Scaling limits: d-dimensional models with conductances, velocity, reservatories and random environment

Transcrição

1 Instituto acional de Matemática Pura e Aplicada Scaling its: d-dimensional models with conductances, velocity, reservatories and random environment Fábio Júlio da Silva Valentim Tese apresentada para obtenção do título de Doutor em Ciências Orientador: Claudio Landim Rio de Janeiro Junho, 2

2 Resumo esta tese consideramos três modelos de processo de exclusão em dimensão d : Processo de Exclusão com Condutâncias, com Condutâncias em Ambiente Aleatório e com Bordos e Velocidades. Para o primeiro, obtemos o ite hidrodinâmico, no segundo obtemos ite hidrodinâmico e as flutuações no equilíbrio, e no último provamos o princípio dos grandes desvios. Keywords: Exclusion Processes, Boundary Driven Exclusion Processes, Hydrodynamic Limit, Equilibrium Fluctuations, Large Deviations, Conductances, Random Environment, Homogenization.

3 i Aos amores de minha vida: minha mãe, minha esposa Tatiana, a meus filhos Arthur (Tuco) e Estela (Teia) e em memória do meu pai, meu grande herói.

4 Agradecimentos Agradeço a Deus por ter me abençoado com a intensidade de virtudes superior aos meus defeitos de modo que consegui finalizar com sucesso o programa de Doutorado. Isto também, somente foi possível, pela força, amizade, companheirismo e incentivo que inúmeras pessoas sempre depositaram em mim durante toda a minha formação acadêmica. Em especial, destaco a minha amada e companheira esposa Tatiana, minha querida mãe Maria da Penha e ao estimado amigo professor Carlos Roberto Alves dos Santos, meu sincero muito obrigado. Agradeço a meu orientador de Mestrado e Doutorado, professor Claudio Landim, pelas oportunidades, os ensinamentos, paciência, pela segurança e competência com que conduziu toda minha orientação. 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 - YU, de forma que tenho a agradecer a excelente recepção e suporte com que professores, alunos e funcionários deste renomado instituto me acolheram. Em especial, sou muito grato ao meu orientador no exterior, professor S.R.S. Varadhan, por ter me propiciado inesquecíveis e valiosos momentos de ensinamento de conduta como matemático e pessoa e ao amigo Antonio Carlos Auffinger (Tuca) pelo apoio e companheirismo desde os tempos de graduação. Agradeço aos parceiros de pesquisa Alexandre de Bustamante Simas e Jonathan Farfan que, efetivamente, contribuiram para o sucesso desta etapa de minha vida, dividimos momentos de descontração, concentração, tristeza e alegria. A matemática ficou mais prazerosa com as inúmeras e intermináveis discurssões no IMPA, em casa, nos ônibus, na praia,... Agradeço a todos os professores que contribuiram para minha formação. Em especial, destaco no CEFETES os professores Oscar Rezende. a UFES os professores José Gilvan, Valmecir Bayer, Ademir Sartim, Luzia Casati e Luiz Fernando Camargo (também orientador de Iniciação Científica, que além do rigor matemático, me instruía em minha formação pessoal), sou muito grato. o IMPA, os professores Fernado Codá, Carlos Gustavo Moreira (Gugu), Manfredo do Carmo, Milton Jara que com sua brilhante capacidade de conduzir as aulas tornavam a Matemática mais bela e inspiradora. Agradeço aos funcionários e alunos do IMPA que, durante minha estada, fizeram com que o ambiente de trabalho fosse agradável e prazeroso. De fato, um local que fiz muitas amizades. Em especial, agradeço a Alexandre Oliveira (Xandão), Fátima Russo, José Paulo Fahl (Paulinho), Fábio do Santos, aos funcionários 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. Agradeço aos professores Leandro Pimentel, Rob Morris, Serguei Popov e Vladas Sidoravicius por terem aceitado participar de minha banca de defesa de tese contribuindo com sugestões, críticas e elogios. Agradeço ao CPq pelo financiamento de minha bolsa de doutorado no país e no exterior. ii

5 Contents Introduction Exclusion process with conductances 3. otation and Results The hydrodynamic equation The operator L W Some remarks on the one-dimensional case The d-dimensional case Random walk with conductances Discrete approximation of the operator L W Semigroups and resolvents Scaling it Tightness Uniqueness of it points Replacement lemma Energy estimate W -Sobolev spaces W -Sobolev spaces The W -Sobolev space Approximation by smooth functions and the energetic space A Rellich-Kondrachov theorem The space H W (Td ) W -Generalized elliptic equations W -Generalized parabolic equations Uniqueness of weak solutions of the parabolic equation W -Generalized Sobolev spaces: Discrete version Connections between the discrete and continuous Sobolev spaces Homogenization H-convergence Random environment Homogenization of random operators Hydrodynamic it of processes with conductances in random environment The exclusion processes with conductances in random environments The hydrodynamic equation Tightness Uniqueness of it points Equilibrium fluctuations otation and results The space S W (T d ) Equilibrium Fluctuations Martingale Problem Generalized Ornstein-Uhlenbeck Processes iii

6 3.4 Tightness Boltzmann-Gibbs Principle Appendix: Stochastic differential equations on nuclear spaces Countably Hilbert nuclear spaces Stochastic differential equations Dynamical Large Deviations otation and Results The boundary driven exclusion process Mass and momentum Dynamical large deviations Hydrodynamics The rate function I T ( γ) I T ( γ)-density Large deviations Superexponential estimates Energy estimates Upper Bound Lower Bound iv

7 Introduction O ite hidrodinâmico permite obter uma descrição das características termodinâmicas (por exemplo, temperatura, densidade, pressão) de sistemas infinitos assumindo que a dinâmica das partículas é estocástica. Seguindo a abordagem da mecânica estatística introduzida por Boltzmann, deduzimos o comportamento macroscópico de um sistema a partir da iteração microscópica entre as partículas. Considera-se a dinâmica microscópica consistindo de caminhos aleatórios sobre um grafo submetida a alguma iteração local, denominado sistema de partículas interagentes introduzido por Spitzer [36], veja também [24]. Ademais, esta abordagem justifica rigorosamente um método algumas vezes utilizado pelos físicos para estabelecer equações diferenciais parciais que descrevem a evolução de características termodinâmicas de um fluido. Assim, a existência de soluções fracas de tais EDPs podem ser vistas como um dos objetivos do ite hidrodinâmico. Um conhecido sistema de partículas interagentes é o processo de exclusão simples. Informalmente é um processo onde apenas uma partícula por sitio é permitida (dai o nome exclusão), e o salto das partículas somente ocorrem para os vizinhos próximos. esta tese consideramos o processo de exclusão simples sobre o toro discreto d-dimensional, T d, e obtemos o comportamento hidrodinâmico nos seguintes modelos: o capítulo, consideramos o processo de exclusão com condutâncias induzida por uma classe de funções W e obtemos que, sobre uma escala difusiva, a evolução das densidades empíricas do processo de exclusão sobre o toro d-dimensional, T d, é descrita pela única solução fraca da equação diferencial parcial generalizada não-linear t ρ = xk Wk Φ(ρ), (..) k= Onde a função Φ : [l, r] R é fixada e suave, definida sobre um intervalo [l, r] de R. Esta função está associada a um fator na taxa de salto das partículas no processo microscópico e depende das configurações do sistema. O adjetivo generalizada decorre do termo Wk cuja definição e referências são dadas na Seção.2. Em Particular, se considerarmos W k (x) = x k, obtemos que (..) é a equação do calor. Para a prova do ite hidrodinâmico, nós também obtemos algumas propriedades do operador elíptico do lado direito de (..). Ultimamente, a evolução de processos de exclusão uni-dimensional com condutâncias tem atraído atenção [3, 4, 8, 2]. Um dos propósitos desta tese é estender esta análise para dimensões maiores. Este processo pode, por exemplo, modelar difusões de partículas em um meio com membranas permeáveis, nos pontos de descontinuidade de W, tendendo a refletir partículas, criando espaços de descontinuidade nos perfis de densidade. as primeiras linhas do capítulo, encontra-se uma detalhamento maior desta aplicação e da real conexão deste operador com os famosos operadores diferenciais de Féller. o capítulo 2, consideramos um processo de exclusão com condutâncias em ambiente aleatório e obtemos o ite hidrodinâmico. A condutância é a mesma considerada no capítulo, no entanto a novidade neste capítulo não se resume ao ambiente aleatório. Isto porque a prova do comportamento hidrodinâmico no capítulo é baseada em estimativas do semigrupo e resolventes entre o processo original e um corrigido. O elo entre o casos d = [8, 4] e d é então estabelecido via uma classe especial de funções W, a saber: W (x,..., x d ) = W k (x k ) x R, k= onde cada W k é da forma considerada em [8]. Enquanto no capítulo 2, usando as propriedades obtidas do operador elíptico em (..), construímos o espaço W -Sobolev, o qual consiste das funções f tendo

8 gradiente generalizado fraco W f = ( W f,..., Wd f). Obtemos varias propriedades para este espaço, que são análogas aos clássicos resultados para espaços de Sobolev. Equações W -generalizada elíptica e parabólica são consideradas, alcançando resultados de existência e unicidade de soluções fracas para estas equações. Resultados de homogenização para uma classe de operadores aleatórios são investigados, finalmente, como primeira aplicação desta teoria desenvolvida, nos provamos o ite hidrodinâmico para o processo em questão. Em particular, substituindo a análise de semigrupos e resolventes feita no capítulo, por homogenização. A motivação para este enfoque foi o artigo [2]. ele os autores consideram um processo de exclusão gradiente em ambiente aleatório e usam a teoria de homogenização, desenvolvida em [3], para obterem o ite hidrodinâmico e flutuações. o capítulo 3, nos obtemos as flutuações do equilíbrio para o processo considerado no capítulo 2. Esta foi a segunda aplicação da teoria previamente desenvolvida. os obtemos que a distribuição empírica é governada pela única solução de uma equação diferencial estocástica, tomando valores em um certo espaço Frechet uclear. o capítulo 4, nos provamos os grandes desvios dinâmicos para um processo boundary driven, i.e. um sistema que possui dois reservatórios infinitos de partículas na fronteira com partículas que podem ter diferentes velocidades. Este resultado baseia-se na recente abordagem introduzida em [5]. Cada capítulo desta tese resultou em um artigo, os quais salvo alguns cortes para evitar excessivas repetições, são os próprios artigos. Em particular cada início de capítulo tem uma pequena introdução que complementa esta. Ressalto que o capítulo 2 é um trabalho conjunto com Alexandre Bustamante de Simas e os capítulos 3 e 4 são em parceria com Jonathan Farfan e Alexandre Bustamante de Simas. 2

9 Chapter Hydrodynamic it of a d-dimensional exclusion process with conductances The evolution of one-dimensional exclusion processes with random conductances has attracted some attention recently [2, 3, 4, 8]. The purpose of this chapter is to extend this analysis to higher dimension. Let W : R d R be a function such that W (x,..., x d ) = d k= W k(x k ), where d and each function W k : R R is strictly increasing, right continuous with left its (càdlàg), and periodic in the sense that W k (u + ) W k (u) = W k () W k () for all u R. Informally, the exclusion process with conductances associated to W is an interacting particle systems on the d-dimensional discrete torus T d, 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 j 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 t ρ = xk Wk Φ(ρ), (..) k= where Φ is a smooth function, strictly increasing in the range of ρ, and such that < b Φ b. Furthermore, we denote by Wk the generalized derivative with respect to W k, see [8, 8] and a revision in Section.2. The partial differential equation (..) appears naturally as, for instance, scaling its 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 it relies strongly on some properties of the differential operator d k= x k Wk presented in Theorem..2. We prove, among other properties: that the operator d k= x k Wk, 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 complete 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, [6, 7, 26]). The operator (d/dx)(d/du), considered in [8], is the formal adjoint of (d/du)(d/dv) in the particular case v(x) = x (as in [7]). 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. we state the main results of the chapter; in Section.2 we prove the main properties of the operator L W = d k= x k Wk ; in Section.3 we prove the convergence of random walks with random conductances to Markov processes with generator given 3

10 by L W ; in Section.4 we prove the scaling it of the exclusion process with conductances given by W ; and, finally, in Section.5 we show that the unique solution of (..) has finite energy.. otation and Results We examine the hydrodynamic behavior of a d-dimensional exclusion process, with d, with conductances induced by a special class of functions W : R d R such that: W (x,..., x d ) = W k (x k ) (..) where W k : R R are strictly increasing right continuous functions with left its (càdlàg), and periodic in the sense that W k (u + ) W k (u) = W k () W k () for all u R and k =,..., d. To keep notation simple, we assume that W k vanishes at the origin, that is, W k () =. Denote by T d = [, ) d the d-dimensional torus and by e,..., e d the canonical basis of R d. For this class of functions we have: W () = ; W is strictly increasing on each coordinate: for all j d, a >, x R d ; W is continuous from above: k= W (x + ae j ) > W (x) W (x) = where we say that y x if y j x j for all j d; W is defined on the torus T d : W (y), y x, y x W (x,..., x j,, x j+,..., x d ) = W (x,..., x j,, x j+,..., x d ) W (e j ), for all j d, (x,..., x j, x j+,..., x d ) T d. Unless explicitly stated W belongs to this class. Let T d = (Z/Z)d = {,..., } d be the d-dimensional discrete torus with points. Distribute particles throughout T d in such a way that each site of T d is occupied at most by one particle. Denote by η the configurations of the state space {, } Td, so that η(x) = if site x is vacant and η(x) = if site x is occupied. Fix b > /2 and W. For x = (x,..., x d ) T d let where all sums are modulo, and let c x,x+ej (η) = + b{η(x e j ) + η(x + 2 e j )}, ξ x,x+ej = [W ((x + e j )/) W (x/)] = [W j ((x j + )/) W j (x j /)]. We now describe the stochastic evolution of the process. Let x = (x,..., x d ) T d. At rate ξ x,x+ej c x,x+ej (η) the occupation variables η(x), η(x + e j ) are exchanged. If W is differentiable at x/ [, ) d, the rate at which particles are exchanged is of order for each direction, but if some W j is discontinuous at x j /, it no longer holds. In fact, assume, to fix ideas, that W j is discontinuous at x j /, and smooth on the segments (x j /, x j / + εe j ) and (x j / εe j, x j /). Assume, also, that W k is differentiable in a neighborhood of x k / for k j. In this case, the rate at which particles jump over the bonds {y e j, y}, with y j = x j, is of order /, whereas in a neighborhood of size of these bonds, 4

11 particles jump at rate. Thus, note that a particle at site y e j jumps to y at rate / and jumps at rate to each one of the 2d other options. Particles, therefore, tend to avoid the bonds {y e j, y}. However, since time will be scaled diffusively, and since on a time interval of length 2 a particle spends a time of order at each site y, particles will be able to cross the slower bond {y e j, y}. Then, this process models membranes that obstruct passages of particles. ote that these membranes are (d )-dimensional hyperplanes embedded in a d-dimensional environment. Moreover, if we consider W j having more than one discontinuity point for more than one j, these membranes will be more sophisticated manifolds, for instance, unions of (d )-dimensional boxes. The effect of the factor c x,x+ej (η) is the following: if the parameter b is positive, the presence of particles in the neighboring sites of the bond {x, x + e j } 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 : {, } Td R as L f(η) = ξ x,x+ej c x,x+ej (η) {f(σ x,x+ej η) f(η)}, (..2) j= where σ x,x+ej η is the configuration obtained from η by exchanging the variables η(x) and η(x + e j ): η(x + e j ) if y = x, (σ x,x+ej η)(y) = η(x) if y = x + e j, (..3) η(y) otherwise. A straightforward computation shows that the Bernoulli product measures {να : α } are invariant, and in fact reversible, for the dynamics. The measure να is obtained by placing a particle at each site, independently from the other sites, with probability α. Thus, να is a product measure over {, } Td with marginals given by να {η : η(x) = } = α, for x in T d. For more details see [23, chapter 2]. We will often omit the index on ν α. Denote by {η t : t } the Markov process on {, } Td associated to the generator L speeded up by 2. Let D(R +, {, } Td ) be the path space of càdlàg trajectories with values in {, } T d. For a measure µ on {, } Td, denote by Pµ the probability measure on D(R +, {, } Td ) induced by the initial state µ, and the Markov process {η t : t }. Expectation with respect to P µ is denoted by E µ... The hydrodynamic equation Fix W = d k= W k as in (..). In [8] it was shown that there exist self-adjoint operators L Wk : D Wk L 2 (T) L 2 (T). The domain D Wk is completely characterized in the following proposition: Proposition... The domain D Wk consists of all functions f in L 2 (T) such that y f(x) = a + bw k (x) + W k (dy) f(z) dz (,x] for some function f in L 2 (T) that satisfies f(z) dz = and (,] { W k (dy) b + y } f(z) dz =. The proof and further details can be found in [8]. Further, the set A Wk forms a complete orthonormal system in L 2 (T). Let of the eigenvectors of L Wk d A W = {f : T d R; f(x,..., x d ) = f k (x k ), f k A Wk, k =,..., d}, (..4) 5 k=

12 and denote by span(a) the space of finite linear combinations of the set A, and let D W := span(a W ). Define the operator L W : D W L 2 (T d ) as follows: for f = d k= f k A W, we have L W (f)(x,... x d ) = d k= j=,j k f j (x j )L Wk f k (x k ), (..5) and then extend to D W by linearity. Lemma.2.2, in Section.2, shows that: L W is symmetric and non-positive; D W is dense in L 2 (T d ); and the set A W forms a complete, orthonormal, countable system of eigenvectors for the operator L W. Let A W = {h k } k, {α k } k be the corresponding eigenvalues of L W, and consider D W = {v = v k h k L 2 (T d ); vkα 2 k 2 < + }. (..6) k= Define the operator L W : D W L 2 (T d ) by L W v = + k= k= α k v k h k (..7) The operator L W is clearly an extension of the operator L W, and we present in Theorem..2 some properties of this operator. Theorem..2. The operator L W : D W L 2 (T d ) enjoys the following properties: (a) The domain D W is dense in L 2 (T d ). In particular, the set of eigenvectors A W = {h k } k forms a complete orthonormal system; (b) The eigenvalues of the operator L W form a countable set {α k } k. All eigenvalues have finite multiplicity, and it is possible to obtain a re-enumeration {α k } k such that = α α and n α n = ; (c) The operator I L W : D W L 2 (T d ) is bijective; (d) L W : D W L 2 (T d ) is self-adjoint and non-positive: (e) L W is dissipative. L W f, f ; In view of (a), (b) and (d), we may use Hille-Yosida theorem to conclude that L W is the generator of a strongly continuous contraction semigroup {P t : L 2 (T d ) L 2 (T d ) } t. Denote by {G λ : L 2 (T d ) L 2 (T d ) } λ> the semigroup of resolvents associated to the operator L W : G λ = (λ L W ). G λ can also be written in terms of the semigroup {P t ; t }: G λ = e λt P t dt. In Section.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 thermodynamical quantities of the model in a fluid. A sequence of probability measures {µ : } on {, } Td is said to be associated to a profile ρ : T d [, ] if µ H(x/)η(x) H(u)ρ (u)du > δ = (..8) 6

13 for every δ >, and every continuous function H : T d R. For details, see [23, chapter 3]. For a positive integer m, denote by C m (T d ) the space of continuous functions H : T d 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, < B Φ (x) B, for all x [l, r]. Let γ : T d [l, r] be a bounded density profile, and consider the parabolic differential equation { t ρ = L W Φ(ρ). (..9) ρ(, ) = γ( ) A bounded function ρ : R + T d [l, r] is said to be a weak solution of the parabolic differential equation (..9) if ρ t, G λ H γ, G λ H = t Φ(ρ s ), L W G λ H ds for every continuous function H : T d R, all t > and all λ >. Existence of these weak solutions follows from tightness of the sequence of probability measures Q W, µ introduced in Section.4. The proof of uniquenesses of weak solutions is analogous to [8]. Theorem..3. Fix a continuous initial profile ρ : T d [, ], and consider a sequence of probability measures µ on {, } Td associated to ρ, in the sense of (..8). Then, for any t, P µ H(x/)η t (x) H(u)ρ(t, u) du > δ = for every δ > and every continuous function H. Here, ρ is the unique weak solution of the non-linear equation (..9) with l =, r =, γ = ρ, and Φ(α) = α + aα 2. Remark..4. As noted in [8, remark 2.3], the specific form of the rates c x,x+ei 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 i ); 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+ei to obtain any polynomial Φ of the form Φ(α) = α + 2 j m a jα j, m, with + 2 j m ja j >. Let, for instance, m = 3. Then the rates ĉ x,x+ei (η) = c x,x+ei (η) + b {η(x 2e i )η(x e i ) + η(x e i )η(x + 2e i ) + η(x + 2e i )η(x + 3e i )}, satisfy the above three conditions, where c x,x+ei is the rate defined at the beginning of Section 2 and b, b are such that + 2b + 3b >. An elementary computation shows that Φ(α) = α + bα 2 + b α 3. In Section.5 we prove that any it point Q W of the sequence QW, µ is concentrated on trajectories ρ(t, u)du, with finite energy in the following sense: for each j d, there is a Hilbert space L 2 x j W j, associated to W j, such that t ds d Φ(ρ(s,.)) 2 x dw j W j <, j where. xj W j is the norm in L 2 x j W j, and d/dw j is the derivative, which must be understood in the generalized sense..2 The operator L W The operator L W : D W L 2 (T d ) L 2 (T d ) is a natural extension, for the d-dimensional case, of the self-adjoint operator obtained for the one-dimensional case in [8]. We begin by presenting one of the main results obtained in [8], and we then present the necessary modifications to conclude similar results for the d-dimensional case. 7

14 .2. Some remarks on the one-dimensional case Let T R be the one-dimensional torus. Denote by, the inner product of L 2 (T): f, g = f(u) g(u) du. T Let W : R R be a strictly increasing right continuous function with left its (càdlàg), and periodic in the sense that W (u + ) W (u) = W () W () for all u in R. Let D W be the set of functions f in L 2 (T) such that y f(x) = a + bw (x) + W (dy) f(z) dz, for a, b R and some function f in L 2 (T) that satisfies: f(z) dz =, (,] (,x] ( W (dy) b + Define the operator L W : D W L 2 (T) by L W f = f. Formally y ) f(z) dz =. L W f = d dx d dw f, (.2.) where the generalized derivative d/dw is defined as if the above it exists and is finite. df f(x + ɛ) f(x) (x) = dw ɛ W (x + ɛ) W (x), (.2.2) Theorem.2.. Denote by I the identity operator in L 2 (T). The operator L W : D W L 2 (T) enjoys the following properties: (a) D W is dense in L 2 (T); (b) The operator I L W : D W L 2 (T) is bijective; (c) L W : D W L 2 (T) is self-adjoint and non-positive: L W f, f ; (d) L W is dissipative i.e., for all g D W and λ >, we have λg (λi L W )g ; (e) The eigenvalues of the operator L W form a countable set {λ n : n }. All eigenvalues have finite multiplicity, = λ λ, and n λ n = ; (f) The eigenvectors {f n } n of the operator L W form a complete orthonormal system. The proof can be found in [8]..2.2 The d-dimensional case Consider W as in (..). Let A Wk be the countable complete orthonormal system of eigenvectors of the operator L Wk : D Wk L 2 (T) R given in Theorem.2.. Let A W be as in (..4), and let the operator L W : D W := span(a W ) L 2 (T d ) be as in (..5). By Fubini s theorem, the set A W is orthonormal in L 2 (T d ), and the constant functions are eigenvectors of the operator L Wk. Moreover, A Wk A W, in the sense that f k (x,..., x d ) = f k (x k ), f k A Wk. 8

15 By (.2.), the operators L Wk can be formally extended to functions defined on T d as follows: given a function f : T d R, we define L Wk f as L Wk f = xk Wk f, (.2.3) where the generalized derivative Wk is defined by f(x,..., x k + ɛ,..., x d ) f(x,..., x k,..., x d ) Wk f(x,..., x k,..., x d ) =, (.2.4) ɛ W k (x k + ɛ) W k (x k ) if the above it exists and is finite. Hence, by (..5), if f D W L W f = L Wk f. (.2.5) k= ote that if f = d k= f k, where f k A Wk is an eigenvector of L Wk associated to the eigenvalue λ k, then f is an eigenvector of L W, with eigenvalue d k= λ k. Lemma.2.2. The following statements hold: (a) The set D W is dense in L 2 (T d ); (b) The operator L W : D W L 2 (T d ) is symmetric and non-positive: L W f, f. Proof. The strategy to prove the above Lemma is the following. We begin by showing that the set is dense in S = span({f L 2 (T d ); f(x,..., x d ) = S = span({f L 2 (T d ); f(x,..., x d ) = d f k (x k ), f k D Wk }) k= d f k (x k ), f k L 2 (T)}). We then show that D W is dense in S. Since S is dense in L 2 (T d ), 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 d k= f k L 2 (T d ), where f k D Wk, by functions of D W. By Theorem.2., the set D Wk is dense in L 2 (T), thus, there exists a sequence (fn) k n converging to f k in L 2 (T). Thus, let d f n (x,..., x d ) = fn(x k k ). By the triangle inequality and Fubini s theorem, the sequence (f n ) converges to d k= f k. Fix ɛ >, and let d h(x,..., x d ) = h k (x k ), h k D Wk. k= Since, for each k =..., d, A Wk D Wk is a complete orthonormal set, there exist sequences gj k A W k, and αj k R, such that n(k) h k αj k gj k L2 (T) < δ, j= where δ = ɛ/dm d and M := + sup k=:n h k. Let g(x,..., x d ) = k= k= n(k) d αj k gj k (x k ) D W. k= j= 9

16 An application of the triangle inequality, and Fubini s theorem, yields h g < ɛ. This proves (a). To prove (b), let f(x,..., x d ) = d f k (x k ) and g(x,..., x d ) = k= be functions belonging to A W. We have that f, L W g = d f k, k= d k= j=,j k g j L Wk g k = d d g k (x k ) k= k= j=,j k f j g j, f k L Wk g k, where, denotes the inner product in L 2 (T d ). Since, by Theorem.2., L Wk is self-adjoint, we have d f j g j, g k L Wk f k = L W f, g. k= j=,j k In particular, the operator L Wk is non-positive, and, therefore, d f, L W f = fj 2, f k L Wk f k. k= j=,j k Item (b) follows by linearity. Lemma.2.2 implies that the set A W forms a complete, orthonormal, countable, system of eigenvectors for the operator L W. Let L W : D W L 2 (T d ) be the operator defined in (..7). The operator L W is clearly an extension of the operator L W. Formally, by (.2.5), L W f = L Wk f, (.2.6) k= where L Wk f = xk Wk f. We are now in conditions to prove Theorem..2. Proof of Theorem..2. By Lemma.2.2, D W is dense in L 2 (T d ). Since D W D W, we conclude that D W is dense in L 2 (T d ). If α k are eigenvalues of L W, we may find eigenvalues λ j, associated to some f j A Wj, such that α k = d j= λ j. By item (e) of Theorem.2., (b) follows. Let {α k } k be the set of eigenvalues of L W. Then, the set of eigenvalues of I L W is {γ k } k, where γ k = α k +, and the eigenvectors are the same as the ones of L W. By item (b), we have Thus, I L W is injective. For = γ γ and n γ n =. v = + k= v k h k L 2 (T d ), such that vk 2 < +, k= let u = + k= v k γ k h k. Then u D W and (I L W )u = v. Hence, item (c) follows.

17 Let L W : D W L2 (T d ) L 2 (T d ) be the adjoint of L W. Since L W is symmetric, we have D W D W. So, to show the equality of the operators it suffices to show that D W D W. Given ϕ = + k= ϕ k h k D W, let L W ϕ = ψ L 2 (T d ). Therefore, for all v = + k= v kh k D W, Hence In particular, v, ψ = v, L W ϕ = L W v, ϕ = + k= ψ = + k= α k ϕ k h k. + k= α 2 kϕ 2 k < + and ϕ D W. α k v k ϕ k. Thus, L W is self-adjoint. Let v = + k= v kh k D W. From item (b), α k, and L W v, v = + k= α k v 2 k. Therefore L W is non-positive, and item (d) follows. Fix a function g in D W, λ >, and let f = (λi L W )g. Taking inner product, with respect to g, on both sides of this equation, we obtain λ g, g + L W g, g = g, f g, g /2 f, f /2. Since g belongs to D W, by (d), the second term on the left hand side is non-negative. Thus, λg f = (λi L W )g..3 Random walk with conductances Recall the decomposition obtained in (.2.6) for the operator L W. In next subsection, we present the discrete version L of L W and we describe, informally, the Markovian dynamics generated by L..3. Discrete approximation of the operator L W Consider the random walk {Xt } t in Td, which jumps from x/ (resp. (x + e j)/) to (x + e j )/ (resp. x/) with rate 2 ξ x,x+ej = /{W j ((x j + )/) W j (x j /)}. The generator L of this Markov process acts on local functions f : Td R as L f(x/) = j= L j f(x/), (.3.) where L j f(x/) = 2{ ξ x,x+ej [f((x + e j )/) f(x/)] + ξ x ej,x[f((x e j )/) f(x/)] }. ote that L j f(x/) is, in fact, a discrete version of the operator L W j. The counting measure m on T d is reversible for this process. The following estimate is a key ingredient for proving the results in Section.4:

18 Lemma.3.. Let f be a function on Td. Then, for each j =,..., d: ( L j f(x/) ) 2 ( L f(x/)) 2. Proof. Let X be the linear space of functions f on T d over the field R. ote that the dimension of X is. Denote by, the following inner product in X : f, g = f(x/)g(x/). For each j =,..., d, consider the linear operators L j on X (i.e., d = ) given by where x and W j are the difference operators: L j f = x W j f, x f(x/) = [f((x + )/) f(x/)] W j f(x/) = f((x + )/) f(x/) W j ((x + )/) W j (x/). and The operators L j are symmetric and non-positive. In fact, a simple computation shows that L j f, g = ( ) W j ((x + )/) W j (x/) W j f(x/) W j g(x/). x T Using the spectral theorem, we obtain an orthonormal basis A j = {hj,..., hj } of X formed by the eigenvectors of L j, i.e., L j hj i = αj i hj i and h j i, hj k = δ i,k, where δ i,k is the Kronecker s delta, which equals if i k, and equals if i = k. Since L j is non-positive, we have that the eigenvalues α j i are non-positive: αj i, j =,..., d and i =,...,. Let A = {φ,..., φ } X be set of functions of the form φ i (x,..., x d ) = d j= hj (x j ), with h j A j. Let α j be the eigenvalue of h j, i.e., L j hj = α j h j. The linear operator L on X, defined in (.3.), is such that L j φ i = α j φ i and L φ i = d j= αj φ i. Furthermore, if φ i (x,..., x d ) = d j= hj (x j ) and φ k (x,..., x d ) = d j= gj (x j ), φ i, φ k A, we have that φ i, φ k = d h j, g j = δ i,k, j= for i, k =,...,. So, the set A is an orthonormal basis of X formed by the eigenvectors of L and L j. In particular, for each f X, there exist β i R such that f = i= β iφ i. Thus, ( ) 2 L j f(x/) = L j f 2 = d Lj β i φ i 2 ( α j i )2 (β i ) 2 = L f 2 = d i= j= i= d = i= ( L f(x/)) 2, (α j i β i) 2 where α j i is the eigenvalue of the operator Lj associated to the eigenvector φ i. This concludes the proof of the lemma. 2

19 .3.2 Semigroups and resolvents. In this subsection we introduce families of semigroups and resolvents associated to the generators L and L W. We present some properties and results regarding the convergence of these operators. Denote by {Pt : t } (resp. {G λ : λ > }) the semigroup (resp. the resolvent) associated to the generator L, by {P,j t : t } the semigroup associated to the generator L j, by {P j t : t } the semigroup associated to the generator L Wj and by {P t : t } (resp. {G λ : λ > }) the semigroup (resp. the resolvent) associated to the generator L W. Since the jump rates from x/ (resp. (x + e j )/) to (x + e j )/ (resp. x/) are equal, Pt is symmetric: Pt (x, y) = Pt (y, x). Using the decompositions (.3.) and (.2.6), we obtain P t (x, y) = d j= By definition, for every H : T d R, P,j t (x j, y j ) and P t (x, y) = d P j t (x j, y j ). j= where I is the identity operator. G λ H = dt e λt P t H = (λi L W ) H, Lemma.3.2. Let H : T d R be a continuous function. Then + P t H(x/) P t H(x/) =. (.3.2) Proof. If H : T d R has the form H(x,..., x d ) = d j= H j(x j ), we have P t H(x) = d j= P,j t H j (x j ) and P t H(x) = d P j t H j (x j ). (.3.3) j= ow, for any continuous function H : T d R, and any ɛ >, we can find continuous functions H j,k : T R, such that H : T d R, which is given by satisfies H H ɛ. Thus, H (x) = m j= k= d H j,k (x k ), P t H(x/) P t H(x/) 2ɛ + P t H (x/) P t H (x/). By (.3.3) and similar identities for P t H and P,j t H, the sum on the right hand side in the previous inequality is less than or equal to m d x T d j= k= m x T d j= P,k t C j H j,k (x k /) k= d Pt k H j,k (x k /) k= P,k t H j,k (x k /) Pt k H j,k (x k /), 3

20 where C j is a constant that depends on the product d k= H j,k. The previous expressions can be rewritten as m j= C j k= m j= x T d C j k= i= P,k t H j,k (i/) P k t H j,k (i/) = i= P,k t H j,k (i/) Pt k H j,k (i/). Moreover, by [4, Lemma 4.5 item iii], when, the last expression converges to. Corollary.3.3. Let H : T d R be a continuous function. Then + G λ H(x/) G λ H(x/) =. (.3.4) Proof. By the definition of resolvent, for each, the previous expression is less than or equal to dt e λt Corollary now follows from the previous lemma. P t H(x/) P t H(x/). Let f : Td R be any function. Then, whenever needed, we consider f : Td R as the extension of f to T d given by: f(y) = f (x), if x T d, y x and y x <. Let H : T d R be a continuous function. Then the extension of Pt H : T d R to Td belongs to L (T d ), and by symmetry of the transition probability Pt (x, y) we have dupt H(u) = T H(x/). (.3.5) d x T d The next Lemma shows that H can be approximated by Pt obtain an approximation result involving the resolvent. H. As an immediate consequence, we Lemma.3.4. Let H : T d R be a continuous function. Then, t + P t H(x/) H(x/) =, (.3.6) and λ + + λg λ H(x/) H(x/) =. (.3.7) Proof. Fix ɛ >, and consider H as in the proof of Lemma.3.2. Thus, P t H(x/) H(x/) 2ɛ + where the second term on the right hand side is less than or equal to C sup j,k P,k t H j,k (x k /) H j,k (x k /), P t H (x/) H (x/), 4

21 with C being a constant that depends on H. By [4, Lemma 4.6], the last expression converges to, when, and then t. This proves the first equality. To obtain the second it, note that, by definition of the resolvent, the second expression is less than or equal to dtλe λt Pt H(x/) H(x/). By (.3.5) the sum is uniformly bounded in t and. Furthermore, it vanishes as and t. This proves the second part. Fix a function H : T d R. For λ >, let H λ = G λ H be the solution of the resolvent equation λh λ L H λ = H. (.3.8) Taking inner product on both sides of this equation with respect to Hλ,we obtain λ (H λ (x/)) 2 = H λ (x/)h(x/). H λ (x/)l H λ A simple computation shows that the second term on the left hand side is equal to j= ξ x,x+ej [,j H λ (x/)] 2, where,j H(x/) = [H((x + e j )/) H(x/)] is the discrete derivative of the function H in the direction of the vector e j. In particular, by Schwarz inequality,.4 Scaling it H λ (x/) 2 λ 2 j= ξ x,x+ej [,j H λ (x/)] 2 H(x/) 2 λ and H(x/) 2. (.3.9) Let M be the space of positive measures on T d with total mass bounded by one, and endowed with the weak topology. Recall that πt M stands for the empirical measure at time t. This is the measure on T d obtained by rescaling space by, and by assigning mass / to each particle: π t = η t (x) δ x/, (.4.) where δ u is the Dirac measure concentrated in u. For a continuous function H : T d R, π t, H stands for the integral of H with respect to π t : π t, H = H(x/)η t (x). This notation is not to be mistaken with the inner product in L 2 (T d ) introduced earlier. Also, when π t has a density ρ, π(t, du) = ρ(t, u)du, we sometimes write ρ t, H for π t, H. 5

22 For a local function g : {, } Zd R, let g : [, ] R be the expected value of g under the stationary states: g(α) = E να [g(η)]. For l and d-dimensional integer x = (x,..., x d ), denote by η l (x) the empirical density of particles in the box B l +(x) = {(y,..., y d ) Z d ; y i x i < l}: η l (x) = l d y B l + (x) η(y). Fix T >, and let D([, T ], M) be the space of M-valued càdlàg trajectories π : [, T ] M endowed with the uniform topology. For each probability measure µ on {, } Td, denote by Q W, µ the measure on the path space D([, T ], M) induced by the measure µ and the process πt introduced in (.4.). Fix a continuous profile ρ : T d [, ], and consider a sequence {µ : } of measures on {, } Td associated to ρ in the sense (..8). Further, we denote by Q W be the probability measure on D([, T ], M) concentrated on the deterministic path π(t, du) = ρ(t, u)du, where ρ is the unique weak solution of (..9) with γ = ρ, l k =, r k =, k =,..., d, and Φ(α) = α + aα 2. In subsection.4. we show that the sequence {Q W, µ : } is tight, and in subsection.4.2 we characterize the it points of this sequence..4. Tightness The proof of tightness of sequence {Q W, µ : } is motivated by [2, 8]. We consider, initially, the auxiliary M-valued Markov process {Π λ, t : t }, λ >, defined by Π λ, t (H) = πt, G λ H = ( G λ H ) (x/)η t (x), for H in C(T d ), where {G λ : λ > } is the resolvent associated to the random walk {X t : t } introduced in Section.3. We first prove tightness of the process {Π λ, t : t T } for every λ >, and we then show that {λπ λ, t : t T } and {πt : t T } are not far apart if λ is large. It is well-known [23, proposition 4..7] that to prove tightness of {Π λ, t : t T } it is enough to show tightness of the real-valued processes {Π λ, t (H) : t T } for a set of smooth functions H : T d R dense in C(T d ) for the uniform topology. Fix a smooth function H : T d R. Denote by the same symbol the restriction of H to T d. Let H, and keep in mind that Πλ, t (H) = πt, Hλ,λ. Denote by Mt the martingale defined by H λ = G λ M,λ t Clearly, tightness of Π λ, t functional t ds 2 L π s, H λ. by: = Π λ, t (H) Π λ, (H) x Z d t (H) follows from tightness of the martingale M,λ t ds 2 L π s, H λ. (.4.2) and tightness of the additive A simple computation shows that the quadratic variation M,λ t of the martingale M,λ t 2d j= In particular, by (.3.9), x T d ξ x,x+ej [,j H λ (x/)] 2 M,λ t C t 2d j= t c x,x+ej (η s ) [η s (x + e j ) η s (x)] 2 ds. ξ x,x+ej [(,j H λ )(x/)] 2 C(H)t λ, is given for some finite constant C(H) which depends only on H. Thus, by Doob inequality, for every λ >, δ >, [ ] P µ M,λ t > δ =. (.4.3) sup t T 6

23 In particular, the sequence of martingales {M,λ t : } is tight for the uniform topology. It remains to be examined the additive functional of the decomposition (.4.2). The generator of the exclusion process L can be decomposed in terms of generators of the random walks L j. By (.3.) and a long but simple computation, we obtain that 2 L π, Hλ is equal to j= { + b b (L j H λ )(x/) η(x) [ (L j H λ )((x + e j )/) + (L j H λ )(x/) ] (τ x h,j )(η) (L j H λ )(x/)(τ x h 2,j )(η) }, where {τ x : x Z d } is the group of translations, so that (τ x η)(y) = η(x + y) for x, y in Z d, and the sum is understood modulo. Also, h,j, h 2,j are the cylinder functions For all s < t T, we have h,j (η) = η()η(e j ), h 2,j (η) = η( e j )η(e j ). t s dr 2 L πr, Hλ ( + 3 b )(t s) j= L j H λ (x/), from Schwarz inequality and Lemma.3., the right hand side of the previous expression is bounded above by ( + 3 b )(t s)d ( ) 2. L Hλ (x/) Since Hλ is the solution of the resolvent equation (.3.8), we may replace L Hλ by U λ = λh λ H in the previous formula. In particular, It follows from the first estimate in (.3.9), that the right hand side of the previous expression is bounded above by dc(h, b)(t s) uniformly in, where C(H, b) is a finite constant depending only on b and H. This proves that the additive part of the decomposition (.4.2) is tight for the uniform topology and therefore that the sequence of processes {Π λ, t : } is tight. Lemma.4.. The sequence of measures {Q W, µ : } is tight for the uniform topology. Proof. It is enough to show that for every smooth function H : T R, and every ɛ >, there exists λ > such that [ ] P µ Π λ, t (λh) πt, H > ɛ =, sup t T since, in this case, tightness of πt follows from tightness of Π λ, t. Since there is at most one particle per site, the expression inside the absolute value is less than or equal to λh λ (x/) H(x/). By Lemma.3.4, this expression vanishes as and then λ..4.2 Uniqueness of it points We prove in this subsection that all it points Q of the sequence Q W, µ 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 (..9) with l = < r = and Φ(α) = α + aα 2. 7

24 Let Q be a it point of the sequence Q W, µ and assume, without loss of generality, that Q W, µ converges to Q. Since there is at most one particle per site, it is clear that Q is concentrated on trajectories π t (du) which are absolutely continuous with respect to the Lebesgue measure, π t (du) = ρ(t, u)du, and whose density ρ is non-negative and bounded by. Fix a continuously differentiable function H : T d R, and λ >. Recall the definition of the martingale M,λ t introduced in the previous section. By (.4.2) and (.4.3), for fixed < t T and δ >, [ π t ] QW, µ t, G λ H π, G λ H ds 2 L πs, G λ H > δ =. Since there is at most one particle per site, we may use Corollary.3.3 to replace G λ H by G λh in the expressions πt, G λ H, π, G λ H above. On the other hand, the expression 2 L πs, G λ H has been computed in the previous subsection. Since E να [h i,j ] = α 2, i =, 2 and j =,..., d, Lemma.3. and the estimate (.3.9), permit us use Corollary.4.4 to obtain, for every t >, λ >, δ >, i =, 2, P µ ε t ds { L j H λ (x/) τ x h i,j (η s ) [ η ε s (x) ] } 2 > δ =. Recall that L G λ H = λg λ H H. As before, we may replace G λ H by G λh. Let U λ = λg λ H H. Since ηs ε (x) = ε d πs ( d j= [x j/, x j / + εe j ]), we obtain, from the previous considerations, that ε QW, π, G λ H µ [ π t, G λ H t ds Φ ( ε d πs ( d [, + εe j ]) ), U λ > δ =. Since H is a smooth function, G λ H and U λ can be approximated, in L (T d ), by continuous functions. Since we assumed that Q W, µ converges in the uniform topology to Q, we have that [ Q πt, G λ H ε t ds Φ ( ε d π s ( j= π, G λ H d [, + εe j ]) ), U λ > δ =. j= Using the fact that Q is concentrated on absolutely continuous paths π t (du) = ρ(t, u)du, with positive density bounded by, ε d π s ( d j= [, + εe j]) converges in L (T d ) to ρ(s,.) as ε. Thus, [ πt t ] Q, G λ H π, G λ H ds Φ(ρ s ), L W G λ H > δ =, because U λ = L W G λ H. Letting δ, we see that, Q a.s., π t, G λ H π, G λ H = t ds Φ(ρ s ), L W G λ 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 t, we obtain that it holds for all 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 λ H n converges to G λ H in L (T d ), if H n converges to H in the uniform topology. Similarly, we can show that it holds for all λ >, since, for any continuous function H, G λn H converges to G λ H in L (T d ), as λ n λ. Proposition.4.2. As, the sequence of probability measures Q W, µ converges in the uniform topology to Q W. 8

25 Proof. In the previous subsection we showed that the sequence of probability measures Q W, µ is tight for the uniform topology. Moreover, we just proved that all it points of this sequence are concentrated on weak solutions of the parabolic equation (..9). The proposition now follows from a straightforward adaptation of the uniquenesses of weak solutions proved in [8] for the d-dimensional case. Proof of Theorem..3. Since Q W, µ converges in the uniform topology to Q W, a measure which is concentrated on a deterministic path. For each t T and each continuous function H : T d R, πt, H converges in probability to du ρ(t, u) H(u), where ρ is the unique weak solution of (..9) T with l k =, r k =, γ = ρ and Φ(α) = α + aα Replacement lemma We will use some results from [23, Appendix A]. Denote by H (µ ν α ) the relative entropy of a probability measure µ with respect to a stationary state ν α, see [23, Section A.8] for a precise definition. By the explicit formula given in [23, Theorem A.8.3], we see that there exists a finite constant K, depending only on α, such that H (µ ν α ) K, (.4.4) for all measures µ. Denote by, να the inner product of L 2 (ν α ) and denote by I ξ the convex and lower semicontinuous [23, Corollary A..3] functional defined by I ξ (f) = L f, f να, for all probability densities f with respect to ν α (i.e., f and fdν α = ). By [23, A..], an elementary computation shows that proposition I ξ (f) = I ξ x,x+e j (f), where j= { } 2 I ξ x,x+e j (f) = (/2) ξ x,x+ej c x,x+ej (η) f(σ x,x+e j η) f(η) dνα. By [23, Theorem A.9.2], if {St : t } stands for the semigroup associated to the generator 2 L, t H (µ St ν α ) I ξ (f s ) ds H (µ ν α ), where fs stands for the Radon-ikodym derivative of µ Ss with respect to ν α. Recall the definition of B l +(x) in begin of this section. For each y B l +(x), such that y > x, let Λ l x+e,y = (z y k ) k M(y) (.4.5) be a path from x + e to y such that:. Λ l x+e,y begins at x + e and ends at y, i.e.: z y = x + e and z y M(y) = y; 2. The distance between two consecutive sites of the Λ l x+e,y = (z y k ) k M(y) is equal to, i.e.: z y k+ = zy k + e j; for some j =..., d and for all k =,..., M(y) ; 3. Λ l x+e,y is injective: z y i zy j for all i < j M(y); 4. The path begins by jumping in the direction of e. Furthermore, the jump in the direction of e j+ is only allowed when it is not possible to jump in the direction of e j, for j =,..., d. 9