∈T

f(x/N ) = 0.

The following results are analogous to the weak solutions of generalized elliptic equations for this

discrete version. We remark that the proofs of these lemmas are identical to the ones in the continuous

case. Furthermore, the weak solution for the case λ = 0 is unique in H

⊥

Lemma 2.4.2. The equation

∇

A∇

u = f,

has weak solution u : N

−1

→ R if and only if



x∈T

f(x) = 0.

In this case we have uniqueness of the solution disregarding addition by constants. Moreover, if u ∈

⊥

) we have the bound

u

1,W

)

≤ Cf

)

, and u

)

≤ λ

−1

f

)

where C > 0 does not depend on f nor N.

Lemma 2.4.3. Let λ > 0. There exists a unique weak solution u : N

−1

→ R of the equation

λu − ∇

A∇

u = f. (2.4.4)

Moreover,

u

1,W

)

≤ Cf

)

, and u

)

≤ λ

−1

f

)

where C > 0 does not depend neither on f nor N.

Remark 2.4.4. Note that in the set of functions in T

we have a “Dirac measure” concentrated in a

point x as a function: the function that takes value N

in x and zero elsewhere. Therefore, we may

integrate these weak solutions with respect to this function to obtain that every weak solution is, in fact,

a strong solution.

2.4.1 Connections between the discrete and continuous Sobolev spaces

Given a function f ∈ H

1,W

), we can deﬁne its restriction f

to the lattice N

−1

(x) = f(x) if x ∈ N

−1

However, given a function f : N

−1

→ R it is not straightforward how to deﬁne an extension

belonging to H

1,W

). To do so, we need the deﬁnition of W -interpolation, which we give below.

Let f

: N

−1

→ R and W : R → R, a strictly increasing right continuous function with left limits

(c`adl`ag), and periodic. The W -interpolation f

∗

of f

is given by:

∗

(x + t) :=

W ((x + 1)/N ) − W ((x + t)/N )

W ((x + 1)/N ) − W (x/N )

f(x) +

W ((x + t)/N ) − W (x/N )

W ((x + 1)/N ) − W (x/N )

f(x + 1)

for 0 ≤ t < 1. Note that

∂f

∗

∂W

(x + t) =

f(x + 1) − f(x)

W ((x + 1)/N ) − W (x/N )

= ∂

f(x).

Using the standard construction of d-dimensional linear interpolation, it is possible to deﬁne the

W -interpolation of a function f

: T

→ R, with W (x) =



i=1

) as deﬁned in (1.1.1).

We now establish the connection between the discrete and continuous Sobolev spaces by showing how

a sequence of functions deﬁned in T

can converge to a function in H

1,W

We say that a family f

∈ L

) converges strongly (resp. weakly) to the function f ∈ L

) as

N → ∞ if f

∗

converges strongly (resp. weakly) to the function f . From now on we will omit the symbol

“

∗

” in the W -interpolated function, and denoting them simply by f

The convergence in H

−1

) can be deﬁned in terms of duality. Namely, we say that a functional f

on T

converges to f ∈ H

−1

) strongly (resp. weakly) if for any sequence of functions u

: T

→ R

and u ∈ H

1,W

) such that u

→ u weakly (resp. strongly) in H

1,W

), we have

, u

)

−→ (f, u), as N → ∞.

Remark 2.4.5. Suppose in Lemma 2.4.3 that f ∈ L

), and let u be a weak solution of the problem

(2.4.4), then we have the following bound

u

1,W

)

≤ Cf

)

since f

)

→ f

)

as N → ∞.

2.5 Homogenization

In this “brief” Section we prove a homogenization result for the W -generalized diﬀerential operator. We

follow the approach considered in [31]. The study of homogenization is motivated by several applications

in mechanics, physics, chemistry and engineering. The focus of our approach is to study the asymptotic

behavior of eﬀective coeﬃcients for a family of random diﬀerence schemes whose coeﬃcients can be

obtained by the discretization of random high-contrast lattice structures.

This Section is structured as follows: in subsection 6.1 we deﬁne the concept of H-convergence

together with some properties; subsection 6.2 deals with a description of the random environment along

with some deﬁnitions, whereas the main result is proved in subsection 6.3.

2.5.1 H-convergence

We say that the diagonal matrix A

= (a

) H-converges to the diagonal matrix A = (a

), denoted by

−→ A, if, for every sequence f

∈ H

−1

) such that f

→ f as N → ∞ in H

−1

), we have

• u

→ u

weakly in H

1,W

) as N → ∞,

• a

∂

→ a

∂

weakly in L

⊗W

) for each j = 1, . . . , d,

where u

: T

→ R is the solution of the problem

λu

− ∇

∇

= f

and u

∈ H

1,W

) is the solution of the problem

λu

− ∇A∇

= f.

The notion of convergence used in both items above was deﬁned in subsection 2.4.1.

We now obtain a property regarding H-convergence.

Proposition 2.5.1. Let A

−→ A, as N → ∞, with u

being the solution of

λu

− ∇

∇

= f,

where f ∈ H

−1

) is ﬁxed. Then, the following limit relations hold true:



x∈T

(x) →



(x)dx,

and

d−1



j=1



x∈T

(x)(∂

(x))

((x

+ 1)/N ) − W

/N )]

→



j=1



(x)(∂

(x))

d(x

⊗ W

as N → ∞.

Proof. We begin by noting that



x∈T

f(u

− u

) → 0, (2.5.1)

as N → ∞ since u

− u

converges weakly to 0 in H

1,W

). On the other hand, we have



x∈T

f(u

− u

) =



x∈T

(λu

− ∇

∇

)(u

− u

)



x∈T

−



x∈T

∇

−



x∈T



x∈T

∇

Using the weak convergences of u

and a

∂

, and the convergence in (2.5.1), we obtain, after a

summation by parts in the above expressions,



x∈T

d−1



j=1



x∈T

(∂

)

((x

+ 1)/N ) − W

)]

N→∞

−→ λ



dx +



j=1



(∂

)

d(x

⊗ W

). (2.5.2)

By Lemma 2.4.3, the sequence u

is  · 

1,W

bounded uniformly. Suppose, now, that u

does not

converge to u

in L

). That is, there exist  > 0 and a subsequence (u

) such that

u

− u



)

> ,

for all k. By Rellich-Kondrachov Theorem (Proposition 2.1.7), we have that there exists v ∈ L

) and

a further subsequence (also denoted by u

) such that

k→∞

−→ v, in L

This implies that

→ v, weakly in L

but this is a contradiction, since

→ u

, weakly in L

and v−u



)

≥ . Therefore, u

→ u

in L

). The proof thus follows from expression (2.5.2).

This Proposition shows that even though the H-convergence only requires weak convergence in its

deﬁnition, it yields a convergence in the strong sense (convergence in the L

-norm).

2.5.2 Random environment

In this subsection we introduce the statistically homogeneous rapidly oscillating coeﬃcients that will be

used to deﬁne the random W -generalized diﬀerence elliptic operators, where the W-generalized diﬀerence

elliptic operator was given in Section 2.4.

Let (Ω, F, µ) be a standard probability space and {T

: Ω → Ω; x ∈ Z

} be a group of F-measurable

and ergodic transformations which preserve the measure µ:

• T

: Ω → Ω is F-measurable for all x ∈ Z

• µ(T

A) = µ(A), for any A ∈ F and x ∈ Z

• T

= I , T

◦ T

= T

x+y

• For any f ∈ L

(Ω) such that f(T

ω) = f(ω) µ-a.s for each x ∈ Z

, is equal to a constant µ-a.s.

The last condition implies that the group T

is ergodic.

Let us now introduce the vector-valued F-measurable functions {a

(ω); j = 1, . . . , d} such that there

exists θ > 0 with

−1

≤ a

(w) ≤ θ,

for all ω ∈ Ω and j = 1, . . . , d. Then, deﬁne the diagonal matrices A

whose elements are given by

(x) := a

= a

ω) , x ∈ T

, j = 1, . . . , d. (2.5.3)

2.5.3 Homogenization of random operators

Let λ > 0, f

be a functional on the space of functions h

: T

→ R, f ∈ H

−1

) (see also, subsection

2.1.4), u

be the unique weak solution of

λu

− ∇

∇

= f

and u

be the unique weak solution of

λu

− ∇A∇

= f. (2.5.4)

For more details on existence and uniqueness of such solutions see Sections 2.2 and 2.4.

We say that the diagonal matrix A is a homogenization of the sequence of random matrices A

the following conditions hold:

• For each sequence f

→ f in H

−1

), u

converges weakly in H

1,W

to u

, when N → ∞;

• a

∂

→ a

∂

u, weakly in L

⊗W

) when N → ∞.

Note that homogenization is a particular case of H-convergence.

We will now state and prove the main result of this Section.

Theorem 2.5.2. Let A

be a sequence of ergodic random matrices, such as the one that deﬁnes our

random environment. Then, almost surely, A

(ω) admits a homogenization, where the homogenized

matrix A does not depend on the realization ω.

Proof. Fix f ∈ H

−1

), and consider the problem

λu

− ∇

∇

= f.

Using Lemma 2.4.3 and Remark 2.4.5, there exists a unique weak solution u

of the problem above,

such that its H

1,W

norm is uniformly bounded in N. That is, there exists a constant C > 0 such that

u



1,W

)

≤ Cf

)

Thus, the L

)-norm of a

∂

is uniformly bounded.

From W -interpolation (see subsection 2.4.1) and the fact that H

1,W

) is a Hilbert space (Lemma

2.1.2), there exists a convergent subsequence of u

(which we will also denote by u

) such that

→ u

, weakly in H

1,W

and

∂

→ v

weakly in L

), (2.5.5)

as N → ∞; v

being some function in L

⊗W

First, observe that the weak convergence in H

1,W

) implies that

∂

N→∞

−→ ∂

u weakly in L

⊗W

). (2.5.6)

From Birkhoﬀ’s ergodic theorem, we obtain the almost sure convergence, as N tends to inﬁnity, of the

random coeﬃcients:

−→ a

, (2.5.7)

where a

= E[a

], for any N

∈ N.

From convergences in (2.5.5), (2.5.6) and (2.5.7), we obtain that

= a

∂

where, from the weak convergences, u

clearly solves problem (2.5.4).

To conclude the proof it remains to be shown that we can pass from the subsequence to the sequence.

This follows from uniquenesses of weak solutions of the problem (2.5.4).

Remark 2.5.3. At ﬁrst sight, one may think that we are dealing with a very special class of matrices A

(diagonal matrices). Nevertheless, the random environment for random walks proposed in [31, Section

2.3], which is also exactly the same random environment employed in [20], results in diagonal matrices.

This is essentially due to the fact that in symmetric nearest-neighbor interacting particle systems (for

example, the zero-range dynamics considered in [20]), a particle at a site x ∈ T

may jump to the sites

x ± e

, j = 1, . . . , d. In such a case, the jump rate from x to x + e

determines the jth element of the

diagonal matrix.

Remark 2.5.4. Note that if u ∈ D

is a strong solution (or weak, in view of Remark 2.4.4) of

λu − ∇A∇

u = f

and u

is strong solution of the discrete problem

λu

− ∇

∇

= f

then, the homogenization theorem also holds, that is, u

also converges weakly in H

1,W

to u.

2.6 Hydrodynamic limit of processes with conductances in ran-

dom environment

Lastly, as an application of all the theory developed in the previous sections, we prove a hydrodynamic

limit for a process with conductances in random environments. Hydrodynamic limits for processes with

conductances have been obtained in [18] for the one-dimensional setup and in Chapter 1 for the d-

dimensional setup. However, the proof given here is much simpler and more natural, in view of the

theory developed here, than the proofs given in [18] and Chapter 1. Furthermore, the proof of this

hydrodynamic limit also provides an existence theorem for the W -generalized parabolic equations in

(2.3.1).

The hydrodynamic limit allows one to deduce the macroscopic behavior of the system from the micro-

scopic interaction among particles. Moreover, this approach justiﬁes rigorously a method often used by

physicists to establish the partial diﬀerential equations that describe the evolution of the thermodynamic

characteristics of a ﬂuid.

This Section is structured as follows: in subsection 7.1 we present the model, derive some properties

and ﬁx the notations; subsection 7.2 deals with the hydrodynamic equation; ﬁnally, subsections 7.3 and

7.4 are devoted to the proof of the hydrodynamic limit.

2.6.1 The exclusion processes with conductances in random environments

Fix a typical realization ω ∈ Ω of the random environment deﬁned in Section 2.5. For each x ∈ T

and

j = 1, . . . , d, deﬁne the symmetric rate ξ

x,x+e

= ξ

x+e

x,x+e

(x)

N[W ((x + e

)/N ) − W (x/N )]

(x)

N[W

((x

+ 1)/N ) − W

/N )]

. (2.6.1)

where a

(x) is given by (2.5.3), and e

, . . . , e

is the canonical basis of R

. Also, let b > −1/2 and

recall that

x,x+e

(η) = 1 + b{η(x − e

) + η(x + 2 e

)} ,

where all sums are modulo N.

Distribute particles on T

in such a way that each site of T

is occupied at most by one particle.

Denote by η the conﬁgurations of the state space {0, 1}

so that η(x) = 0 if site x is vacant, and

η(x) = 1 if site x is occupied.

The exclusion process with conductances in a random environment is a continuous-time Markov

process {η

: t ≥ 0} with state space {0, 1}

= {η : T

→ {0, 1}}, whose generator L

acts on

functions f : {0, 1}

→ R as

f)(η) =



j=1



x∈T

x,x+e

(η) {f(σ

x,x+e

η) − f(η)} ,

where σ

x,x+e

η is the conﬁguration obtained from η by exchanging the variables η(x) and η(x + e

(σ

x,x+e

η)(y) =











η(x + e

) if y = x,

η(x) if y = x + e

η(y) otherwise.

We consider the Markov process {η

: t ≥ 0} on the conﬁgurations {0, 1}

associated to the generator

in the diﬀusive scale, i.e., L

is speeded up by N

We now describe the stochastic evolution of the process. After a time given by an exponential

distribution, a random choice of a point x ∈ T

is made. At rate ξ

x,x+e

the occupation variables η(x),

η(x + e

) are exchanged. Note that only nearest neighbor jumps are allowed. The conductances are

induced by the function W , whereas the random environment is given by the matrix A

:= (a

(x))

d×d

The dynamics informally presented describes a Markov evolution. A computation shows that the

Bernoulli product measures {ν

: 0 ≤ α ≤ 1} are invariant, in fact reversible, for the dynamics.

Consider the random walk {X

}

t≥0

of a particle in T

induced by the generator L

given as follows.

Let ξ

x,x+e

given by (2.6.1). If the particle is on a site x ∈ T

, it will jump to x+e

with rate N

x,x+e

Furthermore, only nearest neighbor jumps are allowed. The generator L

of the random walk {X

}

t≥0

acts on functions f : T

→ R as







j=1





where,





= N



x,x+e





x + e



− f





+ ξ

x−e





x − e



− f





It is not diﬃcult to see that the following equality holds:

f(x/N ) =



j=1

∂

f)(x) := ∇

∇

f(x). (2.6.2)

Note that several properties of the above operator have been obtained in Section 2.4. The counting

measure m

on N

−1

is reversible for this process. This random walk plays an important role in the

proof of the hydrodynamic limit of the process η

, as we will see in subsection 7.3.

Recall that D(R

, {0, 1}

) is the path space of c`adl`ag trajectories with values in {0, 1}

. For a

measure µ

on {0, 1}

, denote by P

the probability measure on D(R

, {0, 1}

) induced by the

initial state µ

and the Markov process {η

: t ≥ 0}. Expectation with respect to P

is denoted by

2.6.2 The hydrodynamic equation

Let A = (a

)

d×d

be a diagonal matrix with a

> 0, j = 1, . . . , d, and consider the operator

∇A∇



j=1

∂

deﬁned on D

A sequence of probability measures {µ

: N ≥ 1} on {0, 1}

is said to be associated to a proﬁle

: T

→ [0, 1] if

lim

N→∞











x∈T

H(x/N )η(x) −



H(u)ρ

(u)du



> δ







= 0 (2.6.3)

for every δ > 0 and every function H ∈ D

Let γ : T

→ [l, r] be a bounded density proﬁle and consider the parabolic diﬀerential equation



∂

ρ = ∇A∇

Φ(ρ)

ρ(0, ·) = γ(·)

, (2.6.4)

where the function Φ : [l, r] → R is given as in the beginning of Section 1.5, and t ∈ [0, T ], for T > 0

ﬁxed.

Recall, from Section 2.3, that a bounded function ρ : [0, T ] ×T

→ [l, r] is said to be a weak solution

of the parabolic diﬀerential equation (1.1.9) if the following conditions hold. Φ(ρ(·, ·)) and ρ(·, ·) belong

to L

([0, T ], H

1,W

)), and we have the integral identity



ρ(t, u)H(u)du −



ρ(0, u)H(u)du =



Φ(ρ(s, u))∇A∇

H(u)du ds ,

for every function H ∈ D

and all t ∈ [0, T ].

Existence of such weak solutions follow from the tightness of the process proved in subsection 2.6.3,

and from the energy estimate obtained in Lemma 1.5.2. Uniquenesses of weak solutions was proved in

subsection 2.3.1.

Theorem 2.6.1. Fix a continuous initial proﬁle ρ

: T

→ [0, 1] and consider a sequence of probability

measures µ

on {0, 1}

associated to ρ

, in the sense of (2.6.3). Then, for any t ≥ 0,

lim

N→∞











x∈T

H(x/N )η

(x) −



H(u)ρ(t, u) du



> δ







= 0

for every δ > 0 and every function H ∈ D

. Here, ρ is the unique weak solution of the non-linear

equation (1.1.9) with l = 0, r = 1, γ = ρ

and Φ(α) = α + aα

Let M be the space of positive measures on T

with total mass bounded by one endowed with the

weak topology. Recall that π

∈ M stands for the empirical measure at time t. This is the measure on

obtained by rescaling space by N and by assigning mass 1/N

to each particle:



x∈T

(x) δ

x/N

, (2.6.5)

where δ

is the Dirac measure concentrated on u.

For a function H : T

→ R, π

, H stands for the integral of H with respect to π

π

, H =



x∈T

H(x/N )η

(x) .

This notation is not to be mistaken with the inner product in L

) introduced earlier. Also, when π

has a density ρ, π(t, du) = ρ(t, u)du.

Fix T > 0 and let D([0, T ], M) be the space of M-valued c`adl`ag trajectories π : [0, T ] → M endowed

with the uniform topology. For each probability measure µ

on {0, 1}

, denote by Q

W,N

the measure

on the path space D([0, T ], M) induced by the measure µ

and the process π

introduced in (2.6.5).

Fix a continuous proﬁle ρ

: T

→ [0, 1] and consider a sequence {µ

: N ≥ 1} of measures on

{0, 1}

associated to ρ

in the sense (2.6.3). Further, we denote by Q

the probability measure on

D([0, T ], M) concentrated on the deterministic path π(t, du) = ρ(t, u)du, where ρ is the unique weak

solution of (2.6.4) with γ = ρ

, l

= 0, r

= 1, k = 1, . . . , d and Φ(α) = α + bα

In subsection 2.6.3 we show that the sequence {Q

W,N

: N ≥ 1} is tight, and in subsection 2.6.4 we

characterize the limit points of this sequence.

2.6.3 Tightness

The goal of this subsection is to prove tightness of sequence {Q

W,N

: N ≥ 1}. We will do it by showing

that the set of equicontinuous paths of the empirical measures (2.6.5) has probability close to one.

Fix λ > 0 and consider, initially, the auxiliary M-valued Markov process {Π

λ,N

: t ≥ 0} deﬁned by

λ,N

(H) = π

, H

 =



x∈Z

(x/N)η

(x),

for H in D

, where H

is the unique weak solution in H

1,W

) (see Section 2.4) of

λH

− ∇

∇

= λH − ∇A∇

with the right-hand side being understood as the restriction of the function to the lattice T

(see

subsection 2.4.1).

We ﬁrst prove tightness of the process {Π

λ,N

: 0 ≤ t ≤ T },then we show that {Π

λ,N

: 0 ≤ t ≤ T }

and {π

: 0 ≤ t ≤ T } are not far apart.

It is well known [23] that to prove tightness of {Π

λ,N

: 0 ≤ t ≤ T } it is enough to show tightness

of the real-valued processes {Π

λ,N

(H) : 0 ≤ t ≤ T } for a set of smooth functions H : T

→ R dense in

C(T

) for the uniform topology.

Fix a smooth function H : T

→ R. Keep in mind that Π

λ,N

(H) = π

, H

, and denote by M

N,λ

the martingale deﬁned by

N,λ

= Π

λ,N

(H) − Π

λ,N

(H) −



ds N

π

, H

 . (2.6.6)

Clearly, tightness of Π

λ,N

(H) follows from tightness of the martingale M

N,λ

and tightness of the additive

functional



ds N

π

, H

.

A long computation, albeit simple, shows that the quadratic variation M

N,λ



of the martingale

N,λ

is given by:

2d−1



j=1



x∈T

[∂

W,j

(x/N )]

[W ((x + e

)/N ) − W (x/N )]×



x,x+e

(η

) [η

(x + e

) − η

(x)]

ds .

In particular, by Lemma 2.4.3,

M

N,λ



≤

2d−1



j=1

H



≤

C(H)t

λN

for some ﬁnite constant C(H), which depends only on H. Thus, by Doob inequality, for every λ > 0,

δ > 0,

lim

N→∞



sup

0≤t≤T



N,λ



> δ



= 0 . (2.6.7)

In particular, the sequence of martingales {M

N,λ

: N ≥ 1} is tight for the uniform topology.

It remains to be examined the additive functional of the decomposition (2.6.6). The generator of the

exclusion process L

can be decomposed in terms of the generator of the random walk L

. A simple

computation, we obtain that N

π

, H

 is equal to



j=1





x∈T

)(x/N ) η(x)



x∈T



)((x + e

)/N ) + (L

)(x/N)



(τ

1,j

)(η)

−



x∈T

)(x/N)(τ

2,j

)(η)



where {τ

: x ∈ Z

} is the group of translations, so that (τ

η)(y) = η(x + y) for x, y in Z

, and the sum

is understood modulo N. Also, h

1,j

, h

2,j

are the cylinder functions

1,j

(η) = η(0)η(e

) , h

2,j

(η) = η(−e

)η(e

) .

For all 0 ≤ s < t ≤ T , we have





dr N

π

, H





≤

(1 + 3|b|)(t − s)



j=1



x∈T

(x/N)| ,

from Schwarz inequality and Lemma 1.3.1, the right hand side of the previous expression is bounded

above by

(1 + 3|b|)(t − s)d









x∈T



(x/N )



Since H

is the weak solution of the discrete equation, we have by Remark 2.4.4 that it is also a

strong solution. Then, we may replace L

by U

= λH

−H in the previous formula. In particular,

It follows from the estimate given in Lemma 2.4.3, that the right hand side of the previous expression is

bounded above by dC(H, b)(t−s) uniformly in N , where C(H, b) is a ﬁnite constant depending only on b

and H. This proves that the additive part of the decomposition (2.6.6) is tight for the uniform topology

and therefore that the sequence of processes {Π

λ,N

: N ≥ 1} is tight.

Lemma 2.6.2. The sequence of measures {Q

W,N

: N ≥ 1} is tight for the uniform topology.

Proof. Fix λ > 0. It is enough to show that for every function H ∈ D

and every  > 0, we have

lim

N→∞



sup

0≤t≤T

|Π

λ,N

(H) − π

, H| > 



= 0,

whence tightness of π

follows from tightness of Π

λ,N

. By Chebyshev’s inequality, the last expression

is bounded above by



sup

0≤t≤T

|Π

λ,N

(H) − π

, H|



≤ 2H

− H

since there exists at most one particle per site. By Theorem 2.5.2 and Proposition 2.5.1, H

−H

→ 0

as N → ∞, and the proof follows.

2.6.4 Uniqueness of limit points

We prove in this subsection that all limit points Q

∗

of the sequence Q

W,N

are concentrated on absolutely

continuous trajectories π(t, du) = ρ(t, u)du, whose density ρ(t, u) is a weak solution of the hydrodynamic

equation (1.1.9) with l = 0, r = 1 and Φ(α) = α + aα

We now state a result necessary to prove the uniqueness of limit points. Recall that, for a local

function g : {0, 1}

→ R, ˜g : [0, 1] → R be the expected value of g under the stationary states:

˜g(α) = E

[g(η)] .

For  ≥ 1 and d-dimensional integer x = (x

, . . . , x

), denote by η



(x) the empirical density of particles

in the box B



(x) = {(y

, . . . , y

) ∈ Z

; 0 ≤ y

− x

< }:



(x) =





y∈B



(x)

η(y) .

Let Q

∗

be a limit point of the sequence Q

W,N

and assume, without loss of generality, that Q

W,N

converges to Q

∗

Since there is at most one particle per site, it is clear that Q

∗

is concentrated on trajectories π

(du)

which are absolutely continuous with respect to the Lebesgue measure, π

(du) = ρ(t, u)du, and whose

density ρ is non-negative and bounded by 1.

Fix a function H ∈ D

and λ > 0. Recall the deﬁnition of the martingale M

N,λ

introduced in the

previous section. From (2.6.7) we have, for every δ > 0,

lim

N→∞



sup

0≤t≤T



N,λ



> δ



= 0 ,

and from (1.4.2), for ﬁxed 0 < t ≤ T and δ > 0, we have

lim

N→∞

W,N





π

, H

 − π

, H

 −



ds N

π

, H





> δ



= 0.

Note that the expression N

π

, H

 has been computed in the previous subsection in terms

of generator L

. On the other hand, L

= λH

− λH + ∇A∇

H. Since there is at most one

particle per site, we may apply Theorem 2.5.2 to replace π

, H

 and π

, H

 by π

, H and π

, H,

respectively, and replace L

by ∇A∇

H plus a term that vanishes as N → ∞.

Since E

i,j

] = α

, i = 1, 2 and j = 1, . . . , d, we have by Proposition 1.4.4 that, for every t > 0,

λ > 0, δ > 0, i = 1, 2,

lim

ε→0

lim

N→∞









x∈T

(x/N )×



i,j

(η

) −



εN

(x)







> δ



= 0.

Since η

εN

(x) = ε

−d

(



j=1

/N, x

/N + εe

]), we obtain, from the previous considerations, that

lim

ε→0

lim

N→∞

W,N





π

, H −

− π

, H −







−d

(



j=1

[·, · + εe

])



, ∇A∇





> δ





= 0 .

Using the fact that Q

W,N

converges in the uniform topology to Q

∗

, we have that

lim

ε→0

Q ∗





π

, G

H − π

, G

H −

−







−d

(



j=1

[·, · + εe

])



, ∇A∇





> δ





= 0 .

Recall that Q

∗

is concentrated on absolutely continuous paths π

(du) = ρ(t, u)du with positive density

bounded by 1. Therefore, ε

−d

(



j=1

[·, · + εe

]) converges in L

) to ρ(s, .) as ε ↓ 0. Thus,

∗





π

, H − π

, H −



ds Φ(ρ

) , ∇A∇

H



> δ



= 0.

Letting δ ↓ 0, we see that, Q

∗

a.s.,



ρ(t, u)H(u)du −



ρ(0, u)H(u)du =



Φ(ρ(s, u))∇A∇

H(u)du ds .

This identity can be extended to a countable set of times t. Taking this set to be dense we obtain,

by continuity of the trajectories π

, that it holds for all 0 ≤ t ≤ T .

From Lemma 1.5.2, we may conclude that all limit points have, almost surely, ﬁnite energy, and

therefore, by Lemma 2.3.1, Φ(ρ(·, ·)) ∈ L

([0, T ], H

1,W

)). Analogously, it is possible to show that

ρ(·, ·) has ﬁnite energy and hence it belongs to L

([0, T ], H

1,W

)).

Proposition 2.6.3. As N ↑ ∞, the sequence of probability measures Q

W,N

converges in the uniform

topology to Q

Proof. In the previous subsection, we showed that the sequence of probability measures Q

W,N

is tight for

the uniform topology. Moreover, we just proved that all limit points of this sequence are concentrated

on weak solutions of the parabolic equation (2.6.4). The proposition now follows from the uniqueness

proved in subsection 2.3.1.

Proof of Theorem 2.6.1. Since Q

W,N

converges in the uniform topology to Q

, a measure which is

concentrated on a deterministic path, for each 0 ≤ t ≤ T and each continuous function H : T

→ R,

π

, H converges in probability to



duρ(t, u)H(u), where ρ is the unique weak solution of (2.6.4)

with l

= 0, r

= 1, γ = ρ

and Φ(α) = α + bα

Chapter 3

Equilibrium ﬂuctuations for

exclusion processes with

conductances in random

environments

In this Chapter we study the equilibrium ﬂuctuations for exclusion processes with conductances in

random environments, which can be viewed as a central limit theorem for the empirical distribution of

particles when the system starts from an equilibrium measure.

Let W : R

→ R be a function such that W (x

, . . . , x

) =



k=1

), where d ≥ 1 and each

function W

: R → R is strictly increasing, right continuous with left limits (c`adl`ag), and periodic in the

sense that W

(u + 1) − W

(u) = W

(1) − W

(0), for all u ∈ R. The inverse of the increments of the

function W will play the role of conductances in our system.

The random environment that we considered is governed by the coeﬃcients of the discrete formulation

of the model on the lattice. Moreover, we will assume the underlying random ﬁeld is ergodic, stationary

and satisﬁes an ellipticity condition.

The purpose of this Chapter is to study the density ﬂuctuation ﬁeld of this system as N → ∞, and

also the inﬂuence of the randomness in this limit. For any realization of the random environment, the

scaling limit depends on the randomness only through some constants which depend on the distribution

of the random transition rates, but not on the particular realization of the random environment.

The evolution of one-dimensional exclusion processes with random conductances has attracted some

attention recently [12, 13, 14, 18, 21], with the hydrodynamic limit proved in [21] being also obtained in

[12], independently. In all of these papers, a hydrodynamic limit was proved. The hydrodynamic limit

may be interpreted as a law of large numbers for the empirical density of the system. Our goal is to go

beyond the hydrodynamic limit and provide a new result for such processes, which is the equilibrium

ﬂuctuations and can be seen as a central limit theorem for the empirical density of the process.

To prove the equilibrium ﬂuctuations, we would like to call attention to the main tools we needed: (i)

the theory of nuclear spaces and (ii) homogenization of diﬀerential operators. The ﬁrst one followed the

classical approach of Kallianpur and Perez-Abreu [22] and Gel’fand and Vilenkin [19]. Nuclear spaces are

very suitable to attain existence and uniqueness of solutions for a general class of stochastic diﬀerential

equations. Furthermore, tightness of processes on such spaces was established by Mitoma [29]. A wide

literature on these spaces can be found cited inside the fourth volume of the amazing collection by

Gel’fand [19]. The second tool is motivated by several applications in mechanics, physics, chemistry and

engineering. We will consider stochastic homogenization. In the stochastic context, several works on

homogenization of operators with random coeﬃcients have been published (see, for instance, [30, 31] and

references therein). In homogenization theory, only the stationarity of such random ﬁelds is used. The

notion of stationary random ﬁeld is formulated in such a manner that it covers many objects of non-

probabilistic nature, e.g., operators with periodic or quasi-periodic coeﬃcients. We follow the approach

given in Chapter 2, which was introduced by [31].

The focus of our approach is to study the asymptotic behavior of eﬀective coeﬃcients for a family

of random diﬀerence schemes, whose coeﬃcients can be obtained by the discretization of random high-

contrast lattice structures. Furthermore, the introduction of a corrected empirical measure was needed.

The corrected empirical measure was used in the literature, for instance, by [21, 18, 20] and also Chapters

1 and 2. It can be understood as a version of Tartar’s compensated compactness lemma in the context of

particle systems. In this situation, the averaging due to the dynamics and the inhomogeneities introduced

by the random media factorize after introducing the corrected empirical process, in such a way that we

can average them separately. It is noteworthy that we managed to prove an equivalence between the

asymptotic behavior with respect to both the corrected empirical measure and the uncorrected one. This

equivalence was helpful in the sense that whenever the calculation with the corrected empirical measure

turned cumbersome, we changed to a calculation with respect to the uncorrected one, and the other way

around. This whole approach made the proof more simpler than the usual one with respect solely to the

corrected empirical measure developed in the articles mentioned above.

We now describe the organization of the Chapter. In Section 3.1 we state the main results of the

article; in Section 3.2 we deﬁne the nuclear space needed in our context; in Section 3.3 we recall some

results obtained in [34] about homogenization, and then we prove the equilibrium ﬂuctuations by showing

that the density ﬂuctuation ﬁeld converges to a process that solves the martingale problem. We also show

that the solution of the martingale problem corresponds to a generalized Ornstein-Uhlenbeck process.

In Section 3.4 we prove tightness of the density ﬂuctuation ﬁeld, as well as tightness of other related

quantities. In Section 3.5 we prove the Boltzmann-Gibbs principle, which is a key result for proving the

equilibrium ﬂuctuations. Finally, the Appendix contains some known results about nuclear spaces and

stochastic diﬀerential equations evolving on topologic dual of such spaces.

3.1 Notation and results

Fix a function W : R

→ R as (1.1.1):

W (x

, . . . , x

) =



k=1

where each W

: R → R is a strictly increasing right continuous function with left limits (c`adl`ag),

periodic in the sense that for all u ∈ R

(u + 1) − W

(u) = W

(1) − W

(0).

Recall in subsection 1.2 the deﬁnitions and properties of the generalized gradient of a function f :

∇

f = (∂

f, . . . , ∂

f) .

We now recall the random environment introduced in Chapter 2. The statistically homogeneous

rapidly oscillating coeﬃcients that will be used to deﬁne the random rates of the exclusion process

with conductances of which we want to study the equilibrium ﬂuctuations. Let (Ω, F, µ) be a standard

probability space and {T

: Ω → Ω; x ∈ Z

} be an ergodic group of F-measurable transformations which

preserve the measure µ:

• T

: Ω → Ω is F-measurable for all x ∈ Z

• µ(T

A) = µ(A), for any A ∈ F and x ∈ Z

• T

= I , T

◦ T

= T

x+y

• Any f ∈ L

(Ω) such that f(T

ω) = f(ω) µ-a.s. for each x ∈ Z

, is equal to a constant µ-a.s..

The last condition implies that the group T

is ergodic.

Let the vector-valued F-measurable functions {a

(ω); j = 1, . . . , d} be such that satisﬁes an ellipticity

condition: there exists θ > 0 such that

−1

≤ a

(ω) ≤ θ,

for all ω ∈ Ω and j = 1, . . . , d. Then, the diagonal matrices A

whose elements are given by

(x) := a

= a

ω) , x ∈ T

, j = 1, . . . , d. (3.1.1)

Fix a typical realization ω ∈ Ω of the random environment. For each x ∈ T

and j = 1, . . . , d,

remember the symmetric rate ξ

x,x+e

= ξ

x+e

x,x+e

(x)

N[W ((x + e

)/N ) − W (x/N )]

(x)

N[W

((x

+ 1)/N ) − W

/N )]

, (3.1.2)

where e

, . . . , e

is the canonical basis of R

Distribute particles on T

in such a way that each site of T

is occupied at most by one particle.

Denote by η the conﬁgurations of the state space {0, 1}

so that η(x) = 0 if site x is vacant, and

η(x) = 1 if site x is occupied.

The exclusion process with conductances in a random environment is the continuous-time Markov

process {η

: t ≥ 0} with state space {0, 1}

= {η : T

→ {0, 1}}, whose generator L

acts on functions

f : {0, 1}

→ R as

f)(η) =



j=1



x∈T

x,x+e

(η) {f(σ

x,x+e

η) − f(η)} , (3.1.3)

We consider the Markov process {η

: t ≥ 0} on the conﬁgurations {0, 1}

associated to the generator

in the diﬀusive scale, i.e., L

is speeded up by N

. A describe of the stochastic evolution of the

process can be found in Section 2.6.

Consider the random walk {X

}

t≥0

of a particle in T

induced by the generator L

given as follows.

Let ξ

x,x+e

given by (3.1.2). If the particle is on a site x ∈ T

, it will jump to x+e

with rate N

x,x+e

Furthermore, only nearest neighbor jumps are allowed. The generator L

of the random walk {X

}

t≥0

acts on functions f : N

−1

→ R as







j=1





where,





= N



x,x+e





x + e



− f





+ ξ

x−e





x − e



− f





It is not diﬃcult to see that the following equality holds:

f(x/N ) =



j=1

∂

f)(x) := ∇

∇

f(x), (3.1.4)

where, ∂

is the standard diﬀerence operator:

∂





= N





x + e



− f







and ∂

is the W

-diﬀerence operator:

∂







x+e



− f







x+e



− W





for x ∈ T

. Several properties of the above operator have been obtained in Chapter 2.

Now we state a central limit theorem for the empirical measure, starting from an equilibrium measure

. Fix ρ > 0 and denote by S

) the generalized Schwartz space on T

, whose deﬁnition as well as

some properties are given in Section 3.2.

Denote by Y

the density ﬂuctuation ﬁeld, which is the bounded linear functional acting on functions

G ∈ S

) as

(G) =

d/2



x∈T

G(x)[η

(x) − ρ]. (3.1.5)

Let D([0, T ], X) be the path space of c`adl`ag trajectories with values in a metric space X. In this

way we have deﬁned a process in D([0, T ], S



)), where S



) is the topologic dual of the space

Theorem 3.1.1. Consider the ﬂuctuation ﬁeld Y

deﬁned above. Then, Y

converges weakly to the

unique S



)-solution, Y

∈ D([0, T ], S



)), of the stochastic diﬀerential equation

= φ



(ρ)∇A∇

dt +



2χ(ρ)φ



(ρ)AdN

, (3.1.6)

where χ(ρ) = ρ(1 − ρ), φ(ρ) = ρ + bρ

, and φ



is the derivative of φ, φ



(ρ) = 1 + 2bρ; further A is a

constant diagonal matrix with jth diagonal element given by a

:= E(a

), for any N ∈ N; and N

is a



)-valued mean-zero martingale, with quadratic variation

N(G)

= t



j=1





∂

G(x)



d(x

⊗ W

where d(x

⊗W

) is the product measure dx

⊗···⊗dx

j−1

⊗dW

⊗dx

j+1

⊗···⊗dx

. Furthermore, N

is a Gaussian process with independent increments. More precisely, for each G ∈ S

), N

(G) is a

time deformation of a standard Brownian motion.

The proof of this theorem is given in Section 3.3.

Remark 3.1.2. The process Y

is known in the literature as the generalized Ornstein-Uhlenbeck process

with characteristics φ



(ρ)∇A∇

and



2χ(ρ)φ



(ρ)A∇

3.2 The space S

)

Recall the properties of the operator L

introduced in Section 1.2. In this Section we build the countably

Hilbert nuclear space S

), which is associated the the self-adjoint operator L

= ∇∇

. This space,

as we shall see, is a natural environment to attain existence and uniqueness of solutions of the stochastic

diﬀerential equation (3.1.6). Several lemmas are obtained to fulﬁll the conditions to ensure existence and

uniqueness of such solutions. The reader is referred to Appendix.

Let {ϕ

}

j≥1

be the complete orthonormal set of the eigenvectors of the operator L = I − L

, and

{λ

}

j≥1

the associated eigenvalues. Note that λ

= 1 + α

Consider the following increasing sequence  · 

, n ∈ N, of Hilbertian norms:

f, g

∞



k=1

P

f, P

gλ

where we denote by P

the orthogonal projection on the linear space generated by the eigenvector ϕ

So,

f

∞



k=1

P

f

where  ·  is the L

) norm.

Consider the Hilbert spaces S

which are obtained by completing the space D

with respect to the

inner product ·, ·

The set

) =

∞



n=0

endowed with the metric (3.6.2) is our countably Hilbert space, and even more, it is a countably Hilbert

nuclear space, see the Appendix for further details. In fact, ﬁxed n ∈ N and m > n + 1/2, we have that

{

(jλ

)

}

j≥1

is a complete orthonormal set in S

. Therefore,

∞



j=1



(jλ

)



≤

∞



j=1

2(m−n)

< ∞,

where the above formula corresponds to formula (3.6.3) in Appendix.

Lemma 3.2.1. Let L

: D

→ L

) be the operator obtained in Theorem 1.1.2. We have

(a) L

is the generator of a strongly continuous contraction semigroup {P

: L

) → L

)}

t≥0

;

(b) L

is a closed operator;

), t → P

f is a continuous function from [0, ∞) to L

);

(d) L

f = P

f for each f ∈ D

and t ≥ 0;

(e) (I − L

)

f = P

(I − L

)

f for each f ∈ D

, t ≥ 0 and n ∈ N;

Proof. Item (a) follows from Theorem 1.1.2 and Hille-Yosida’s theorem. Items (b), (c) and (d) follows

from item (a), see, for instance, [10, chapter 1]. Item (e) follows from item (d) and from the fact that

f = L

f if f ∈ D

The next Lemma permits to conclude that the semigroup {P

: t ≥ 0} acting on the domain S

)

is a C

0,1

-semigroup, whose deﬁnition is recalled in Appendix 3.6.2.

Lemma 3.2.2. Let {P

: t ≥ 0} be the semigroup whose inﬁnitesimal generator is L

. Then for each

q ∈ N we have:

P

f

≤ f

for all f ∈ S

). In particular, {P

: t ≥ 0} is a C

0,1

-semigroup.

Proof. Let f ∈ D

, then

f =



j=1

for some k ∈ N, and some constants β

, . . . , β

. A simple calculation shows that

f =



j=1

t(1−λ

)

Therefore, for f ∈ D

P

f

= 



j=1

t(1−λ

)





j=1

β

t(1−λ

)



≤



j=1

β



= f

Since D

is dense in S

), we conclude the proof of the lemma.

Lemma 3.2.3. The operator L

belongs to L(S

), S

)), the space of linear continuous operators

from S

) into S

Proof. Let f ∈ S

), and {ϕ

}

j≥1

be the complete orthonormal set of eigenvectors of L

, with

{(1 − λ

)}

j≥1

being their respectively eigenvalues. We have that

f =

∞



j=1

, with

∞



j=1

< ∞.

We also have that

f =

∞



j=1

(1 − λ

)β

For every n ∈ N:

L

f

∞



k=1

P

f)

∞



k=1

β

(1 − λ

)ϕ



∞



k=1

β



(1 − λ

)

≤ 2

∞



k=1

P

f

+ 2

∞



k=1

P

f

2(n+1)

= 2(f 

+ f

n+1

Therefore, by the deﬁnition of S

), L

f belongs to S

). Furthermore, L

is continuous

from S

) to S

3.3 Equilibrium Fluctuations

We begin by stating some results on homogenization of diﬀerential operators obtained in Chapter 2,

which will be very useful along this section.

Let L

⊗W

) be the space of square integrable functions with respect to the product measure

d(x

⊗W

) = dx

⊗···⊗dx

i−1

⊗dW

⊗dx

i+1

⊗···⊗dx

, and H

1,W

) be the Sobolev space of functions

with W -generalized derivatives. More precisely, H

1,W

) is the space of functions g ∈ L

) such that

for each i = 1, . . . , d there exist functions G

∈ L

⊗W

) satisfying the following integral by parts

identity.





∂



g dx = −



(∂

f) G

d(x

⊗ W

), (3.3.1)

for every function f ∈ S

), where L

⊗W

) is the closed subspace of L

⊗W

) consisting of

the functions that have zero mean with respect to the measure d(x

⊗ W



fd(x

⊗ W

) = 0.

. We denote G

simply by ∂

g. See [34] for further details and properties of this space.

Let λ > 0, f be a functional on H

1,W

), u

be the unique weak solution of

λu

− ∇

∇

= f,

and u

be the unique weak solution of

λu

− ∇A∇

= f. (3.3.2)

For more details on existence and uniqueness of such solutions see [34].

In this context, we say that the diagonal matrix A = {a

} = {a

} is a homogenization of the sequence

of random matrices A

, denoted by A

−→ A, if the following conditions hold:

• u

converges weakly in H

1,W

) to u

, when N → ∞;

• a

∂

→ a

∂

u, weakly in L

⊗W

) when N → ∞.

Theorem 3.3.1. Let A

be a sequence of ergodic random matrices, such as the one that deﬁnes our

random environment. Then, almost surely, A

(ω) admits a homogenization, where the homogenized

matrix A does not depend on the realization ω.

The following proposition regards the convergence of energies:

Proposition 3.3.2. Let A

−→ A, as N → ∞, with u

being the solution of

λu

− ∇

∇

= f,

where f is a ﬁxed functional on H

1,W

). Then, the following limit relations hold true:



x∈T

(x) →



(x)dx,

and

d−1



j=1



x∈T

(x)(∂

(x))

((x

+ 1)/N ) − W

/N )]

→



j=1



(x)(∂

(x))

d(x

⊗ W

as N → ∞.

The proofs of these results can be found in Chapter 2.

3.3.1 Martingale Problem

We say that Y

∈ S



) solves the martingale problem with initial condition Y

if for any G ∈ S

)

(G) = Y

(G) − Y

(G) − φ



(ρ)



(∇A∇

G)ds (3.3.3)

is a martingale with quadratic variation

M

(G) = 2tχ(ρ)φ



(ρ)



j=1





∂



d(x

⊗ W

). (3.3.4)

Observe that if Y

is the generalized Ornstein-Uhlenbeck process with characteristics φ



(ρ)∇A∇

and



2χ(ρ)φ



(ρ)A∇

, then Y

solves the martingale problem above.

Recall the deﬁnition of the density ﬂuctuation ﬁeld Y

given in (3.1.5), and denote by Q

the

distribution in D([0, T ], S

)) induced by Y

, with initial distribution ν

. Our goal is to show that

any limit point of Y

solves the martingale problem. To this end, let us introduce the corrected density

ﬂuctuation ﬁeld:

N,λ

(G) =

d/2



x∈T

(x)



(x) − ρ



where G

is the weak solution of the equation

λG

− L

= λG − ∇A∇

G, (3.3.5)

that, via homogenization, converges to G which is the trivial solution of the problem

λG − ∇A∇

G = λG −∇A∇

The processes Y

and Y

N,λ

have the same asymptotic behavior, as we will see. But some calculations

are simpler with one of them than with the other. In this way, we have deﬁned two processes in

D([0, T ], S



)).

Given a process Y

in D([0, T ], S



)), and for t ≥ 0, let F

be the σ-algebra generated by Y

(H)

for s ≤ t and H ∈ S

). Furthermore, set F

∞

= σ





t≥0



. Denote by Q

the distribution on

D([0, T ], S



)) induced by the corrected density ﬂuctuation ﬁeld Y

N,λ

and initial distribution ν

Theorem 3.1.1 is a consequence of the following result about the corrected ﬂuctuation ﬁeld.

Theorem 3.3.3. Let Q be the probability measure on D([0, T ], S



)) corresponding to the generalized

Ornstein-Uhlenbeck process of mean zero and characteristics φ



(ρ)∇·A∇

and



2χ(ρ)φ



(ρ)A∇

. Then

the sequence {Q

}

N≥1

converges weakly to the probability measure Q.

Note also that the above theorem implies that any limit point of Y

solves the martingale problem

(3.3.3)-(3.3.4).

Before proving the Theorem 3.3.3, we will state and prove a lemma. This lemma shows that tightness

of Y

N,λ

follows from tightness of Y

, and even more, that they have the same limit points. So we can

derive our main theorem from Theorem 3.3.3.

Lemma 3.3.4. For all t ∈ [0, T ] and G ∈ S

), lim

N→∞



(G) − Y

N,λ

(G)



= 0.

Proof. By convergence of energies, we have that lim

N→∞

= G in L

), i.e.

G

− G



x∈T

(x/N ) − G(x/N )]

→ 0, as N → ∞. (3.3.6)

Since ν

is a product measure we obtain



(G) − Y

N,λ

(G)



= E





x,y∈T

(x/N ) − G(x/N )][G

(y/N) − G(y/N)](η

(x) − ρ)(η

(y) − ρ)



= E





x∈T

(x/N ) − G(x/N )]

(η

(x) − ρ)



≤

C(ρ)



x∈T

(x/N ) − G(x/N )]

where C(ρ) is a constant that depend on ρ. By (3.3.6) the last expression vanishes as N → ∞.

Proof of Theorem 3.3.3

Consider the martingale

(G) = Y

(G) − Y

(G) −



(G)ds (3.3.7)

associated to the original process and the martingale

N,λ

(G) = Y

N,λ

(G) − Y

N,λ

(G) −



N,λ

(G)ds (3.3.8)

associated to the corrected process.

A long, albeit simple, computation shows that the quadratic variation of the martingale M

N,λ

(G),

M

N,λ

(G)

, is given by:

d−1



j=1



x∈T

[∂

(x/N )]

[W ((x + e

)/N ) − W (x/N )]× (3.3.9)



x,x+e

(η

) [η

(x + e

) − η

(x)]

ds .

Is not diﬃcult see that the quadratic variation of the martingale M

(G), M

(G)

, has the expres-

sion (3.3.9) with G replacing G

. Further,



x,x+e

(η) [η

(x + e

) − η

(x)]



[1 + b(η(x − e

) + η(x))]E

[(η(x + e

) − η(x))

] =

2(1 + 2bρ)ρ(1 − ρ) = 2φ



(ρ)χ(ρ).

Lemma 3.3.5. Fix G ∈ S

) and t > 0, and let M

N,λ

(G)

and M

(G)

be the quadratic

variations of the martingales M

N,λ

(G) and M

(G), respectively. Then,

lim

N→∞



M

N,λ

(G)

− M

(G)



= 0. (3.3.10)

Proof. Fix G ∈ S

) and t > 0. A straightforward calculation shows that



M

N,λ

(G)

− M

(G)



≤

d−1



j=1



x∈T

[



∂

(x/N)



−



∂

G(x/N )



][W (

x + e

) − W (

)]}

where the constant k comes from the integral term. By the convergence of energies (Proposition 2.5.1),

the last term vanishes as N → ∞.

Lemma 3.3.6. Let G ∈ S

) and d ≥ 1. Then

lim

N→∞



d−1





j=1



x∈T



∂

G(x/N)



[W ((x + e

)/N ) − W (x/N )]×



x,x+e

(η

) [η

(x + e

) − η

(x)]

− 2χ(ρ)φ



(ρ)





= 0.

Proof. Fix G ∈ S

) and d > 1. The term in the previous expression is less than or equal to

C(ρ)

d−1

∇

G

W,N,4

, (3.3.11)

where

∇

G

W,N,4

d−1



j=1



x∈T



∂

G(x/N )



[W ((x + e

)/N ) − W (x/N )].

Thus, since for G ∈ S

), ∇

G

W,N,4

is bounded, the term in (3.3.11) converges to zero as N → ∞.

The case d = 1 follows from calculations similar to the ones found in Lemma 12 of [28].

So, by Lemma 3.3.5 and 3.3.6, M

N,λ

(G)

is given by

2tχ(ρ)φ



(ρ)

d−1



j=1



x∈T



∂

(x/N )



[W ((x + e

)/N ) − W (x/N )]

plus a term that vanishes in L

) as N → ∞. By the convergence of energies, Proposition 2.5.1, it

converges, as N → ∞, to

2tχ(ρ)φ



(ρ)



j=1





∂

G(x)



⊗ W

Our goal now consists in showing that it is possible to write the integral part of the martingale as

the integral of a function of the density ﬂuctuation ﬁeld plus a term that goes to zero in L

). After

some simple computations, we obtain that

N,λ

(G) =



j=1



d/2



x∈T

(x/N ) η

(x)

d/2



x∈T



((x + e

)/N ) + L

(x/N )



(τ

1,j

)(η

)

−

d/2



x∈T

(x/N)(τ

2,j

)(η

)



where {τ

: x ∈ Z

} is the group of translations, so that (τ

η)(y) = η(x + y) for x, y in Z

, and the sum

is understood modulo N. Also, h

1,j

, h

2,j

are the cylinder functions

1,j

(η) = η(0)η(e

) , h

2,j

(η) = η(−e

)η(e

) .

Note that inside the expression N

N,λ

we may replace L

by a

∂

G. Indeed, the

expression

ν(ρ)







j=1

d/2



x∈T



(x/N ) − a

∂

G(x/N )





(x) − ρ



d/2



x∈T



((x + e

)/N ) − a

∂

G((x + e

)/N ) +

(x/N ) − a

∂

G(x/N )





(τ

1,j

)(η

) − ρ



−

d/2



x∈T



(x/N ) − a

∂

G(x/N )





(τ

2,j

)(η

) − ρ





is less than or equal to

C(ρ, b)





x∈T



(x/N ) − ∇A∇

G(x/N)



Now, recall that G

is solution of the equation (3.3.5), and therefore, the previous expression is less

than or equal to

t C(ρ, b)

G

− G

thus, by homogenization and energy estimates in Theorem 3.3.1 and Proposition 3.3.2, respectively, the

last expression converges to zero as N → ∞.

By the Boltzmann Gibbs principle, Theorem 3.5.1 below, we can replace (τ

i,j

)(η

)−ρ

by 2ρ[η

(x)−

ρ] for i = 1, 2. Doing so, the martingale (3.3.8) can be written as

N,λ

(G) = Y

N,λ

(G) − Y

N,λ

(G) −



d/2



x∈T

∇A∇

G(x/N )φ



(ρ)



− ρ



ds, (3.3.12)

plus a term that vanishes in L

) as N → ∞.

Notice that, by (3.1.5), the integrand in the previous expression is a function of the density ﬂuctuation

ﬁeld Y

. By Lemma 3.3.4, we can replace the term inside the integral of the above expression by a term

which is a function of the corrected density ﬂuctuation ﬁeld Y

N,λ

From the results of Section 3.4, the sequence {Q

}

N≥1

is tight and let Q

be a limit point of it. Let

be the process in D([0, T ], S



)) induced by the canonical projections under Q

. Taking the limit

as N → ∞, under an appropriate subsequence, in expression (3.3.12), we obtain that

(G) = Y

(G) − Y

(G) −



(φ



(ρ)∇ · A∇

G)ds, (3.3.13)

where M

is some S



)-valued process, in fact, a martingale. To see this, note that for a measurable

set U with respect to the canonical σ-algebra F

, E

N,λ

(G)1

] converges to E

(G)1

]. Since

N,λ

(G) is a martingale, E

N,λ

(G)1

] = E

N,λ

(G)1

]. Taking a further subsequence if

necessary, this last term converges to E

(G)1

], which proves that M

(G) is a martingale for any

G ∈ S

). Since all the projections of M

are martingales, we conclude that M

is a S



)-valued

martingale.

Now, we need obtain the quadratic variation M

(G)

of the martingale M

(G). A simple applica-

tion of Tchebyshev’s inequality shows that M

N,λ

(G)

converges in probability to

2tχ(ρ)φ



(ρ)



j=1





∂



d(x

⊗ W

where χ(ρ) stands for the static compressibility given by χ(ρ) = ρ(1−ρ). By Doob-Meyer’s decomposition

theorem, we need to prove that

(G) := M

(G)

− 2tχ(ρ)φ



(ρ)



j=1





∂



d(x

⊗ W

)

is a martingale. The same argument we used above applies now if we can show that sup

N,λ

(G)

] <

∞ and sup

[M

N,λ

(G)

] < ∞. Both bounds follows easily from the explicit form of M

N,λ

(G)

and (3.3.12).

On the other hand, by a standard central limit theorem, Y

is a Gaussian ﬁeld with covariance



(G)Y

(H)



= χ(ρ)



G(x)H(x)dx.

Therefore, by Theorem 3.3.7, Q

is equal to the probability distribution Q of a generalized Ornstein-

Uhlenbeck process in D([0, T ], S



)) (and it does not depend on λ). By existence and uniqueness of

the generalized Ornstein-Uhlenbeck processes (also due to Theorem 3.3.7), the sequence {Q

}

N≥1

has

at most one limit point, and from tightness, it does have a unique limit point. This concludes the proof

of Theorem 3.3.3.

3.3.2 Generalized Ornstein-Uhlenbeck Processes

In this subsection we show that the generalized Ornstein-Uhlenbeck process obtained as the solution

martingale problem which we are interested, is also a S



)-solution of a stochastic diﬀerential equation,

and then we apply the theory in Appendix to conclude that there is at most one solution of the martingale

problem. Moreover, we also conclude that this process is a Gaussian process.

Theorem 3.3.7. Let Y

be a Gaussian ﬁeld on S



). Then the unique S



)-solution, Y

, of the

stochastic diﬀerential equation

= φ



(ρ)∇A∇

dt +



2χ(ρ)φ



(ρ)AdN

, (3.3.14)

solves the martingale problem (3.3.3)-(3.3.4) with initial condition Y

, where N

is a mean-zero S



valued martingale with quadratic variation given by

N(G)

= t



j=1





∂



d(x

⊗ W

Moreover, Y

is a Gaussian process.

Proof. In view of the deﬁnition of solutions of stochastic diﬀerential equations (see Appendix), Y

is a



)-solution of (3.3.14). In fact, by hypothesis Y

satisﬁes the integral identity (3.3.3), and is also

an additive functional of a Markov process.

We now check the conditions in Proposition 3.6.1 to ensure uniqueness of S



)-solutions of (3.3.14).

Since by hypothesis Y

is a Gaussian ﬁeld, condition 1 is satisﬁed, and since the martingale M

has

quadratic variation given by (3.3.4), we use Remark 3.6.2 to conclude that condition 2 holds. Condition

3 follows from Lemmas 3.2.2 and 3.2.3. Therefore Y

is unique.

Finally, by Blumenthal’s 0-1 law for Markov processes, M

and Y

are independent, since for measur-

able sets A and B,

P (Y

∈ A, M

∈ B) = E(1

∈A

∈B

) =

E[E(1

∈A

∈B

)] = E[1

∈A

E(1

∈B

)] =

E[1

∈A

P (M

∈ B)] = P (Y

∈ A)P (M

∈ B).

Applying L´evy’s martingale characterization of Brownian motions, the quadratic variation of M

, given

by (3.3.4), yields that M

is a time deformation of a Brownian motion. Therefore, M

is a Gaussian

process with independent increments. Since Y

is a Gaussian ﬁeld, we apply Proposition 3.6.3 to conclude

that Y

is a Gaussian process in D([0, T ], S



)).

3.4 Tightness

In this section we prove tightness of the density ﬂuctuation ﬁeld {Y

}

introduced in Section 1.1. We

begin by stating Mitoma’s criterion [29]:

Proposition 3.4.1. Let Φ

∞

be a nuclear Fr´echet space and Φ



∞

its topological dual. Let {Q

}

be a

sequence of distributions in D([0, T ], Φ



∞

), and for a given function G ∈ Φ

∞

, let Q

N,G

be the distribu-

tion in D([0, T ], R) deﬁned by Q

N,G

[y ∈ D([0, T ], R); y(·) ∈ A] = Q

[Y ∈ D([0, T ], Φ



∞

); Y (·)(G) ∈ A].

Therefore, the sequence {Q

}

is tight if and only if {Q

N,G

}

is tight for any G ∈ Φ

∞

From Mitoma’s criterion, {Y

}

is tight if and only if {Y

(G)}

is tight for any G ∈ S

), since

) is a nuclear Fr´echet space. By Dynkin’s formula and after some manipulations, we see that

(G) = Y

(G)





j=1



d/2



x∈T

(x/N ) η

(x)

d/2



x∈T



((x + e

)/N ) + L

(x/N)



(τ

1,j

)(η

)

−

d/2



x∈T

(x/N)(τ

2,j

)(η

)



ds + M

(G), (3.4.1)

where M

(G) is a martingale of quadratic variation

M

(G)

d−1



j=1



x∈T

[∂

(x/N )]

[W ((x + e

)/N ) − W (x/N )]×



x,x+e

(η

) [η

(x + e

) − η

(x)]

ds .

In order to prove tightness for the sequence {Y

(G)}

, it is enough to prove tightness for {Y

(G)}

and the integral term in (3.4.1). The easiest one is the initial condition: from the usual central

limit theorem, Y

(G) converges to a normal random variable of mean zero and variance χ(ρ)



G(x)

dx,

where χ(ρ) = ρ(1 − ρ). For the other two terms, we use Aldous’ criterion:

Proposition 3.4.2 (Aldous’ criterion). A sequence of distributions {P

} in the path space D([0, T ], R)

is tight if:

i) For any t ∈ [0, T ] the sequence {P

} of distributions in R deﬁned by P

(A) = P

[y ∈ D([0, T ], R) : y(t) ∈ A]

is tight,

ii) For any  > 0,

lim

δ>0

lim

n→∞

sup

τ∈Υ

θ≤δ



y ∈ D([0, T ], R) : |y(τ + θ) − y(τ )| > 



= 0,

where Υ

is the set of stopping times bounded by T and y(τ + θ) = y(T ) if τ + θ > T .

Now we prove tightness of the martingale term. By the optional sampling theorem, we have





τ+θ

(G) − M

(G)



> 



≤





τ+θ

(G)



−



(G)







τ+θ

(G)



−



(G)





d−1



j=1



x∈T

(x)[∂

G(x/N)]

[W ((x + e

)/N ) − W (x)]



t+δ

x,x+e

(η

)[η

(x + e

) − η

(x)]

≤



(1 + 2|b|)θ

d−1



j=1



x∈T

[∂

G(x/N )]

[W ((x + e

)/N ) − W (x)] (3.4.2)

≤



(1 + 2|b|)θ(∇

G

+ δ),

for N suﬃciently large, since the rightmost term on (3.4.2) converges to ∇

G

, as N → ∞, where

∇

G



i=1





∂



d(x

⊗ W

Therefore, the martingale M

(G) satisﬁes the conditions of Aldous’ criterion. The integral term can

be handled in a similar way:

 



τ+δ

d/2



j=1





G(x/N )(η

− ρ)

+ b[L

G((x + e

)/N ) + L

G(x/N )](τ

− ρ

)

− bL

G(x/N )(τ

− ρ

)





≤ δC(b)



j=1



x∈T



G(x/N )



≤ δC(G, b),

where C(b) is a constant that depends on b, and C(G, b) is a constant that depends on C(b) and on the

function G ∈ S

). Therefore, we conclude, by Mitoma’s criterion, that the sequence {Y

}

is tight.

Thus, the sequence of S



)-valued martingales {M

}

is also tight.

3.5 Boltzmann-Gibbs Principle

We show in this section that the martingales M

(G) introduced in Section 3.3 can be expressed in terms

of the ﬂuctuation ﬁeld Y

. This replacement of the cylinder function (τ

i,j

)(η

) − ρ

by 2ρ[η

(x) − ρ]

for i = 1, 2, constitutes one of the main steps toward the proof of equilibrium ﬂuctuations.

Recall that (Ω, F, µ) is a standard probability space where we consider the vector-valued F-measurable

functions {a

(ω); j = . . . , d} that form our random environment (see Sections 1.1 and 3.3 for more de-

tails).

Take a function f : Ω × {0, 1}

→ R. Fix a realization ω ∈ Ω, let x ∈ T

, and deﬁne

f(x, η) = f(x, η, ω) =: f(T

ω, τ

η),

where τ

η is the shift of η to x: τ

η(y) = η(x + y).

We say that f is local if there exists R > 0 such that f(ω, η) depends only on the values of η(y) for

|y| ≤ R. On this case, we can consider f as deﬁned in all the space Ω ×{0, 1}

for N ≥ R.

We say that f is Lipschitz if there exists c = c(ω) > 0 such that for all x, |f(ω, η) − f(ω, η



)| ≤

c|η(x) − η



(x)| for any η, η



∈ {0, 1}

such that η(y) = η



(y) for any y = x. If the constant c can be

chosen independently of ω, we say that f is uniformly Lipschitz.

Theorem 3.5.1. (Boltzmann-Gibbs principle)

For every G ∈ S

), every t > 0 and every local, uniformly Lipschitz function f : Ω×{0, 1}

→ R,

it holds

lim

N→∞





d/2



x∈T

G(x)V

(x, η

)ds



= 0, (3.5.1)

where

(x, η) = f(x, η) − E



f(x, η)



− ∂





f(x, η)dν

(η)





η(x) − ρ



Here, E denotes the expectation with respect to µ, the random environment.

Let f : Ω × {0, 1}

→ R be a local, uniformly Lipschitz function and take f(x, η) = f(θ

ω, τ

η).

Fix a function G ∈ S

) and an integer K that shall increase to ∞ after N. For each N, we subdivide

into non-overlapping boxes of linear size K. Denote them by {B

, 1 ≤ i ≤ M

}, where M = [

More precisely,

= y

+ {1, . . . , K}

where y

∈ T

, and B

∩ B

= ∅ if i = r. We assume that the points y

have the same relative position

on the boxes.

Let B

be the set of points that are not included in any B

, then |B

| ≤ dKN

d−1

. If we restrict the

sum in the expression that appears inside the integral in (3.5.1) to the set B

, then its L

)-norm

clearly vanishes as N → +∞, since f is local, ν

is an invariant product measure, and V

has mean zero

with respect to ν

Let Λ

be the smallest cube centered at the origin that contains the support of f and deﬁne s

the radius of Λ

. Denote by B

the interior of the box B

, namely the sites x in B

that are at a distance

at least s

+ 2 from the boundary:

= {x ∈ B

, d(x, T

\ B

) > s

+ 2}.

Denote also by B

the set of points that are not included in any B

. By construction, it is easy to

see that |B

| ≤ dN

(

c(f)

), where c(f) is a constant that depends on f.

We have that for continuous H : T

→ R,

d/2



x∈T

H(x)V

(x, η

) =

d/2



x∈B

H(x)V

(x, η

d/2



i=1



x∈B



H(x) − H(y

)



(x, η

) +

d/2



i=1

H(y

)



x∈B

(x, η

Note that we may take H continuous, since the continuous functions are dense in L

). The ﬁrst step

is to prove that

lim

K→∞

lim

N→∞





d/2



x∈B

H(x)V

(x, η

)ds



= 0.

As ν

is an invariant product measure and V

has mean zero with respect to the measure ν

, the last

expectation is bounded above by



x,y∈B

|x−y|≤2s

H(x)H(y)E



(x, η)V

(y, η)



Since V

belongs to L

) and |B

| ≤ dN

(

c(f)

), the last expression vanishes by taking ﬁrst

N → +∞ and then K → +∞.

From the continuity of H, and applying similar arguments, one may show that

lim

N→∞





d/2



i=1



x∈B



H(x) − H(y

)



(x, η

)ds



= 0.

In order to conclude the proof it remains to be shown that

lim

K→∞

lim

N→∞





d/2



i=1

H(y

)



x∈B

(x, η

)ds



= 0. (3.5.2)

To this end, recall proposition A 1.6.1 of [23]:





V (η

)ds



≤ 20θtV 

−1

, (3.5.3)

where  · 

−1

is given by

V 

−1

= sup

F ∈L

(ν

)





V (η)F (η)dν

− F, L

F 



and ·, ·

denotes the inner product in L

(ν

Let

be the generator of the exclusion process without the random environment, and without the

conductances (that is, taking a(ω) ≡ 1, and W

) = x

, for j = 1, . . . , d, in (1.1.2)), and also without

the diﬀusive scaling N

g(η) =



j=1



x∈T

x,x+e

(η)



g(η

x,x+e

) − g(η)



for cylindric functions g on the conﬁguration space {0, 1}

For each i = 1, .., M

denote by ζ

the conﬁguration {η(x), x ∈ B

} and by

the restriction of the

generator

to the box B

, namely:

h(η) =



x,y∈B

|x−y|=1/N

x,y

(η)



h(η

x,y

) − h(η)



We would like to emphasize that we introduced the generator

because it is translation invariant.

Now we introduce some notation. Let L

(P ⊗ ν

) the set of measurable functions g such that



g(ω, η)

dν

] < ∞. Fix a local function h : Ω ×{0, 1}

→ R in L

(P ⊗ν

), measurable with respect

to σ(η(x), x ∈ B

), and let h

be the translation of h by y

− y

: h

(x, η) = h(θ

−y

ω, τ

−y

η).

Consider

H,h

(η) =

d/2



i=1

H(y

)

(ζ

The strategy of the proof (3.5.2) is the following: we show that V

H,h

vanishes in some sense as

N → ∞, and then, that the diﬀerence between V

and V

H,h

also vanishes, as N → ∞. The result follows

a simple triangle inequality. The ﬁrst part is done by obtaining estimates on boxes, whereas the second

part mainly considers the projections of V

on some appropriate Hilbert spaces, plus ergodicity of the

environment.

Let

W,B

h(η) =



j=1



x∈B

x,x+e

(η)

(x)

W (x + e

) − W (x)

[h(η

x,x+e

) − h(η)].

Note that the following estimate holds



i=1

h, −L

W,B

h

≤ h, −L

h

Furthermore,

f, −

h ≤ max

1≤k≤d

(1) − W

(0)}

θh, −L

W,B

h

Using the Cauchy-Schwartz inequality, we have, for each i,



, F 

≤

2γ

−

, h



F, −

F 

where γ

is a positive constant.

Therefore,



H,h

(η)F (η)dν

≤

d/2



i=1

H(y

)



2γ

−

, h



F, −

F 



. (3.5.4)

Choose

1+d/2

θ max

1≤k≤d

(1) − W

(0)}|H(y

and observe that the generator L

is already speeded up by the factor N

. We, thus, obtain

d/2



i=1

H(y

)

F, −

F 

≤ F, −

F 

The above bound and (3.5.4) allow us to use inequality (3.5.2) on V

H,h

, with the generator L

W,B

Therefore, we have that the expectation in (3.5.3) with V

H,h

is bounded above by

20θt

d/2



i=1

|H(y

−

, h



which in turn is less than or equal to

20tH

∞

d+1

max

1≤k≤d

(1) − W

(0)}



i=1

−

, h



By Birkhoﬀ’s ergodic theorem, the sum in the previous expression converges to a ﬁnite value as N → ∞.

Therefore, this whole expression vanishes as N → ∞. This concludes the ﬁrst part of the strategy of the

proof.

To conclude the proof of the theorem it is enough to show that

lim

K→∞

inf

h∈L

(ν

⊗P )

lim

N→∞





d/2



i=1

H(y

)





x∈B

(x, η

) −

(ζ

(s))



= 0.

To this end, observe that the expectation in the previous expression is bounded by



i=1

||H||

∞





x∈B

(x, η) −

(ζ

)



because the measure ν

is invariant under the dynamics and the supports of V

(x, η) −

(ζ

) and

(y, η) −

(ζ

) are disjoint for x ∈ B

and y ∈ B

, with i = r.

By the ergodic theorem, as N → ∞, this expression converges to

||H||

∞









x∈B

(x, η) −

h(ω, η)



dν



. (3.5.5)

So, it remains to be shown that

lim

K→∞

||H||

∞

inf

h∈L

(ν

⊗P )









x∈B

(x, η) −

h(ω, η)



dν



= 0.

Denote by R(

) the range of the generator

in L

(ν

⊗P ) and by R(

)

⊥

the space orthogonal

to R(

). The inﬁmum of (3.5.5) over all h ∈ L

(ν

⊗ P ) is equal to the projection of



x∈B

(x, η)

into R(

)

⊥

The set R(

)

⊥

is the space of functions that depend on η only through the total number of particles

on the box B

. So, the previous expression is equal to

lim

K→∞

||H||

∞











x∈B

(x, η)





dν



, (3.5.6)

where η

= K

−d



x∈B

η(x).

Let us call this last expression I

. Deﬁne ψ(x, ρ) = E

[f(θ

ω)]. Notice that V

(x, η) = f(x, η) −

ψ(x, ρ) −E[∂

ψ(x, ρ)]



η(x) −ρ



, since in the last term the partial derivative with respect to ρ commutes

with the expectation with respect to the random environment. In order to estimate the expression (3.5.6),

we use the elementary inequality (x + y)

≤ 2x

+ 2y

. Therefore, we obtain I

≤ 4(I

+ I

), where









x∈B



f(x, η)|η



− ψ(x, η

)



dν











x∈B

ψ(x, η

) − ψ(x, ρ) − ∂

ψ(x, ρ)[η

− ρ]



dν



and







x∈B



∂

ψ(x, ρ) − E[∂

ψ(x, ρ)]



− ρ







Recall the equivalence of ensembles (see Lemma A.2.2.2 in [23]):

Lemma 3.5.2. Let h : {0, 1}

→ R be a local uniformly Lipschitz function and S ∈ {1, . . . , N }. Then,

there exists a constant C that depends on h only through its support and its Lipschitz constant, such that



[h(η)|η

] − E

[h(η)]



≤

and

(x) =



y∈Λ

η(y),

with Λ

= {0, . . . , S − 1}

Applying Lemma 3.5.2, we get









x∈B



f(x, η)|η



− ψ(x, η

)



dν



≤

which vanishes as K → ∞.

Using a Taylor expansion for ψ(x, ρ), we obtain that









x∈B

ψ(x, η

) − ψ(x, ρ) − ∂

ψ(x, ρ)[η

− ρ]



dν



≤

and also goes to 0 as K → ∞.

Finally, we see that

= E



(η(0) − ρ)



· E





x∈B

(∂

ψ(x, ρ) − E[∂

ψ(x, ρ)]





and it goes to 0 as K → ∞ by the L

-ergodic theorem. This concludes the proof of Theorem 3.5.1.

3.6 Appendix: Stochastic diﬀerential equations on nuclear spaces

3.6.1 Countably Hilbert nuclear spaces

In this subsection we introduce countably Hilbert nuclear spaces which will be the natural environment

for the study of the stochastic evolution equations obtained from the martingale problem. We will begin

by recalling some basic deﬁnitions on these spaces. To this end, we follow the ideas of Kallianpur and

Perez-Abreu [22] and Gel’fand and Vilenkin [19].

Let Φ be a (real) linear space, and let  · 

, r ∈ N be an increasing sequence of Hilbertian norms.

Deﬁne Φ

as the completion of Φ with respect to  · 

. Since for n ≤ m

f

≤ f

, for all f ∈ Φ, (3.6.1)

we have,

⊂ Φ

, for all m ≥ n.

Let

∞



r=1

Then Φ

∞

is a Fr´echet space with respect to the metric

ρ(f, g) =

∞



r=1

−r

f − g

1 + f − g

, (3.6.2)

and (Φ

∞

, ρ) is called a countably Hilbert space.

A countably Hilbert space Φ

∞

is called nuclear if for each n ≥ 0, there exists m > n such that the

canonical injection π

m,n

: Φ

→ Φ

is Hilbert-Schmidt, i.e., if {f

}

j≥1

is a complete orthonormal system

in Φ

, we have

∞



j=1

f



< ∞. (3.6.3)

We now characterize the topologic dual Φ



∞

of the countably Hilbert nuclear space Φ

∞

in terms of

the topologic dual of the auxiliary spaces Φ

Let Φ



be the dual (Hilbert) space of Φ

, and for φ ∈ Φ



let

φ

−n

= sup

f

≤1

|φ[f]|,

where φ[f] means the value of φ at f. Equation (3.6.1) implies that



⊂ Φ



for all m ≥ n.

Let Φ



∞

be the topologic dual of Φ

∞

with respect to the strong topology, which is given by the

complete system of neighborhoods of zero given by sets of the form, {φ ∈ Φ



∞

: φ

< }, where

φ

= sup{|φ[f]| : f ∈ B} and B is a bounded set in Φ

∞

. So,



∞



r=1



3.6.2 Stochastic diﬀerential equations

The aim of this subsection is to recall some results about existence and uniqueness of stochastic evolution

equations in nuclear spaces.

We denote by L(Φ

∞

, Φ

∞

) (resp. L(Φ



∞

, Φ



∞

)) the class of continuous linear operators from Φ

∞

(resp.Φ



∞

to Φ



∞

A family {S(t) : t ≥ 0} of the linear operators on Φ

∞

is said to be a C

0,1

-semigroup if the following

three conditions are satisﬁed:

• S(t

)S(t

) = S(t

+ t

) for all t

, t

≥ 0, S(0) = I;

• The map t → S(t)f is Φ

∞

-continuous for each f ∈ Φ

∞

;

• For each q ≥ 0 there exist numbers M

> 0, σ

> 0 and p ≥ q such that

S(t)f

≤ M

f

for all f ∈ Φ

∞

, t > 0.

Let A in L(Φ

∞

, Φ

∞

) be inﬁnitesimal generator of the semigroup {S(t) : t ≥ 0} in L(Φ

∞

, Φ

∞

). The

relations

φ[S(t)f] := (S



(t)φ)[f] for all t ≥ 0, f ∈ Φ

∞

and φ ∈ Φ



∞

;

φ[Af] := (A



φ)[f] for all f ∈ Φ

∞

and φ ∈ Φ



∞

;

deﬁne the inﬁnitesimal generator A



in L(Φ



∞

, Φ



∞

) of the semigroup {S



(t) : t ≥ 0} in L(Φ



∞

, Φ



∞

Let (Σ, U, P ) be a complete probability space with a right continuous ﬁltration (U

)

t≥0

, U

containing

all the P -null sets of U, and M = (M

)

t≥0

be a Φ



∞

-valued martingale with respect to U

, i.e., for each

f ∈ Φ

∞

, M

[f] is a real-valued martingale with respect to U

, t ≥ 0. We are interested in results of

existence and uniqueness of the following Φ



∞

-valued stochastic evolution equation:

dξ

= A



dt + dM

, t > 0,

= γ,

(3.6.4)

where γ is a Φ



∞

-valued random variable, and A is the inﬁnitesimal generator of a C

0,1

-semigroup on

∞

We say that ξ = (ξ

)

t≥0

is a Φ



∞

-solution of the stochastic evolution equation (3.6.4) if the following

conditions are satisﬁed:

• ξ

is Φ



∞

-valued, progressively measurable, and U

-adapted;

• the following integral identity holds:

[f] = γ[f] +



[Af]ds + M

[f],

for all f ∈ Φ

∞

, t ≥ 0 a.s..

It is proved in [22, Corollary 2.2] the following result on existence and uniqueness of solutions of the

stochastic diﬀerential equation (3.6.4):

Proposition 3.6.1. Assume the conditions below:

1. γ is a Φ



∞

-valued U

-measurable random element such that, for some r

> 0, E|γ|

−r

< ∞;

2. M = (M

)

t≥0

is a Φ



∞

-valued martingale such that M

= 0 and, for each t ≥ 0 and f ∈

Φ, E(M

[f])

< ∞;

3. A is a continuous linear operator on Φ

∞

, and is the inﬁnitesimal generator of a C

0,1

-semigroup

{S(t) : t ≥ 0} on Φ

∞

Then, the Φ



∞

-valued homogeneous stochastic evolution equation (3.6.4) has a unique solution ξ = (ξ

)

t≥0

given explicitly by the “evolution solution”:

= S



(t)γ +





(t − s)dM

Remark 3.6.2. The statement E(M

[f])

< ∞ in condition 2 of Proposition 3.6.1 is satisﬁed if

E(M

[f])

= tQ(f, f), where f ∈ Φ

∞

, and Q(·, ·) is a positive deﬁnite continuous bilinear form on

∞

× Φ

∞

We now state a proposition, whose proof can be found in Corollary 2.1 of [22], that gives a suﬃcient

condition for the solution ξ

of the equation (3.6.4) be a Gaussian process.

Proposition 3.6.3. Assume γ is a Φ



∞

-valued Gaussian element independent of the Φ



∞

-valued Gaussian

martingale with independent increments M

. Then, the solution ξ = (ξ

) of (3.6.4) is a Φ



∞

-valued

Gaussian process.

Chapter 4

Dynamical large deviations for a

boundary driven stochastic lattice

gas model with many conserved

quantities

In the last years there has been considerable progress in understanding stationary non equilibrium states:

diﬀusive systems in contact with diﬀerent reservoirs at the boundary imposing a gradient on the conserved

quantities of the system. In these systems there is a ﬂow of matter through the system and the dynamics

is not reversible. The main diﬀerence with respect to equilibrium (reversible) states is the following: in

equilibrium, the invariant measure, which determines the thermodynamic properties, is given for free

by the Gibbs distribution speciﬁed by the Hamiltonian; on the other hand, in non equilibrium states

the construction of the stationary state requires the solution of a dynamical problem. One of the most

striking typical property of these systems is the presence of long-range correlations. For the symmetric

simple exclusion this was already shown in a pioneering paper by Spohn [37]. We refer to [5, 7] for two

recent reviews on this topic.

We discuss this issue in the context of stochastic lattice gases in a box of linear size N with birth

and death processes at the boundary modeling the reservoirs. We consider the case when there are

many thermodynamic variables: the local density denoted by ρ, and the local momentum denoted by

, k = 1, . . . , d, d being the dimension of the box.

Let the set of possible velocities, V, be a ﬁnite subset of R

, and for a point x = (x

, . . . , x

) ∈ R

let ˜x = (x

, . . . , x

). The model which we will study can be informally described as follows: ﬁx a velocity

v ∈ V, an integer N ≥ 1, and boundary densities 0 < α

(·) < 1 and 0 < β

(·) < 1; at any given time,

each site of the set {1, . . . , N − 1} × {0, . . . , N − 1}

d−1

is either empty or occupied by one particle at

velocity v. In the bulk, each particle attempts to jump at any of its neighbors at the same velocity, with

a weakly asymmetric rate. To respect the exclusion rule, the particle jumps only if the target site at the

same velocity v is empty; otherwise nothing happens. At the boundary, sites with ﬁrst coordinates given

by 1 or N − 1 have particles being created or removed in such a way that the local densities are α

(˜x)

and β

(˜x): at rate α

(˜x/N ) a particle is created at {1}×{˜x} if the site is empty, and at rate 1 −α

(˜x)

the particle at {1} × {˜x} is removed if the site is occupied, and at rate β

(˜x) a particle is created at

{N −1}×{˜x} if the site is empty, and at rate 1 −β

(˜x) the particle at {N −1}×{˜x} is removed if the

site is occupied. Superposed to this dynamics, there is a collision process which exchange velocities of

particles in the same site in a way that momentum is conserved. Similar models have been studied by

[1, 9, 32]. In fact, the model we consider here is based on the model of Esposito et al. [9] which was used

to derive the Navier-Stokes equation. It is also noteworthy that the derivation of hydrodynamic limits

and macroscopic ﬂuctuation theory for a system with two conserved quantities have been studied in [4].

The hydrodynamic limit for the above model has been proved in [33]. The hydrodynamic equation

is derived from the underlying stochastic dynamics through an appropriate scaling limit in which the

microscopic time and space coordinates are rescaled diﬀusively. The hydrodynamic equation thus rep-

resents the law of large numbers for the empirical density of the stochastic lattice gas. The convergence

has to be understood in probability with respect to the law of the stochastic lattice gas. Once it is

established a natural question is to consider large deviations from the hydrodynamic limit.

In this Chapter thus provides a derivation of the dynamical large deviations for this model. As usual,

the main diﬃculty appears in the proof of the lower bound where one needs to show that any trajectory

, 0 ≤ t ≤ T , with ﬁnite rate function, I

(λ) < ∞, can be approximated by a sequence of regular

trajectories {λ

: n ≥ 1} such that

−→ λ and I

(λ

) −→ I

(λ) . (4.0.1)

To avoid this diﬃculty, we follow the method introduced in [15]. It is well known that if I

(λ) < ∞,

then there exists an external ﬁeld H associated to λ, in the sense that λ solves a hydrodynamic equation

perturbed by the external ﬁeld H. The strategy of [15] is to approximate the external ﬁeld H by

a sequence of smooth functions, H

, and then to show that the corresponding weak solutions of the

hydrodynamical equations perturbed by H

converge to λ in the sense (4.0.1).

The main diﬀerence of our proof with respect to theirs, is that their proof of the convergence (4.0.1)

relied on some energy estimates that we were not able to achieve due to the presence of velocities.

Therefore, we had to overcome this problem by taking an alternative approach at that part. More

speciﬁc details are given in Section 4.4.

The Chapter is organized as follows: in Section 4.1 we establish the notation and state the main

results of the article; in Section 4.2, we review the hydrodynamics for this model, that was obtained in

[33]; in Section 4.3, several properties of the rate function are derived; Section 4.4 proves the I

(·|γ)-

density, which is a key result for proving the lower bound; ﬁnally, in Section 4.5 the proofs of the upper

and lower bounds of the dynamical large deviations are given.

4.1 Notation and Results

Fix a positive integer d ≥ 1, and denote by D

the open set (0, 1) ×T

d−1

, where T

is the k-dimensional

torus (R/Z)

= [0, 1)

, and by Γ the boundary of D

: Γ = {(u

, . . . , u

) ∈ [0, 1] × T

d−1

; u

= 0 or 1}.

For an open subset Λ of R × T

d−1

, C

(Λ), 1 ≤ m ≤ +∞, stands for the space of m-continuously

diﬀerentiable real functions deﬁned on Λ. Let C

(Λ) (resp. C

(Λ)), 1 ≤ m ≤ +∞, be the subset of

functions in C

(Λ) which vanish at the boundary of Λ (resp. with compact support in Λ).

For each integer N ≥ 1, denote by T

d−1

= (Z/N Z)

d−1

= {0, . . . , N − 1}

d−1

, the discrete (d − 1)-

dimensional torus of length N. Let D

= {1, . . . , N − 1} × T

d−1

be the cylinder in Z

of length N − 1

and basis T

d−1

and let Γ

= {(x

, . . . , x

) ∈ Z ×T

d−1

; x

= 1 or (N − 1)} be the boundary of D

Let V ⊂ R

be a ﬁnite set of velocities v = (v

, . . . , v

). Assume that V is invariant under reﬂexions

and permutations of the coordinates:

, . . . , v

i−1

, −v

, v

i+1

, . . . , v

) and (v

σ(1)

, . . . , v

σ(d)

) (4.1.1)

belong to V for all 1 ≤ i ≤ d, and all permutations σ of {1, . . . , d}, provided (v

, . . . , v

) belongs to V.

On each site of D

, at most one particle for each velocity is allowed. We denote: the number of

particles with velocity v at x, v ∈ V, x ∈ D

, by η(x, v) ∈ {0, 1}; the number of particles in each velocity

v at a site x by η

= {η(x, v); v ∈ V}; and a conﬁguration by η = {η

; x ∈ D

}. The set of particle

conﬁgurations is X



{0, 1}



On the interior of the domain, the dynamics consists of two parts: (i) each particle of the system

evolves according to a nearest neighbor weakly asymmetric random walk with exclusion among particles

of the same velocity, and (ii) binary collision between particles of diﬀerent velocities. Let p(x, v) be an

irreducible probability transition function of ﬁnite range, and mean velocity v:



xp(x, v) = v.

The jump law and the waiting times are chosen so that the jump rate from site x to site x + y for a

particle with velocity v is

(y, v) =



j=1

(δ

y,e

+ δ

y,−e

) +

p(y, v),

where δ

x,y

stands for the Kronecker delta, which equals one if x = y and 0 otherwise, and {e

, . . . , e

} is

the canonical basis in R

4.1.1 The boundary driven exclusion process

Our main interest is to examine the stochastic lattice gas model given by the generator L

which is the

superposition of the boundary dynamics with the collision and exclusion:

= N

+ L

}, (4.1.2)

where L

stands for the generator which models the part of the dynamics at which a particle at the

boundary can enter or leave the system, L

stands for the generator which models the collision part of

the dynamics and lastly, L

models the exclusion part of the dynamics. Let f be a function on X

The generator of the exclusion part of the dynamics, L

, is given by

f)(η) =



v∈V



x,z∈D

η(x, v)[1 − η(z, v)]P

(z − x, v) [f (η

x,z,v

) − f(η)] ,

where

x,y,v

(z, w) =







η(y, v) if w = v and z = x,

η(x, v) if w = v and z = y,

η(z, w) otherwise.

The generator of the collision part of the dynamics, L

, is given by

f)(η) =



y∈D



q∈Q

p(y, q, η) [f(η

y,q

) − f(η)] ,

where Q is the set of all collisions which preserve momentum:

Q = {q = (v, w, v



, w



) ∈ V

; v + w = v



+ w



the rate p(y, q, η) is given by

p(y, q, η) = η(y, v)η(y, w)[1 − η(y, v



)][1 − η(y, w



)],

and for q = (v

, v

), the conﬁguration η

y,q

after the collision is deﬁned as

y,q

(z, u) =



η(y, v

j+2

) if z = y and u = v

for some 0 ≤ j ≤ 3,

η(z, u) otherwise,

where the index of v

j+2

should be taken modulo 4. Particles of velocities v and w at the same site collide

at rate one and produce two particles of velocities v



and w



at that site.

Finally, the generator of the boundary part of the dynamics is given by

f)(η) =



x∈D



v∈V

[α

(˜x/N )[1 − η(x, v)] + (1 − α

(˜x/N ))η(x, v)][f(σ

x,v

η) − f(η)]



x∈D

=N−1



v∈V

[β

(˜x/N )[1 − η(x, v)] + (1 − β

(˜x/N ))η(x, v)][f(σ

x,v

η) − f(η)],

where ˜x = (x

, . . . , x

x,v

η(y, w) =



1 − η(x, w), if w = v and y = x,

η(y, w), otherwise.

and for every v ∈ V, α

, β

∈ C

d−1

). We also assume that, for every v ∈ V, α

and β

have images

belonging to some compact subset of (0, 1). The functions α

and β

, which aﬀect the birth and death

rates at the two boundaries, represent the densities of the reservoirs.

Note that time has been speeded up diﬀusively in (4.1.2). Let {η(t); t ≥ 0} be the Markov process with

generator L

, and let D(R

, X

) be the set of right continuous functions with left limits taking values

on X

. For a probability measure µ on X

, denote by P

the measure on the path space D(R

, X

)

induced by {η(t); t ≥ 0} and the initial measure µ. Expectation with respect to P

is denoted by E

4.1.2 Mass and momentum

For each conﬁguration ξ ∈ {0, 1}

, denote by I

(ξ) the mass of ξ and by I

(ξ), k = 1, . . . , d, the

momentum of ξ:

(ξ) =



v∈V

ξ(v), I

(ξ) =



v∈V

ξ(v).

Set I(ξ) := (I

(ξ), . . . , I

(ξ)). Assume that the set of velocities is chosen in such a way that the

unique quantities conserved by the random walk dynamics described above are mass and momentum:



x∈D

I(η

). Two examples of sets of velocities satisfying these conditions can be found at [9].

For each chemical potential λ = (λ

, . . . , λ

) ∈ R

d+1

, denote by m

the measure on {0, 1}

given by

(ξ) =

Z(λ)

exp {λ · I(ξ)}, (4.1.3)

where Z(λ) is a normalizing constant. Note that m

is a product measure on {0, 1}

, i.e., that the

variables {ξ(v); v ∈ V} are independent under m

Denote by µ

the product measure on X

, with marginals given by

{η; η(x, ·) = ξ} = m

(ξ),

for each ξ in {0, 1}

and x ∈ D

. Note that {η(x, v); x ∈ D

, v ∈ V} are independent variables under

, and that the measure µ

is invariant for the exclusion process with periodic boundary condition.

The expectation under µ

of the mass and momentum are given by

ρ(λ) := E

(η

)] =



v∈V

(λ),

(λ) := E

(η

)] =



v∈V

(λ).

In this formula θ

(λ) denotes the expected value of the density of particles with velocity v under m

(λ) := E

[ξ(v)] =

exp





k=1



1 + exp





k=1



Denote by (ρ, p)(λ) := (ρ(λ), p

(λ), . . . , p

(λ)) the map that associates the chemical potential to the

vector of density and momentum. It is possible to prove that (ρ, p) is a diﬀeomorphism onto U ⊂ R

d+1

the interior of the convex envelope of



I(ξ); ξ ∈ {0, 1}



. Denote by Λ = (Λ

, . . . , Λ

) : U → R

d+1

the

inverse of (ρ, p). This correspondence allows one to parameterize the invariant states by the density and

momentum: for each (ρ, p) in U we have a product measure ν

ρ,p

= µ

Λ(ρ,p)

on X

4.1.3 Dynamical large deviations

Fix T > 0, let M

be the space of ﬁnite positive measures on D

endowed with the weak topology,

and let M be the space of bounded variation signed measures on D

endowed with the weak topology.

Let M

× M

be the cartesian product of these spaces endowed with the product topology, which is

metrizable. Let also M

be the subset of M

×M

of all absolutely continuous measures with respect

to the Lebesgue measure satisfying:



π ∈ M

× M

; π(du) = (ρ, p)(u)du, and (ρ, p) ∈ U, a.e.



Note that if (ρ, p) ∈ U, then 0 ≤ ρ(u) ≤ |V| , |p

(u)| ≤ ˘v|V|, k = 1, . . . , d, where ˘v = max

v∈V

. Let

D([0, T ], M

×M

) be the set of right continuous functions with left limits taking values on M

×M

endowed with the Skorohod topology. M

is a closed subset of M

×M

and D([0, T ], M

) is a closed

subset of D([0, T ], M

×M

). For a measure π ∈ M, denote by π, G the integral of a function G with

respect to π.

Let Ω

= (0, T ) × D

and Ω

= [0, T ] × D

. For 1 ≤ m, n ≤ +∞, denote by C

m,n

(Ω

) the space of

functions G = G

(u) : Ω

→ R with m continuous derivatives in time and n continuous derivatives in

space. We also denote by C

m,n

(Ω

) (resp. C

∞

(Ω

)) the set of functions in C

m,n

(Ω

) (resp. C

∞,∞

(Ω

))

which vanish at [0, T ] × Γ (resp. with compact support in Ω

Let the energy Q : D([0, T ], M

) → [0, ∞] be given by

Q(π) =



k=0



i=1

sup

G∈C

∞

(Ω

)





dt p

k,t

, ∂

 −



G(t, u)



where p

k,t

(u) = p

(t, u) and p

0,t

(u) = ρ(t, u).

Let C

1,2

(Ω

) be the set of vector valued function G = (G

, . . . , G

) : [0, T ] × D

→ R

d+1

, with

each coordinate G

in C

1,2



Ω



, k = 0, . . . , d. For each G ∈ C

1,2

(Ω

) and each measurable function

γ = (ρ

, p

) : D

→ U, let

G,γ,T

: D([0, T ], M

) → R be the functional given by

(π) =



G(T, u) · (ρ, p)(T, u)du −



G(0, u) · (ρ

, p

)(u)du

−





(ρ, p)(t, u) · ∂

G(t, u) +

(ρ, p)(t, u) ·



i=1

∂

G(t, u)





{1}×T

d−1

dS b(˜u) ·∂

G(t, u) −



{0}×T

d−1

dS a(˜u) ·∂

G(t, u)





v∈V

˜v · χ

(ρ, p)



i=1

∂

G(t, u)

−





v∈V





k=0

∂

(u)



(ρ, p),

where χ(r) = r(1 − r) is the static compressibility, χ

(·) = χ(θ

(Λ(·))), for u = (u

, . . . , u

) ∈ R

, ˜u =

, . . . , u

), π

(du) = (ρ, p)(t, u)du, and dS is the Lebesgue measure on T

d−1

. Deﬁne J

= J

G,γ,T

D([0, T ], M

× M

) → R by

(π) =



(π), if π ∈ D([0, T ], M

+∞, otherwise .

We deﬁne the rate functional I

(·|γ) : D([0, T ], M

× M

) → [0, +∞] as

(π|γ) =







sup

G∈C

1,2

(Ω

)



(π)



, if Q(π) < ∞,

+∞, otherwise .

We now present the main result of this article, whose proof is given in Section 4.5, which is the

dynamical large deviations for this boundary driven exclusion process with many conserved quantities.

Theorem 4.1.1. Fix T > 0 and a measurable function γ = (ρ

, p

) : D

→ U. Consider a sequence η

of conﬁgurations in X

associated to γ in the sense that:

lim

N→∞

π

(η

), G =



G(u)ρ

(u) du,

and

lim

N→∞

π

(η

), G =



G(u)p

(u) du, k = 1, . . . , d,

for every continuous function G : D

→ R. Then, the measure Q

= P

(π

)

−1

on D([0, T ], M

) satisﬁes a large deviation principle with speed N

and rate function I

(·|γ). Namely, for each

closed set C ⊂ D([0, T ], M

× M

lim

N→∞

log Q

π∈C

(π|γ)

and for each open set O ⊂ D([0, T ], M

× M

lim

N→∞

log Q

(O) ≥ − inf

π∈O

(π|γ) .

Moreover, the rate function I

(·|γ) is lower semicontinuous and has compact level sets.

4.2 Hydrodynamics

Fix T > 0 and let (B,  · 

) be a Banach space. We denote by L

([0, T ], B) the Banach space of

measurable functions U : [0, T ] → B for which

U

([0,T ],B)



U



dt < ∞.

Moreover, we denote by H

) the Sobolev space of measurable functions in L

) that have gener-

alized derivatives in L

For x = (x

, ˜x) ∈ {0, 1} × T

d−1

, let

d(x) =







a(˜x) =



v∈V

(α

(˜x), v

(˜x), . . . , v

(˜x)), if x

= 0,

b(˜x) =



v∈V

(β

(˜x), v

(˜x), . . . , v

(˜x)), if x

= 1.

(4.2.1)

Fix a bounded density proﬁle ρ

: D

→ R

, and a bounded momentum proﬁle p

: D

→ R

. A

bounded function (ρ, p) : [0, T ] × D

→ R

× R

is a weak solution of the system of parabolic partial

diﬀerential equations







∂

(ρ, p) +



v∈V

˜v [v · ∇χ

(ρ, p)] =

∆(ρ, p),

(ρ, p)(0, ·) = (ρ

, p

)(·) and (ρ, p)(t, x) = d(x), x ∈ {0, 1} × T

d−1

(4.2.2)

if for every vector valued function H ∈ C

1,2

(Ω

), we have



H(T, u) · (ρ, p)(T, u)du −



H(0, u) · (ρ

, p

)(u)du





(ρ, p)(t, u) · ∂

H(t, u) +

(ρ, p)(t, u) ·



i=1

∂

H(t, u)



−



{1}×T

d−1

dS b(˜u) ·∂

H(t, u) +



{0}×T

d−1

dS a(˜u) ·∂

H(t, u)

−





v∈V

˜v · χ

(ρ, p)



i=1

∂

H(t, u).

We say that that the solution (ρ, p) has ﬁnite energy if its components belong to L

([0, T ], H

)):







∇ρ(s, u)



< ∞,

and







∇p

(s, u)



< ∞,

for k = 1, . . . , d, where ∇f represents the generalized gradient of the function f.

In [33] the following theorem was proved:

Theorem 4.2.1. Let (µ

)

be a sequence of probability measures on X

associated to the proﬁle

(ρ

, p

) in the sense of Theorem 4.1.1. Then, for every t ≥ 0, for every continuous function H : D

→ R

vanishing at the boundary Γ, and for every δ > 0,

lim

N→∞









x∈D





(η

(t)) −



H(u)ρ(t, u)du



> δ





= 0,

and for 1 ≤ k ≤ d

lim

N→∞









x∈D





(η

(t)) −



H(u)p

(t, u)du



> δ





= 0,

where (ρ, p) has ﬁnite energy and is the unique weak solution of equation (4.2.2).

4.3 The rate function I

(·|γ)

We examine in this section the rate function I

(·|γ). The main result, presented in Theorem 4.3.6

below, states that I

(·|γ) has compact level sets. The proof relies on two ingredients. The ﬁrst one,

stated in Lemma 4.3.2, is an estimate of the energy and of the H

−1

norm of the time derivative of

a trajectory in terms of the rate function. The second one, stated in Lemma 4.3.5, establishes that

sequences of trajectories, with rate function uniformly bounded, which converge weakly in L

converge

in fact strongly. We follow the strategy introduced in [15].

Let V be an open neighborhood of D

, and consider, for each v ∈ V, smooth functions κ

: V → (0, 1)

in C

(V ), for k = 0, . . . , d. We assume that the restriction of κ =



v∈V

(κ

, v

, . . . , v

) to {0}×T

d−1

equals the vector valued function a(·) deﬁned in (4.2.1), and that the restriction of κ to {1}×T

d−1

equals

the vector valued function b(·), also deﬁned in (4.2.1), in the sense that κ(x) = d(x

, ˜x) if x ∈ {0, 1}×T

d−1

Let L

) be the Hilbert space of functions G : D

→ R such that



|G(u)|

du < ∞ equipped

with the inner product

G, F 



Ω

G(u) F (u) du ,

and the norm of L

) is denoted by  · 

Recall that H

) is the Sobolev space of functions G with generalized derivatives ∂

G, . . . , ∂

in L

). H

) endowed with the scalar product ·, ·

1,2

, deﬁned by

G, F 

1,2

= G, F 



j=1

∂

G , ∂

F 

is a Hilbert space. The corresponding norm is denoted by ·

1,2

Recall also that we denote by C

∞

) the set of inﬁnitely diﬀerentiable functions G : D

→ R, with

compact support in D

. Denote by H

) the closure of C

∞

) in H

). Since D

is bounded,

by Poincar´e’s inequality, there exists a ﬁnite constant C such that for all G ∈ H

)

G

≤ C



j=1

∂

G , ∂

G

This implies that, in H

)

G

1,2,0





j=1

∂

G , ∂

G



1/2

is a norm equivalent to the norm ·

1,2

. Moreover, H

) is a Hilbert space with inner product given

G , J

1,2,0



j=1

∂

G , ∂

J

To assign boundary values along the boundary Γ of D

to any function G in H

), recall, from the

trace Theorem ([39], Theorem 21.A.(e)), that there exists a continuous linear operator Tr : H

) →

(Γ), called trace, such that Tr(G) = G



if G ∈ H

) ∩ C(D

). Moreover, the space H

) is the

space of functions G in H

) with zero trace ([39], Appendix (48b)):

) =



G ∈ H

); Tr(G) = 0



Finally, denote by H

−1

) the dual of H

). H

−1

) is a Banach space with norm ·

−1

given

v

−1

= sup

G∈C

∞

)



2v, G

−1,1

−



∇G(u)



where v, G

−1,1

stands for the values of the linear form v at G.

For each G ∈ C

∞

(Ω

) and each integer 1 ≤ i ≤ d, let Q

i,k

: D([0, T ], M

) → R be the functional

given by

i,k

(π) = 2



dt π

, ∂

 −



du G(t, u)

where π = (π

, π

, . . . , π

). Recall, from subsection 2.2, that the energy Q(π) is given by

Q(π) =



k=0



i=1

i,k

(π), with Q

i,k

(π) = sup

G∈C

∞

(Ω

)

i,k

(π) .

The functional Q

i,k

is convex and continuous in the Skorohod topology. Therefore Q

i,k

and Q are

convex and lower semicontinuous. Furthermore, it is well known that a measure π(t, du) = (ρ, p)(t, u)du

in D([0, T ], M

× M

) has ﬁnite energy, Q(π) < ∞, if and only if its density ρ and its momentum p

belong to L

([0, T ], H

)). In which case

Q(π) :=



k=0



du ∇p

k,t

(u)

< ∞,

where p

0,t

(u) = ρ(t, u).

Let D

= D

γ,d

be the subset of C([0, T ], M

) consisting of all paths π(t, du) = (ρ, p)(t, u)du with

initial proﬁle γ(·) = (ρ

, p

)(·), ﬁnite energy Q(π) (in which case ρ

and p

belong to H

) for almost

all 0 ≤ t ≤ T and so Tr(ρ

) is well deﬁned for those t) and such that Tr(ρ

) = d

and Tr(p

k,t

) = d

k = 1, . . . , d, for almost all t in [0, T ], where d(·) = (d

(·), d

(·), . . . , d

(·)).

Lemma 4.3.1. Let π be a trajectory in D([0, T ], M

×M

) such that I

(π|γ) < ∞. Then π belongs to

Proof. Fix a path π in D([0, T ], M

×M

) with ﬁnite rate function, I

(π|γ) < ∞. By deﬁnition of I

π belongs to D([0, T ], M

). Denote its density and momentum by (ρ, p): π(t, du) = (ρ, p)(t, u)du.

The proof that (ρ, p)(0, ·) = γ(·) is similar to the one of Lemma 3.5 in [6], and the proof that

Tr(ρ

) = d

, Tr(p

k,t

) = d

, k = 1, . . . , d, is similar to the one found in Lemma 4.1 in [15].

We deal now with the continuity of π. We claim that there exists a positive constant C

such that,

for any g ∈ [C

∞

)]

d+1

, and any 0 ≤ s < r < T ,

|π

, g − π

, g| ≤ C

(r − s)

1/2



+ I

(π|γ) + g

1,2,0

+ (r − s)

1/2

∆g



. (4.3.1)

Indeed, for each δ > 0, let ψ

: [0, T ] → R be the function given by

(r − s)

1/2

(t) =











0 if 0 ≤ t ≤ s or r + δ ≤ t ≤ T ,

t−s

if s ≤ t ≤ s + δ ,

1 if s + δ ≤ t ≤ r ,

1 −

t−r

if r ≤ t ≤ r + δ ,

and let G



(t, u) = ψ



(t)g(u), where ψ



(·) is the standard -molliﬁcation of ψ

(·). Since G



is in C

1,2

(Ω

we have

(r − s)

1/2

lim

δ→0

lim

→0



(π) = π

, g − π

, g −



dt π

, ∆g





v∈V

˜v · χ

(ρ, p)



i=1

∂

g(u)

−

(r − s)

1/2





v∈V





k=0

∂

(u)



(ρ, p).

To conclude the proof, we observe that the left-hand side is bounded by (r − s)

1/2

(π|γ), that χ

is positive and bounded above on [0, 1] by 1/4, and ﬁnally, we use the elementary inequality 2ab ≤

+ b

Denote by L

([0, T ], H

))

∗

the dual of L

([0, T ], H

)). By Proposition 23.7 in [39], L

([0, T ], H

))

∗

corresponds to L

([0, T ], H

−1

)) and for v in L

([0, T ], H

))

∗

, G in L

([0, T ], H

)),

v, G

−1,1



v

, G



−1,1

dt , (4.3.2)

where the left hand side stands for the value of the linear functional v at G. Moreover, if we denote by

|||v|||

−1

the norm of v,

|||v|||

−1



v



−1

dt .

Fix a path π(t, du) = (ρ, p)(t, u)du in D

and suppose that for k = 0, . . . , d

sup

G∈C

∞

(Ω

)





dt p

k,t

, ∂



−



du ∇G





< ∞ . (4.3.3)

In this case, for each k, ∂

: C

∞

(Ω

) → R deﬁned by

∂

(G) = −



p

k,t

, ∂



can be extended to a bounded linear operator ∂

: L

([0, T ], H

)) → R. It belongs therefore

to L

([0, T ], H

))

∗

= L

([0, T ], H

−1

)). In particular, there exists v

= {v

; 0 ≤ t ≤ T } in

([0, T ], H

−1

)), which we denote by v

= ∂

k,t

, such that for any G in L

([0, T ], H

)),

∂

, G

−1,1



∂

k,t

, G



−1,1

dt .

Moreover,

|||∂

|||

−1



∂

k,t



−1

= sup

G∈C

∞

(Ω

)





dt p

k,t

, ∂



−



du ∇G





Denote by ∂

(ρ, p), ·

−1,1

: L

([0, T ], [H

)]

d+1

) → R the linear functional given by

∂

(ρ, p), G

−1,1



k=0

∂

, G



−1,1

with G = (G

, . . . , G

), and

|||∂

(ρ, p)|||

−1



k=0

|||∂

|||

−1

Let W be the set of paths π(t, du) = (ρ, p)(t, u)du in D

such that (4.3.3) holds, i.e., such that ∂

belongs to L



[0, T ], H

−1

)



. For G in L



[0, T ], [H

)]

d+1



, let J

: W → R be the functional

given by

(π) = ∂

(ρ, p), G

−1,1





i=1

∂

(ρ, p)(t, u) · ∂

G(t, u)





v∈V

˜v · χ

(ρ, p)



i=1

∂

G(t, u)

−





v∈V

(˜v ·∂

(u))

(ρ, p),

Note that J

(π) = J

(π) for every G in C

∞

(Ω

) × [C

∞

)]

. Moreover, since J

(π) is continuous

in L



[0, T ], [H

)]

d+1



and since C

∞

(Ω

) is dense in C

1,2

(Ω

) and in L

([0, T ], H

)), for every

π in W ,

(π|γ) = sup

G∈C

∞

(Ω

)×[C

∞

)]

(π) = sup

G∈L

(

[0,T ],[H

)]

d+1

)

(π) . (4.3.4)

Lemma 4.3.2. There exists a constant C

> 0 such that if the density and momentum (ρ, p) of some

path π(t, du) = (ρ, p)(t, u)du in D([0, T ], M

) has generalized gradients, ∇ρ and ∇p

, k = 1, . . . , d. Then

|||∂

(ρ, p)|||

−1

≤ C

(π|γ) + Q(π)} , (4.3.5)



k=0



du ∇p

(t, u)

≤ C

(π|γ) + 1} , (4.3.6)

where p

= ρ.

Proof. Fix a path π(t, du) = (ρ, p)(t, u)du in D([0, T ], M

). In view of the discussion presented before

the lemma, we need to show that the left hand side of (4.3.3) is bounded by the right hand side of

(4.3.5). Such an estimate follows from the deﬁnition of the rate function I

(·|γ) and from the elementary

inequality 2ab ≤ Aa

+ A

−1

To prove (4.3.6), observe that

(π|γ) ≥ J

(π) = ∂

π(G) +





i=1

∂

(ρ, p) · ∂





v∈V

(ρ, p)



i=1

˜v · (v

∂

−





v∈V



i=1

(˜v ·∂

(ρ, p)

≥ ∂

π(G) +





i=1

∂

(ρ, p) · ∂

G − C





k=0

∇G



where C is constant obtained from the elementary inequality 2ab ≤ a

+ b

, the fact that V is ﬁnite, and

that χ is bounded above by 1/4 in [0, 1].

Recall the deﬁnition of the function κ given at the beginning of Section 4.3. Now, consider G =

K(π − κ), K > 0 being a constant, and note that π − κ belongs to L

([0, T ], H

)), which implies

that it may be approximated by C

∞

functions. Therefore |∂

π(G)| = K|π

, π

/2 −κ−π

, π

/2 −κ|,

which is bounded from above by some constant C

. We, then, obtain that

I(π) ≥





− C



k=0

∇p



−



i=1

∂

(ρ, p) · ∂

κ − CK



k=0

∇(p

− κ

)



≥





K/4 − 2CK





k=0

∇p



−



k=0

∇κ



− 2CK



k=0

∇κ



− C



where in the last inequality we used the Cauchy-Schwartz inequality and the elementary inequality

2ab ≤ a

+ b

. The proof thus follows from choosing a suitable K, the estimate given in (4.3.5), and the

fact we have a ﬁxed smooth function κ.

Corollary 4.3.3. The density (ρ, p) of a path π(t, du) = (ρ, p)(t, u)du in D([0, T ], M

) is the weak

solution of the equation (4.2.2) and initial proﬁle γ if and only if the rate function I

(π|γ) vanishes.

Moreover, if any of the above conditions hold, π has ﬁnite energy (Q(π) < ∞).

Proof. On the one hand, if the density (ρ, p) of a path π(t, du) = (ρ, p)(t, u)du in D([0, T ], M

) is the

weak solution of equation (4.2.2) with initial condition is γ, in the formula of

(π), the linear part

in G vanishes which proves that the rate functional I

(π|γ) vanishes. On the other hand, if the rate

functional vanishes, the path (ρ, p) belongs to L

([0, T ], [H

)]

d+1

) and the linear part in G of J

(π)

has to vanish for all functions G. In particular, (ρ, p) is a weak solution of (4.2.2). Moreover, if the rate

function is ﬁnite, by the previous lemma, π has ﬁnite energy. Accordingly, if π is a weak solution, we

have from Theorem 4.2.1 that it has ﬁnite energy.

For each q > 0, let E

be the level set of I

(π|γ) deﬁned by



π ∈ D([0, T ], M

× M

); I

(π|γ) ≤ q



By Lemma 4.3.1, E

is a subset of C([0, T ], M

). Thus, from the previous lemma, it is easy to deduce

the next result.

Corollary 4.3.4. For every q ≥ 0, there exists a ﬁnite constant C(q) such that

sup

π∈E



|||∂

(ρ, p)|||

−1



k=0



du ∇p

(t, u)



≤ C(q) .

Next result together with the previous estimates provide the compactness needed in the proof of the

lower semicontinuity of the rate function.

Lemma 4.3.5. Let {ρ

; n ≥ 1} be a sequence of functions in L

(Ω

) such that uniformly on n,



dt ρ



1,2



dt ∂



−1

≤ C

for some positive constant C. Suppose that ρ ∈ L

(Ω

) and that ρ

→ ρ weakly in L

(Ω

). Then ρ

→ ρ

strongly in L

(Ω

Proof. Since H

) ⊂ L

) ⊂ H

−1

) with compact embedding H

) → L

), from Corol-

lary 8.4, [35], the sequence {ρ

} is relatively compact in L



[0, T ], L

)



. Therefore the weak conver-

gence implies the strong convergence in L



[0, T ], L

)



Theorem 4.3.6. The functional I

(·|γ) is lower semicontinuous and has compact level sets.

Proof. We have to show that, for all q ≥ 0, E

is compact in D([0, T ], M

× M

). Since E

⊂

C([0, T ], M

) and C([0, T ], M

) is a closed subset of D([0, T ], M), we just need to show that E

compact in C([0, T ], M

We will show ﬁrst that E

is closed in C([0, T ], M

). Fix q ∈ R and let {π

; n ≥ 1} be a sequence in

converging to some π in C([0, T ], M

). Then, for all G ∈ C(Ω

) × [C(D

)]

lim

n→∞



dt π

, G

 =



dt π

, G

.

Notice that this means that π

n,k

→ π

weakly in L

(Ω

), for each k = 0, . . . , d, which together with

Corollary 4.3.4 and Lemma 4.3.5 imply that π

n,k

→ π

strongly in L

(Ω

). From this fact and the

deﬁnition of J

it is easy to see that, for all G in C

1,2

(Ω

lim

n→∞

(π

) = J

(π) .

This limit, Corollary 4.3.4 and the lower semicontinuity of Q permit us to conclude that Q(π) ≤ C(q)

and that I

(π|γ) ≤ q.

We prove now that E

is relatively compact. To this end, it is enough to prove that for every

continuous function G : D

→ R, and every k = 0, . . . , d,

lim

δ→0

sup

π∈E

sup

0≤s,r≤T

|r−s|<δ

|π

, G − π

, G| = 0 . (4.3.7)

Since E

⊂ C([0, T ], M

), we may assume by approximations of G in L

) that G ∈ C

∞

). In

which case, (4.3.7) follows from (4.3.1).

We conclude this section with an explicit formula for the rate function I

(·|γ). For each π(t, du) =

(ρ, p)(t, u)du in D([0, T ], M

), denote by H

(π) the Hilbert space induced by C

1,2

(Ω

) endowed with

the inner product ·, ·

deﬁned by

H, G



v∈V



i=1



duχ

(ρ, p)[˜v ·∂

H][˜v ·∂

G] . (4.3.8)

Induced means that we ﬁrst declare two functions F, G in C

1,2

(

Ω

) to be equivalent if F −G, F −G

= 0,

and then we complete the quotient space with respect to the inner product ·, ·

. The norm of H

(π)

is denoted by  · 

Fix a path π in D([0, T ], M

) and a function H in H

(π). A measurable function λ : [0, T ] × D

→

× R

is said to be a weak solution of the nonlinear boundary value parabolic equation











∂

λ +



i=1



v∈V

˜v∂

[χ

(λ)(v

− ˜v · ∂

H)] =

∆λ,

λ(0, ·) = γ(·)

λ(t, x) = d(x), x ∈ {0, 1} × T

d−1

(4.3.9)

if it satisﬁes the following two conditions:

(i) For k = 0, . . . , d, λ

belongs to L



[0, T ], H

)









 ∇λ

(s, u) 



< ∞ ;

(ii) For every function G(t, u) = G

(u) in C

1,2

(Ω



G(T, u) · λ(T, u)du −



G(0, u) · γ(u)du





λ(t, u) · ∂

G(t, u) +

λ(t, u) ·



i=1

∂

G(t, u)



−



{1}×T

d−1

dS b(˜u) ·∂

G(t, u) +



{0}×T

d−1

dS a(˜u) ·∂

G(t, u)

−





v∈V

˜v · χ

(λ)



i=1

∂

G(t, u),



v∈V



i=1



duχ

(λ)[˜v ·∂

H][˜v ·∂

G].

Uniqueness of solutions of equation (1.3.9) follows from the same arguments of the uniqueness proved

in [33].

Lemma 4.3.7. Assume that π(t, du) = (ρ, p)(t, u)du in D([0, T ], M

) has ﬁnite rate function: I

(π|γ) <

∞. Then, there exists a function H in H

(π) such that (ρ, p) is a weak solution to (4.3.9). Moreover,

(π|γ) =

H

. (4.3.10)

The proof of this lemma is similar to the one of Lemma 10.5.3 in [3] and is therefore omitted.

4.4 I

(·|γ)-Density

The main result of this section, stated in Theorem 4.4.5, asserts that any trajectory λ

, 0 ≤ t ≤ T , with

ﬁnite rate function, I

(λ|γ) < ∞, can be approximated by a sequence of smooth trajectories {λ

; n ≥ 1}

such that

−→ λ and I

(λ

|γ) −→ I

(λ|γ) .

This is one of the main steps in the proof of the lower bound of the large deviations principle for the

empirical measure. The proof is mainly based on the regularizing eﬀects of the hydrodynamic equation.

This strategy was introduced in [15].

A subset A of D([0, T ], M

× M

) is said to be I

(·|γ)-dense if for every π in D([0, T ], M

× M

)

such that I

(π|γ) < ∞, there exists a sequence {π

; n ≥ 1} in A such that π

converges to π and

(π

|γ) converges to I

(π|γ).

Let Π

be the subset of D([0, T ], M

) consisting of paths π(t, du) = (ρ, p)(t, u)du whose density (ρ, p)

is a weak solution of the hydrodynamic equation (4.2.2) in the time interval [0, δ] for some δ > 0.

Lemma 4.4.1. The set Π

is I

(·|γ)-dense.

Proof. Fix π(t, du) = (ρ, p)(t, u)du in D([0, T ], M

× M

) such that I

(π|γ) < ∞. By Lemma 4.3.1, π

belongs to C([0, T ], M

). For each δ > 0, let (ρ

, p

) be the path deﬁned as

(ρ

, p

)(t, u) =











τ(t, u) if 0 ≤ t ≤ δ ,

τ(2δ − t, u) if δ ≤ t ≤ 2δ ,

(ρ, p)(t − 2δ, u) if 2δ ≤ t ≤ T ,

where τ is the weak solution of the hydrodynamic equation (4.2.2) starting at γ. It is clear that π

(t, du) =

(ρ

, p

)(t, u)du belongs to D

, because so do π and τ and that Q(π

) ≤ Q(π) + 2Q(τ ) < ∞. Moreover,

converges to π as δ ↓ 0 because π belongs to C([0, T ], M

). By the lower semicontinuity of I

(·|γ),

(π|γ) ≤ lim

δ→0

(π

|γ). Then, in order to prove the lemma, it is enough to prove that I

(π|γ) ≥

lim

δ→0

(π

|γ). To this end, decompose the rate function I

(π

|γ) as the sum of the contributions

on each time interval [0, δ], [δ, 2δ] and [2δ, T ]. The ﬁrst contribution vanishes because π

solves the

hydrodynamic equation in this interval. On the time interval [δ, 2δ], ∂

= −∂

2δ−t

= −

∆τ

2δ−t



v∈V

˜v[v ·∇χ

(τ

2δ−t

)] = −

∆(ρ

, p

) +



v∈V

˜v[v ·∇χ

(ρ

, p

)]. In particular, the second contribution

is equal to

sup

G∈C

1,2

(Ω

)



i=1





du ∂

(ρ, p) · ∂

G −



v∈V



duχ

(ρ, p)[˜v ·∂



which, by Lemma 4.5.5 is bounded from above, and therefore this last expression converges to zero as

δ ↓ 0. Finally, the third contribution is bounded by I

(π|γ) because π

in this interval is just a time

translation of the path π.

Recall the deﬁnition of the set U given at the ending of subsection 4.1.2. Let Π

be the set of all

paths π in Π

with the property that for every δ > 0 there exists  > 0 such that, for k = 0, . . . , d,

d(π

(·), ∂U) ≥  for all t ∈ [δ, T ], where ∂U stands for the boundary of U.

We begin by proving an auxiliary lemma.

Lemma 4.4.2. Let π, λ ∈ U, and let π



= (1 − )π + λ, 0 ≤  ≤ 1. Then, for all v ∈ V, we have

(Λ(π



)) = (1 − )θ

(Λ(π)) + θ

(Λ(λ)).

Proof. Fix some λ ∈ U. Observe that





v∈V

(Λ(λ)),



v∈V

(Λ(λ)), . . . ,



v∈V

(Λ(λ))



= (λ

, λ

, . . . , λ

)

is a linear system with d + 1 equations and |V| unknowns (given by θ

(Λ(λ)), for v ∈ V). Therefore, any

solution of this linear system can be expressed as a linear combination of λ

, i = 0, 1, . . . , d. The proof

follows from this fact.

Remark 4.4.3. In the particular case when d = 1 and the set of velocities is V = {v, −v} ⊂ R, a simple

computation gives the unique solution

(Λ(λ

, λ

)) =

and θ

−v

(Λ(λ

, λ

)) =

−

Lemma 4.4.4. The set Π

is I

(·|γ)-dense.

Proof. By Lemma 4.4.1, it is enough to show that each path π(t, du) = (ρ, p)(t, u)du in Π

can be

approximated by paths in Π

. Fix π in Π

and let τ be as in the proof of the previous lemma. For each

0 < ε < 1, let (ρ

, p

) = (1 − ε)(ρ, p) + ετ, π

(t, du) = (ρ

, p

)(t, u)du. Note that Q(π

) < ∞ because

Q is convex and both Q(π) and Q(τ ) are ﬁnite. Hence, π

belongs to D

since both ρ and τ satisfy the

boundary conditions. Moreover, It is clear that π

converges to π as ε ↓ 0. By the lower semicontinuity

of I

(·|γ), in order to conclude the proof, it is enough to show that

lim

N→∞

(π

|γ) ≤ I

(π|γ) . (4.4.1)

By Lemma 4.3.7, there exists H ∈ H

(π) such that (ρ, p) solves the equation (4.3.9). Let P

i,v

(π) =

(ρ, p)



˜v · ∂

H − v



, and note that P

i,v

(τ) = −v

(τ). Let also



i,v

= (1 − )P

i,v

(π) + P

i,v

(τ).

Observe that, by Lemma 4.3.7,

(π|γ) =

H

and that, using the deﬁnition of  · 

in (4.3.8),

H



v∈V



i=1



duχ

(ρ, p)(˜v ·∂



v∈V



i=1



i,v

(π) + v

(ρ, p))

(ρ, p)

A simple computation shows that

(π



) =



v∈V



i=1





i,v

+ χ

(ρ



, p



](˜v ·∂

G) − χ

(ρ



, p



)(˜v ·∂



v∈V



i=1





i,v

+ χ

(ρ



, p



]

(ρ



, p



)

−





i,v

+ χ

(ρ



, p



)



(ρ



, p



)

−



(ρ, p)(˜v ·∂



Let





v∈V



i=1





i,v

+ χ

(ρ



, p



]

(ρ



, p



)

and



(G) =







i,v

+ χ

(ρ



, p



)



(ρ



, p



)

−



(ρ, p)(˜v ·∂



This implies that

(π



|γ) = sup

(π



) = sup





− B



(G)



= A



− inf



(G)

≤ A



where the supremum and inﬁmum are taken over in G in C

∞

(Ω

) × [C

∞

)]

It remains to be shown that A



is uniformly integrable in . However, this is a simple consequence of

Lemma 4.4.2.

Let Π be the subset of Π

consisting of all those paths π which are solutions of the equation (4.3.9)

for some H ∈ C

1,2

(Ω

Theorem 4.4.5. The set Π is I

(·|γ)-dense.

Proof. By the previous lemma, it is enough to show that each path π in Π

can be approximated by

paths in Π. Fix π(t, du) = (ρ, p)(t, u)du in Π

. By Lemma 4.3.7, there exists H ∈ H

(π) such that (ρ, p)

solves the equation (4.3.9). Since π belongs to Π

⊂ Π

, (ρ, p) is the weak solution of (4.2.2) in some

time interval [0, 2δ] for some δ > 0. In particular, ˜v · ∂

H = 0 a.e in [0, 2δ] × D

, i = 1, . . . , d, v ∈ V.

This implies, by equation (4.1.1), that ∇H

= 0 a.e. in [0, 2δ] × D

, k = 0, . . . , d. On the other hand,

since π belongs to Π

, there exists  > 0 such that, for k = 0, . . . , d, d(π

(·), ∂U) ≥  for δ ≤ t ≤ T .

Therefore,



∇H

(u)

du < ∞, k = 0, . . . , d. (4.4.2)

Since H belongs to H

(π), there exists a sequence of functions {H

= (H

n,1

, . . . , H

n,d

); n ≥ 1} in

1,2

(Ω

) converging to H in H

(π). We may assume of course that ∇H

n,k

≡ 0 in the time interval [0, δ],

k = 0, . . . , d. In particular,

lim

n→∞



du ∇H

n,k

(u) − ∇H

(u)

= 0 , k = 0, . . . , d. (4.4.3)

For each integer n > 0, let (ρ

, p

) be the weak solution of (4.3.9) with H

in place of H and set

(t, du) = (ρ

, p

)(t, u)du. By (4.3.10) and since χ is bounded above in [0, 1] by 1/4, we have that

(π

|γ) =



v∈V



i=1



dt χ

(ρ

, p

), (˜v ·∂

)



≤ C



v∈V



i=1



du (˜v · ∂

(u))

In particular, by (4.4.2) and (4.4.3), I

(π

|γ) is uniformly bounded on n. Thus, by Theorem 4.3.6, the

sequence π

is relatively compact in D([0, T ], M

× M

The sequence π

has a subsequence converging to some π

in D([0, T ], M

). To keep notation simple,

we will assume that the sequence π

converges to π

. For every G in C

1,2

(Ω



G(T, u) · (ρ

, p

)(T, u)du −



G(0, u) · γ(u)du





(ρ

, p

)(t, u) · ∂

G(t, u) +

(ρ

, p

)(t, u) ·



i=1

∂

G(t, u)



−



{1}×T

d−1

dS b(˜u) ·∂

G(t, u) +



{0}×T

d−1

dS a(˜u) ·∂

G(t, u)

−





v∈V

˜v · χ

(ρ

, p

)



i=1

∂

G(t, u),



v∈V



i=1



duχ

(ρ

, p

)[˜v ·∂

][˜v ·∂

G].

Letting n → ∞ in this equation, we obtain the same equation with π

and H in place of π

and H

respectively, if

lim

n→∞





v∈V

˜v · χ

(ρ

, p

)



i=1

∂

G(t, u)





v∈V

˜v · χ

(ρ

, p

)



i=1

∂

G(t, u),

lim

n→∞



v∈V



i=1



duχ

(ρ

, p

)[˜v ·∂

][˜v ·∂



v∈V



i=1



duχ

(ρ

, p

)[˜v ·∂

H][˜v ·∂

G].

(4.4.4)

We prove the second claim, the ﬁrst one being simpler. Note ﬁrst that we can replace H

by H in

the previous limit, because χ is bounded in [0, 1] by 1/4, and (4.4.3) holds. Now, (ρ

, p

) converges

to (ρ

, p

) weakly in L

(Ω

) × [L

)]

because π

converges to π

in D([0, T ], M

). Since I

(π

|γ)

is uniformly bounded, by Corollary 4.3.4 and Lemma 4.3.5, (ρ

, p

) converges to (ρ

, p

) strongly in

(Ω

) × [L

)]

which implies (4.4.4). In particular, since (4.4.2) holds, by uniqueness of weak

solutions of equation (4.3.9), π

= π and we are done.

4.5 Large deviations

We prove in this section Theorem 4.1.1, which is the dynamical large deviations principle for the empirical

measure of boundary driven stochastic lattice gas model with many conserved quantities. The proof uses

some of the ideas introduced in [15].

4.5.1 Superexponential estimates

It is well known that one of the main steps in the derivation of the upper bound is a super-exponential

estimate which allows the replacement of local functions by functionals of the empirical density in the

large deviations regime.

Let κ be as in the beginning of Section 4.3. Note that since ν

is not the invariant state, there are

no reasons for −N

f, f 

to be positive. The next statement shows that this expression is almost

positive.

For each function f : X

→ R, let D

(f) be

(f) = D

(f) + D

(f),

where

(f) =



v∈V



x∈D



x+z∈D

(z − x, v)







f(η

x,z,v

) −



f(η)



(dη),

(f) =



q∈Q



x∈D



p(x, q, η)





f(η

x,q

) −



f(η)



(dη),

and

(f) =



v∈V



x∈{1}×T

d−1



[α

(˜x/N )(1 − η(x, v)) + (1 − α

(˜x/N ))η(x, v)]×





f(σ

x,v

η) −



f(η)



(dη) +



v∈V



x∈{N−1}×T

d−1



[β

(˜x/N )(1 − η(x, v)) + (1 − β

(˜x/N ))η(x, v)]×





f(σ

x,v

η) −



f(η)



(dη).

Proposition 4.5.1. There exist constants C

> 0 and C

= C

(α, β) > 0 such that for every density f

with respect to ν

, we have

< L



f >

≤ −C

(f) + C

d−2

The proof of this proposition is elementary and is thus omitted.

Further, we may choose κ for which there exists a constant θ > 0 such that:

κ(u

, ˜u) = d(0, ˜u) if 0 ≤ u

≤ θ ,

κ(u

, ˜u) = d(1, ˜u) if 1 − θ ≤ u

≤ 1 ,

for all ˜u ∈ T

d−1

. In that case, for every N large enough, ν

is reversible for the process with generator

and then −N

f, f 

is positive.

Fix L ≥ 1 and a conﬁguration η, let I

(x, η) := I

(x) = (I

(x), . . . , I

(x)) be the average of the

conserved quantities in a cube of the length L centered at x:

(x) =

|Λ



z∈x+Λ

I(η

where, Λ

= {−L, . . . , L}

and |Λ

| = (2L + 1)

is the discrete volume of box Λ

For each G ∈ C(Ω

) × [C(D

)]

, and each ε > 0, let

G,1

Nε

(s, η) =



k=0



i,j=1



x∈D

∂

(s, x/N )



j,k

Nε



where

j,k

Nε

(η) =

(2 + 1)



y∈Λ

Nε



v∈V



z∈Z

p(z, v)z

(η(0, v)[1 − η(z, v)])

−



v∈V



(0)),

and let

G,2

Nε

(s, η) =



v∈V



x∈D



i=1



j,k=0

∂

(x/N )∂

(x/N ) ×



η(x, v)[1 − η(x + e

, v)] + η(x, v)[1 − η(x − e

, v)] − 2χ



(0))



Let, again, G ∈ C(Ω

) × [C(D

)]

, and consider the quantities

−

(s, η, G) =

d−1



k=0



˜x∈T

d−1

(s, ˜x/N )



(η

(1,˜x)

(s)) −



v∈V

(˜x/N )



(s, η, G) =

d−1



k=0



˜x∈T

d−1

(s, ˜x/N )



(η

(N−1,˜x)

(s)) −



v∈V

(˜x/N )



Proposition 4.5.2. Fix G ∈ C(Ω

) ×[C(D

)]

, H in C([0, T ] ×Γ) ×[C(Γ)]

, a cylinder function Ψ and

a sequence {η

; N ≥ 1} of conﬁgurations with η

in X

. For every δ > 0,

lim

ε→0

lim

N→∞

log P







G,j

Nε

(s, η

) ds



> δ



= −∞,

lim

N→∞







(s, η, G)



> δ



= −∞,

for j = 1, 2.

The proof of the above proposition follows from Proposition 4.5.1, the replacement lemmas proved

in [33], and the computation presented in [3], p. 78, for nonreversible processes.

For each ε > 0 and π in M

×M

, for k = 0, . . . , d, denote by Ξ

(π

) = π

the absolutely continuous

measure obtained by smoothing the measure π

(π

)(dx) = π

(dx) =

(Λ

(x))

|Λ

(x)|

dx ,

where Λ

(x) = {y ∈ D

; |y − x| ≤ ε}, |A| stands for the Lebesgue measure of the set A, and {U

; ε > 0}

is a strictly decreasing sequence converging to 1: U

> 1, U

> U



for ε > ε



, lim

ε↓0

= 1. Let

N,ε



(π

), Ξ

(π

), . . . , Ξ

(π

)



A simple computation shows that π

N,ε

belongs to M

for N suﬃciently large because U

> 1, and that

for each continuous function H : D

→ R

d+1

π

N,ε

, H =



x∈D

H(x/N ) · I

εN

(x) + O(N, ε) ,

where O(N, ε) is absolutely bounded by C

−1

+ ε} for some ﬁnite constant C

depending only on H.

For each H in C

1,2

(Ω

) consider the exponential martingale M

deﬁned by

= exp







, H



−



, H



−



−N

π





∂

+ N



π





Recall from subsection 2.2 the deﬁnition of the functional

. An elementary computation shows that

= exp





(π

N,ε

) + V

N,ε

+ c

(ε) + c

−1

)



. (4.5.1)

In this formula,

N,ε

= −



G,1

Nε

(s, η) ds −



i=1



G,2

Nε

(s, η) ds

+ V

(s, η, ∂

H) − V

−

(s, η, ∂

H) + π

, H

 − γ, H

;

and c

: R

→ R, j = 1, 2, are functions depending only on H such that c

(δ) converges to 0 as δ ↓ 0.

In particular, the martingale M

is bounded by exp



C(H, T )N



for some ﬁnite constant C(H, T )

depending only on H and T . Therefore, Proposition 4.5.2 holds for P

= P

in place of P

4.5.2 Energy estimates

To exclude paths with inﬁnite energy in the large deviations regime, we need an energy estimate. We

state ﬁrst the following technical result.

Lemma 4.5.3. There exists a ﬁnite constant C

, depending on T , such that for every G in C

∞

(Ω

every integer 1 ≤ i ≤ d, 0 ≤ k ≤ d, and every sequence {η

; N ≥ 1} of conﬁgurations with η

in X

lim

N→∞

log E



exp





dt π

N,k

, ∂

G



≤ C



1 +



G





The proof of this proposition follows from Lemma 3.8 in [33], and the fact that dδ

/dν

≤ C

for some positive constant C = C(κ).

For each G in C

∞

(Ω

) and each integer 1 ≤ i ≤ d, let

i,k

: D([0, T ], M

×M

) → R be the function

given by

i,k

(π) =



dt π

, ∂

 − C



du G(t, u)

Notice that

sup

G∈C

∞

(Ω

)



i,k

(π)



i,k

(π)

. (4.5.2)

Fix a sequence {G

; r ≥ 1} of smooth functions dense in L

([0, T ], H

)). For any positive integers

m, l, let

m,l



π ∈ D([0, T ], M

× M

); max

1≤j≤m

1≤i≤d

i,k

(π) ≤ l



Since, for ﬁxed G in C

∞

(Ω

) and 1 ≤ i ≤ d integer, the function

i,k

is continuous, B

m,l

is a closed

subset of D([0, T ], M).

Lemma 4.5.4. There exists a ﬁnite constant C

, depending on T , such that for any positive integers

r, l and any sequence {η

; N ≥ 1} of conﬁgurations with η

in X

lim

N→∞

log Q



m,l

)



≤ −l + C

where k = 0, . . . , d.

Proof. For integers 1 ≤ k ≤ r and 1 ≤ i ≤ d, by Chebychev inequality and by Lemma 4.5.3,

lim

N→∞

log P



i,k

> l



≤ −l + C

Hence, from

lim

N→∞

log(a

+ b

) ≤ max



lim

N→∞

log a

, lim

N→∞

log b



, (4.5.3)

we obtain the desired inequality.

Lemma 4.5.5. There exists a ﬁnite constant C

, depending on T , such that for every G in C

∞

(Ω

) ×

∞

)]

, and every sequence {η

; N ≥ 1} of conﬁgurations with η

in X

lim

N→∞

log E



exp







i=1



k=0

dt π

N,k

, ∂





≤ C



1 +



G





In particular, we have that if (ρ, p) is the solution of (4.2.2), then

sup

G∈C

1,2

(Ω

)



i=1





du ∂

(ρ, p) · ∂

G −



v∈V



duχ

(ρ, p)[˜v ·∂



is ﬁnite, and vanishes if T → 0.

Proof. Applying Feynman-Kac’s formula and using the same arguments of Lemma 3.3 in [33], we have

that

log E





exp











i=1



k=0



x∈D

(η

(s)) − I

(η

x−e

(s)))∂

(s, x/N )











is bounded above by



ds,

where λ

is equal to

sup





i,k



x∈D

(η(x)) − I

(η(x − e

)))∂

(s, x/N ), f



+ N

< L



f >



where the supremum is taken over all densities f with respect to ν

. By Proposition 4.5.1, the expression

inside brackets is bounded above by

−

(f) +



i,k



x∈D



N∂

(s, x/N )



(η

) − I

(η

x−e

)]f(η)ν

(dη)



We now rewrite the term inside the brackets as



v∈V



i=1



x∈D





N(˜v ·∂

G(s, x/N ))[η(x, v) − η(x − e

, v)]f(η)ν

(dη)



Writing η(x, v) −η(x −e

, v) = η(x, v)[1 −η(x −e

, v)] −η(x −e

, v)[1 −η(x, v)], and applying the same

arguments in Lemma 3.8 of [33], we obtain that

N(˜v ·∂

G(s, x/N ))



[η(x, v) − η(x − e

, v)]f(η)ν

(dη)

≤ (˜v ·∂

G(s, x/N ))



η(x, v)[1 − η(x − e

, v)]f(η

x−e

,x,v

)dν



f(η

x−e

,x,v

)





1 −

x−e

, v

x,v



(dη)

+ N



[



f(η

x−e

,x,v

) −



f(η)]

(dη)

+ 2(˜v ·∂

G(s, x/N ))



η(x, v)[1 − η(x − e

, v)](



f(η) +



f(η

x−e

,x,v

))

(dη),

we have that (



f(η) +



f(η

x−e

,x,v

))

≤ 2(f(η) + f(η

x−e

,x,v

)). An application of the replacement

lemma (Lemma 3.7 in [33]) concludes the proof.

4.5.3 Upper Bound

Fix a sequence {F

; j ≥ 1} of smooth functions dense in C(D

) for the uniform topology, with positive

coordinates. For j ≥ 1 and δ > 0, let

j,δ



π ∈ D([0, T ], M

× M

); |π

, F

| ≤ ˘v

|V|



(x) dx + C

δ , k = 0, . . . , d, 0 ≤ t ≤ T



where ˘v

= 1 and ˘v

= ˘v, C

= ∇F



∞

and ∇F is the gradient of F . Clearly, the set D

j,δ

, j ≥ 1,

δ > 0, is a closed subset of D([0, T ], M

× M

). Moreover, if

m,δ



j=1

j,δ

we have that D([0, T ], M

) = ∩

n≥1

∩

m≥1

m,1/n

. Note, ﬁnally, that for all m ≥ 1, δ > 0,

N,ε

belongs to E

m,δ

for N suﬃciently large. (4.5.4)

Fix a sequence of conﬁgurations {η

; N ≥ 1} with η

in X

and such that π

(η

) converges to

γ(u)du in M

× M

. Let A be a subset of D([0, T ], M

× M

log P



∈ A



log E



)

−1

1{π

∈ A}



Maximizing over π

in A, we get from (4.5.1) that the last term is bounded above by

− inf

π∈A

(π

) +

log E



−N

N,ε



− c

(ε) − c

−1

) .

Since π

(η

) converges to γ(u)du in M

×M

and since Proposition 4.5.2 holds for P

= P

place of P

, the second term of the previous expression is bounded above by some C

(ε, N) such that

lim

ε→0

lim

N→∞

(ε, N) = 0 .

Hence, for every ε > 0, and every H in C

1,2

(Ω

lim

N→∞

log P

[A] ≤ − inf

π∈A

(π

) + C



(ε) , (4.5.5)

where lim

ε→0



(ε) = 0. Let

r,l



π ∈ D([0, T ], M

× M

); max

1≤j≤r

1≤i≤d



k=0

i,k

(π) ≤ l



and, for each H ∈ C

1,2

(Ω

), each ε > 0 and any r, l, m, n ∈ Z

, let J

r,l,m,n

H,ε

: D([0, T ], M

× M

) →

R ∪ {∞} be the functional given by

r,l,m,n

H,ε

(π) =



(π

) if π ∈ B

r,l

∩ E

m,1/n

+∞ otherwise .

This functional is lower semicontinuous because so is

◦Ξ

and because B

r,l

, E

m,1/n

are closed subsets

of D([0, T ], M

× M

Let O be an open subset of D([0, T ], M

× M

). By Lemma 4.5.4, (4.5.3), (4.5.4) and (4.5.5),

lim

N→∞

log Q

[O] ≤ max



lim

N→∞

log Q

[O ∩ B

r,l

∩ E

m,1/n

] ,

lim

N→∞

log Q

[(B

r,l

)

]



≤ max



− inf

π∈O∩B

r,l

∩E

m,1/n

(π

) + C



(ε) , −l + C



= − inf

π∈O

r,l,m,n

H,ε

(π) ,

where

r,l,m,n

H,ε

(π) = min



r,l,m,n

H,ε

(π) − C



(ε) , l − C



In particular,

lim

N→∞

log Q

[O] ≤ − sup

H,ε,r,l,m,n

inf

π∈O

r,l,m,n

H,ε

(π) .

Note that, for each H ∈ C

1,2

(Ω

), each ε > 0 and r, l, m, n ∈ Z

, the functional L

r,l,m,n

H,ε

is lower

semicontinuous. Then, by Lemma A2.3.3 in [23], for each compact subset K of D([0, T ], M

× M

lim

N→∞

log Q

[K] ≤ − inf

π∈K

sup

H,ε,r,l,m,n

r,l,m,n

H,ε

(π) .

By (4.5.2) and since D([0, T ], M

) = ∩

n≥1

∩

m≥1

m,1/n

lim

ε→0

lim

l→∞

lim

r→∞

lim

m→∞

lim

n→∞

r,l,m,n

H,ε

(π) =



(π) if Q(π) < ∞ and π ∈ D([0, T ], M

) ,

+∞ otherwise .

This result and the last inequality imply the upper bound for compact sets because

and J

coincide

on D([0, T ], M

). To pass from compact sets to closed sets, we have to obtain exponential tightness

for the sequence {Q

}. This means that there exists a sequence of compact sets {K

; n ≥ 1} in

D([0, T ], M

× M

) such that

lim

N→∞

log Q

) ≤ −n .

The proof presented in [2] for the non interacting zero range process is easily adapted to our context.

4.5.4 Lower Bound

The proof of the lower bound is similar to the one in the convex periodic case. We just sketch it and

refer to [23], Section 10.5. Fix a path π in Π and let H ∈ C

1,2

(Ω

) be such that π is the weak solution

of equation (4.3.9). Recall from the previous section the deﬁnition of the martingale M

and denote by

the probability measure on D([0, T ], X

) given by P

[A] = E

1{A}]. Under P

and for

each 0 ≤ t ≤ T , the empirical measure π

converges in probability to π

. Further,

lim

N→∞







= I

(π|γ) ,

where H(µ|ν) stands for the relative entropy of µ with respect to ν. From these two results we can

obtain that for every open set O ⊂ D([0, T ], M

× M

) which contains π,

lim

N→∞

log P





≥ −I

(π|γ) .

The lower bound follows from this and the I

(·|γ)-density of Π established in Theorem 4.4.5.

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