SCALING LIMITS FOR SLOWED EXCLUSION PROCESS

Transcrição

1 UIVERSIDADE FEDERAL DA BAHIA - UFBA ISTITUTO DE MATEMÁTICA - IM PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA - PGMAT TESE DE DOUTORADO SCALIG LIMITS FOR SLOWED EXCLUSIO PROCESS MARIAA TAVARES DE AGUIAR Salvador-Bahia Jaeiro de 29

2 SCALIG LIMITS FOR SLOWED EXCLUSIO PROCESS MARIAA TAVARES DE AGUIAR Tese de Doutorado apresetada ao Programa de Pós-Graduação em Matemática, da Uiversidade Federal da Bahia, como parte dos requisitos ecessários para a obteção do título de Doutora em Matemática. Orietador: Prof. Dr. Tertuliao Fraco Satos Fraco Salvador-Bahia Jaeiro de 29

3 Sistema Uiversitário de Bibliotecas da UFBA Tavares de Aguiar, Mariaa Scalig Limits for Slowed Exclusio Process / Mariaa Tavares de Aguiar. Salvador, f. : il Orietador: Prof. Dr. Tertuliao Fraco Satos Fraco. Tese Doutorado- Doutorado em Matemática Uiversidade Federal da Bahia, Istituto de Matemática, 29.. Sistema de Partículas. 2. Processo de Exclusão. 3. Limite Hidrodiâmico. 4. Flutuações. I. Fraco Satos Fraco, Tertuliao. II. Título.

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5 SCALIG LIMITS FOR SLOWED EXCLUSIO PROCESS MARIAA TAVARES DE AGUIAR Tese submetida ao corpo docete da pósgraduação e pesquisa do Istituto de Matemática da Uiversidade Federal da Bahia como parte dos requisitos ecessários para a obteção do grau de doutora em Matemática. Baca examiadora: Prof. Dr. Tertuliao Fraco Satos Fraco Orietador UFBA Prof. Dr. Dirk Erhard UFBA Prof. Dr. Paulo César Rodrigues Pito Varadas UFBA Prof. Dr. Claudio Ladim IMPA Prof. Dr. Milto David Jara Valezuela IMPA

6 Aos meus pais, meu marido Aderbal e ossa filha Moaa.

7 Agradecimetos A Deus por ter me dado saúde, força e pricipalmete por ter me eviado a beção que carrego o meu vetre, miha filha Moaa. Com certeza, essa reta fial, apesar de todas limitações que a gravidez ocasioa, tive esse presete ilumiado para me impulsar a fidar esse trabalho. Agradeço também aos meus pais, meus amados pais, por toda a dedicação, ateção e cuidado ao logo de toda a miha vida. Papis, que me mostra todo o camiho do bem, me esiado que a humildade e a hoestidade valem mais que tudo. E mamis, que me ispira com sua força e determiação. Sedo filha de vocês, com toda educação e amor, seria estraho se ão coseguisse ultrapassar cada obstáculo. Também agradeço a toda miha família, em especial miha vó Isabel i memoria, por todo dego que me deu, meu irmão Eduardo, pelas crises de risos a cada perturbação em casa, à miha prima-irmã Isis, por simplesmete ser meu ajo da guarda, miha afilhada Sophia por tato amor e aos meus irmãos emprestados Tai e Iho. Gostaria de agradecer imesamete ao meu marido Aderbal, que me mostra todos os dias o quão é possível ter um amor puro e verdadeiro. Obrigada, meu lido, por me icetivar tato. Você acredita em mim muito mais do que eu. Você me eleva, trasforma todos os meus sohos em realidade e faz com que eu ão desista de absolutamete ada. Obrigada por ser a miha paz e tão parceiro! Sem você, teho certeza, ão teria coseguido essa vitória! Aos meus sogros, Regia e Aderbal, por me icluirem a família e me darem tato apoio e cariho. Um super obrigado a meu orietador Tertu, por todo cohecimeto trasmitido, sempre com muita dedicação e paciêcia. Você me mostrou que ser orietador vai muito além de esiar, estimular e direcioar. Você se importou comigo como ser humao, que tem seus mometos de fraqueza. Lembro muito bem quado uma vez te disse da miha agoia. Me setia só. Tetava tata coisa a tese e parecia que ada daria certo. Você, uma boa, me disse: Calma! Eu teho certeza que vai fucioar. A vida de doutorado às vezes é solitária, casativa, parece que as coisas ão darão certo, mas quado coseguir, você vai setir uma felicidade imesa e vai te eergizar. Obrigada! Eu me sito eergizada! Cotiue assim Tertu! Esse amigo, professor, orietador! Agradeço aos professores Dirk, Paulo, Claudio e Milto que aceitaram fazer parte da miha baca. Gostaria de agradecer a professora Patrícia Goçalves por me recepcioar muito bem em Portugal, me mostrar que as mulheres podem ser icríveis a ciêcia ao mesmo tempo em que são icríveis como seres humaos. Patrícia, você é um misto de calorosidade, etusiasmo, alegria e muita força! Prazer em ter ficado um tempo

8 trabalhado com você. Aproveito para expor o meu cariho aos meus queridos amigos que fiz em Portugal: Rodrigo, Reato, Stefao e Otávio! Obrigada por todo acolhimeto! Rodrigo, em especial, gostaria de agradecer pela ateção com meu trabalho e a votade imesa de me ajudar e de apreder ao mesmo tempo! Foi demais. Aos meus amigos do Vieira, da uiversidade e da vida. Em especial às mihas lidas e aos meus lidos: Aa Carol, loge, perto, sempre muito presete, meu presete! Elaie, Fabi, Lipe e Sara, meus amores de turma! Cheios de sorrisos, sempre protos a ajudar! uca mais largo vocês! Ele, aquela amiga do abraço gostoso. Dri, Carol, Ju e Di, meus feireses prediletos! Sempre muito carihosos comigo. Da e Moa, amigos da graduação, da vida, pra todo o sempre. Muito amor! Vocês todos são os amigos que a pós da UFBA se ecarregou em me dar e que toraram esses últimos aos tão especiais. À CAPES pelo apoio fiaceiro.

9 Do or do ot. There is o try. Master Yoda

10 Resumo O presete trabalho teve o ituito de estudar os seguites problemas: Limite Hidrodiâmico para o processo de exclusão simples simétrico SSEP com uma membraa leta e as Flutuações fora do equilíbrio para o SSEP com um elo leto. Mais precisamete, o modelo em estudo do Limite Hidrodiâmico é o SSEP, o toro d-dimesioal, que possui uma membraa Λ cuja taxa de passagem é dada por α/ β, α >, meor do que a taxa em outros elos. Devido à existêcia desta membraa leta, depededo do regime do parâmetro que regula a letidão desta membraa, aparecem a ível macroscópico codições de froteira. Para β,, a equação hidrodiâmica é dada pela equação de calor o toro cotíuo, sigificado que a membraa leta ão tem efeito o limite. Para β,, a equação hidrodiâmica é dada pela equação de calor com codições de bordo de euma, sigificado que a membraa divide o toro em duas regiões isoladas Λ ad Λ. E, para o valor crítico β =, a equação hidrodiâmica é dada pela equação de calor com codições de froteira de Robi, relacioada com a lei de Fick. o caso das Flutuações, o modelo em estudo é o SSEP uidimesioal que possui um elo leto. A grade dificuldade o trabalho das Flutuações, foi obter as estimativas precisas de probabilidades de trasição de passeios aleatórios de dimesão, quado olhamos para a derivada discreta e de dimesão 2 quado olhamos para a fução correlação. Palavras-chave: Sistema de Partículas, limite hidrodiâmico, flutuações, processo de exclusão.

11 Abstract The preset work aims to study the followig problems: The Hydrodyamic Limit for the simple symmetric exclusio processes SSEP with a slow membrae ad the o-equilibrium fluctuatios for the SSEP with a slow bod. more precisely, the model i study of the Hydrodyamic Limit is the SSEP i the d- dimesioal torus, bods crossig the membrae Λ have jump rate α/ β, α >, lower tha the rate i other bods. Due to the existece of this slow membrae, depedig o the regime of the parameter that regulates the slowess of this membrae, boudary coditios appear ate macroscopic level. For β,, the hydrodyamic equatio is give by the usual heat equatio o the cotiuous torus, meaig that the slow membrae has o effect i the limit. For β,, the hydrodyamic equatio is the heat equatio with euma boudary coditios, meaig that the slow membrae divides the torus ito two isolated regios Λ ad Λ. Ad, for the critical value β =, the hydrodyamic equatio is the heat equatio with certai Robi boudary coditios related to the Fick s Law. I the case of Fluctuatios, the model i study is the SSEP oe-dimesioal with a slow bod. The mai difficulty of this work is a precise estimate of trasitio probabilities of radom walks, i -d whe lookig at the discrete derivative ad i 2-d whe lookig at the correlatio. Keywords: Particle Systems, hydrodyamic limit, fluctuatios, exclusio process.

12 Cotets List of Figures Itroductio 2 2 Hydrodyamic Limit for the SSEP with a slow membrae 3 2. Itroductio Defiitios ad Results Scalig Limit ad Proof s Structure Tightess Replacemet Lemma ad Eergy Estimates Replacemet Lemma for β, Replacemet Lemma for β, Eergy Estimates Characterizatio of limit poits Characterizatio of limit poits for β, Characterizatio of limit poits for β = Characterizatio of limit poits for β, Uiqueess of weak solutios Auxiliary results o-equilibrium Fluctuatios for the SSEP with a slow bod 4 3. Itroductio Statemet of results The model Hydrodyamic limit Space of test fuctios ad semigroup Discrete derivatives ad covariace estimatives Orstei-Uhlebeck process o-equilibrium fluctuatios Estimates o local times Estimates i dimesio two Estimates i dimesio oe Estimates o the discrete derivative ad correlatios Estimate o the discrete derivative Estimate o the correlatio fuctio Commets o the lower boud

13 3.5 Proof of desity fluctuatios Associated martigales Tightess Uiqueess of the Orstei-Uhlebeck process Characterizatio of limit poits Auxiliary results o radom walks Fluctuatios at the iitial time

14 List of Figures 2. The regio i gray represets Λ, ad the white regio represets its complemet Λ. The grid represets T d, the discrete torus embedded o the cotiuous torus T d. By ζu we deote the ormal exterior uitary vector to Λ at the poit u Λ Illustratio i dimesio 2 of a polygoal path joiig the sites x ad y = x + j e + j 2 e 2, with j = j 2 = 3. ote the embeddig i the cotiuous torus T d Illustratio i dimesio two of C x, 2. The sites i C x, 2 are those layig i the gray regio Illustratio i dimesio two of Cu, ε, which is represeted by the regio i gray, while Bu, ε is represeted by the square delimited by the dashed lie. ote that Cu, ε is the cotiuous couterpart of C x, l defied i Illustratio of sites x, y, z Γ. We ote that two adjacet edges to x are slow bods, ad two adjacet edges are ot. Besides, ay opposite vertex to x will be of the form x ± e j I the left, a illustratio of the set Γ,, whose elemets are represeted by black balls. I the right, a illustratio of the sets Γ j,left, ad Γ j,right, for j = 2, whose elemets are represeted by gray ad black balls, respectively Jump rates. The bod {, } has a particular jump rate associated to it, which is give by α/ Sets V, D ad U ad U. Sites of V are the oes layig o the light gray regio. Sites i D lay o the dotted lie ad sites of U are marked as gray balls. Elemets of U are edges marked with a thick black segmet havig jump rate equal to α/ slow bods. Ay other edges have rate Illustratio Relatio betwee X, Y ad X, Y Ilustratio of equivalece relatio i the 2d Step of the proof of Lemma W diag gets idetified with the poits o dashed lies. The four poits marked with black balls compose a sigle equivalece class. o-zero jump rates betwee ay two equivalece classes are everywhere equal to /

15 Chapter Itroductio This thesis is focused o the developmet of two importat cotributios i the area of scalig limits of iteractig particle systems. The first result establishes the hydrodyamic limit for a symmetric simple exclusio process SSEP o the d-dimesioal discrete torus T d with a spatial o-homogeeity give by a slow membrae. The slow membrae is defied here as the boudary of a smooth simple coected regio Λ o the cotiuous d-dimesioal torus T d. I this settig, bods crossig the membrae have jump rate α/ β ad all other bods have jump rate oe, where α >, β,, ad is the scalig parameter. I the diffusive scalig we prove that the hydrodyamic limit presets a dyamical phase trasitio, that is, it depeds o the regime of β. For β,, the hydrodyamic equatio is give by the usual heat equatio o the cotiuous torus, meaig that the slow membrae has o effect i the limit. For β,, the hydrodyamic equatio is the heat equatio with euma boudary coditios, meaig that the slow membrae Λ divides T d ito two isolated regios Λ ad Λ. Ad for the critical value β =, the hydrodyamic equatio is the heat equatio with certai Robi boudary coditios related to the Fick s Law. The secod result is the o-equilibrium fluctuatios for the oe-dimesioal symmetric simple exclusio process with a slow bod. This geeralizes a result of 8,, which dealt with the equilibrium fluctuatios. The foudatio stoe of our proof is a precise estimate o the correlatios of the system. To obtai these estimates, we first deduce a spatially discrete PDE for the covariace fuctio ad we relate it to the local times of a radom walk i a o-homogeeous eviromet via Duhamel s priciple. Projectio techiques ad couplig argumets reduce the aalysis to the problem of studyig the local times of the classical radom walk. We thik that the method developed here ca be applied to a variety of models, ad we provide a discussio o this matter. 2

16 Chapter 2 Hydrodyamic Limit for the SSEP with a slow membrae 2. Itroductio A cetral questio of Statistical Mechaics is about how microscopic iteractios determie the macroscopic behavior of a give system. Uder this guidelie, a etire area o scalig limits of iteractig radom particle systems has bee developed, see 8 ad refereces therei. I the last years, may attetio has bee give to scalig limits of spatially o-homogeeous iteractig systems, see for istace 2, 7 amog may others. Such a attetio is quite atural due to the fact that a o-homogeeity may represet vast physical situatios, as impurities, chagig of desity i the media etc. Amog those iteractig particles systems, processes of exclusio type have special importace: they are, at same time, mathematically tractable ad have a physical iteractio, leadig to precise represetatio of may pheomea. Beig more precise, a radom process is called of exclusio type if it has the hard-core iteractio, that is, at most oe particle is allowed per site of a give graph. The radom evolutio of the system i the symmetric case ca be described as follows: to each edge of the give graph, a Poisso clock is associated, all of them idepedet. At a rig time of some clock, the occupatio values for the vertexes of the correspodig edge are iterchaged. I 2, a quite broad settig for the oe-dimesioal symmetric exclusio process SEP i o-homogeeous medium has bee cosidered, beig obtaied its hydrodyamic limit, that is, the law of large umbers for the time evolutio of the spatial desity of particles. The hydrodyamic equatio there was give by a PDE related to a Krei-Feller operator. Ad i 6, the fluctuatios for the same model were obtaied. The sceario for the SEP i o-homogeeous medium i dimesio d 2 up to ow is far less uderstood. I 25, a geeralizatio of 2 to the d-dimesioal settig was reached. However, the defiitio of model there was very specific to permit a reductio to the oe-dimesioal approach of 2. 3

17 I 3, the hydrodyamic limit i the diffusive scalig for the followig d- dimesioal simple symmetric exclusio process SSEP i o-homogeeous medium was proved, where the term simple meas that oly jumps to earest eighbors are allowed. The uderlyig graph is the discrete d-dimesioal torus, ad all bods of the graph have rate oe, except those layig over a d -dimesioal closed surface, which have rate give by times a costat depedig o the agle betwee the edge ad the ormal vector to the surface, where is the scalig parameter. The hydrodyamic equatio obtaied was give by a PDE related to a d-dimesioal Krei-Feller operator. Despite less broad i certai sese tha the settig of 25, the model i 3 caot be approached by oe-dimesioal techiques, beig truly d-dimesioal. ζu T d u Λ Λ Figure 2.: The regio i gray represets Λ, ad the white regio represets its complemet Λ. The grid represets T d, the discrete torus embedded o the cotiuous torus T d. By ζu we deote the ormal exterior uitary vector to Λ at the poit u Λ. I the preset paper, we cosider a d-dimesioal model close to the oe i 3 ad related to the slow bod phase trasitio behavior of 7, 8, 9. It is fixed a d -dimesioal smooth surface Λ i the cotiuous d-dimesioal torus T d, see Figure 2.. Edges have rates equal to oe, except those itersectig Λ, which have rate α/ β, where α >, β, ad is the scalig parameter. Here we prove the hydrodyamic limit, which depeds o the rage of β, amely, if β,, β = or β,. For β,, the hydrodyamic equatio is give by the usual heat equatio: meaig that, i this regime, the slow bods do ot have ay effect i the cotiuum limit. For β,, the hydrodyamic equatio is the heat equatio with the followig euma boudary coditios over Λ: ρt, u + ζu = ρt, u ζu =, t, u Λ, where ζ is the ormal uitary vector to Λ. This meas that, i this regime, the slow bods are so strog that there o flux of mass through Λ i the cotiuum, despite the existece of flux of particles i the discrete for each. For the 4

18 critical value β =, the hydrodyamic equatio is give by the heat equatio with the followig Robi boudary coditios: ρt, u + ζu = ρt, u ζu d = α ρt, u + ρt, u ζu, e j, t, u Λ, 2. j= where u deotes the limit towards u Λ through poits over Λ while u + deotes the limit towards u Λ through poits over Λ, ad {e..., e d } is the caoical basis of R d. We observe that the Robi boudary coditio above is i agreemet with the Fick s Law: the spatial derivatives are equal due to the coservatio of particles, represetig the rate at which the mass crosses the boudary. Such a rate is proportioal to the differece of cocetratio o each side of the boudary, beig the diffusio coefficiet through the boudary at a poit u Λ give by Du = α d j= ζu, e j. Sice ζu is a uitary vector, the reader ca check via Lagrage multipliers that this diffusio coefficiet satisfies α Du α d i dimesio d 2. Moreover, i this case β =, the hydrodyamic equatio exhibits the pheomea of o-ivariace for isometries. Let us explai this otio. Cosider a isometry T : T d T d, a iitial desity profile ρ : T d, ad deote by Stρ u the solutio of the usual heat equatio with iitial coditio ρ. The, Stρ T u = Stρ Tu. I other words, if we isometrically move the iitial coditio of the usual heat equatio, the solutio of the PDE uder this ew iitial coditio is the equal to the previous solutio moved by the same isometry. O the other had, as we ca see i 2., the diffusio coefficiet Du depeds o how the surface Λ is positioed with respect to the caoical basis. Hece the PDE for β = is ot ivariat for isometries, differetly from the cases β, ad β,. ote that the diffusio coefficiet also says that the uderlyig graph plays a role i the limit. Besides the dyamical phase trasitio itself, this work has the followig features. First of all, i cotrast with some previous works, the hydrodyamic equatios are characterized as classical PDEs, with clear iterpretatio. I the regime β,, the proof relies o a sharp replacemet lemma which compares occupatios of eighbor sites i opposite sides of Λ. For β =, the proof is based o a precise aalysis of the surface itegrals ad the model drops the ad hoc hypothesis adopted i 3: here the rates for bods crossig Λ are all equal to α/, with o extra costat depedig o the icidet agle. Fially, a remark the uiqueess of weak solutios for the cases β = ad β,. Uiqueess of weak solutios are i geeral a delicate ad techical issue, specially for dimesio higher tha oe. I Propositio we provide a geeral statemet which leads to the uiqueess of weak solutios i both cases β = ad β,. The keystoe of the proof is the otio of Friedrichs extesio for strogly mootoe 5

19 symmetric operators. The uiqueess statemet has the feature of beig simple, d-dimesioal ad easily adaptable to may cotexts. However, it is strictly limited to the uiqueess of weak solutios of parabolic liear PDEs with liear boudary coditios. The paper is divided as follows: I Sectio 2.2 we state defiitios ad results. I Sectio 2.3 we draw the strategy of proof for the hydrodyamic limit. I Sectio 2.4 is reserved to the proof of tightess of the processes. I Sectio 3.3 we prove the ecessary replacemet lemmas ad eergy estimates. I Sectio 2.6 we characterize limit poits as cocetrated o weak solutios of the respective PDEs, ad i Sectio 2.7 we assure uiqueess of those weak solutios. 2.2 Defiitios ad Results Let T d be the cotiuous d-dimesioal torus, which is, d with periodic boudary coditios, ad let T d be the discrete torus with d poits, which ca aturally embedded i the cotiuous torus as T d, see Figure 2.. We therefore will ot distiguish otatio for fuctios defied o T d or T d. By η = ηx x T d we deote cofiguratios i the state space Ω = {, } Td, where ηx = meas that the site x is empty, ad ηx = meas that the site x is occupied. By a symmetric simple exclusio process we mea the Markov Process with cofiguratio space Ω ad exchage rates ξx,y > for x, y T d with x y =. This process ca be characterized i terms of the ifiitesimal geerator L actig o fuctios f : Ω R as L fη = d ξx,x+e j fη x,x+e j fη, x T d j= where {e,..., e d } is the caoical basis of R d ad η x,x+e j is the cofiguratio obtaied from η by exchagig the occupatio variables ηx ad ηx + e j, that is, ηx + e j, if y = x, η x,x+e j y = ηx, if y = x + e j, ηy, otherwise. The Beroulli product measures {νθ : θ, } are ivariat ad i fact, reversible, for the symmetric earest eighbor exclusio process itroduced above. amely, νθ is a product measure o Ω whose margial at site x T d is give by ν θ {η : ηx = } = θ. Fix ow two parameters α > ad β, ad a simple coected closed regio Λ T d whose boudary Λ is a smooth d -dimesioal surface. The symmetric simple exclusio process with slow bods over Λ SSEP with slow bods over Λ we defie ow is the particular simple symmetric exclusio process with exchage rates give by ξ x,x+e j = α, if x β Λ ad x+e j, otherwise, 6 Λ, or x Λ ad x + e j Λ, 2.2

20 for all x T d ad j =,..., d. That is, the slow bods of the process will be the bods i T d for which oe of its vertices belogs to Λ ad the other oe belogs to Λ. See Figure 2. for a illustratio. ote that, whe β =, there are o crossigs of particles through the boudary Λ. From ow o, abusig of otatio, we will call the geerator of the SSEP with slow bods over Λ by L, beig uderstood that jump rates will be give by 2.2. Deote by {η t : t } the Markov process with state space Ω ad geerator 2 L, where the 2 factor is the so-called diffusive scalig. This Markov process depeds o, but it will ot be idexed o it to ot overload otatio. Let DR +, Ω be the Skorohod space of càdlàg trajectories takig values i Ω. For a measure µ o Ω, deote by P µ the probability measure o DR +, Ω iduced by the iitial state µ ad the Markov process {η t : t }. Expectatio with respect to P µ will be deoted by E µ. I the sequel, we preset the partial differetial equatios goverig the time evolutio of the desity profile for the differet regimes of β, defiig the otio of weak solutio for each oe of those equatios. Deote by ρ t a fuctio ρt, ad deote by C T d the set of cotiuous fuctios from T d to R with cotiuous derivatives of order up to. Let, ad be the ier product ad orm i L 2 T d, that is, f, g = fu gu du ad f = f, f, f, g L 2 T d. 2.3 T d Fix oce ad for all a measurable desity profile ρ : T d,. ote that ρ is bouded. Defiitio. A bouded fuctio ρ :, T T d R is said to be a weak solutio of the heat equatio { t ρt, u = ρt, u, t, u T d, ρ, u = ρ u, u T d 2.4. if, for all fuctios H C 2 T d ad all t, T, the fuctio ρt, satisfies the itegral equatio ρ t, H ρ, H t ρ s, H ds =. We recall ext the defiitio of Sobolev Space from 5. Let U be a ope set of R d or T d. The Sobolev Space H U cosists of all locally summable fuctios κ : U R such that there exist fuctios uj κ L 2 U, j =,..., d, satisfyig uj Huκu du = Hu uj κu du T d T d for all H C U with compact support. Furthermore, for κ H U, we defie the orm κ H U = uj κ /2. du 2 Fially, we defie the d space j= U 7

21 L 2, T, H U, which cosists of all measurable fuctios τ :, T H U such that T /2 τ L 2,T,H U := τ t 2 H U dt <. ote that U = T d \ Λ is a ope subset of T d. The followig otatio will be used several times alog the text. Give a fuctio f : T d \ Λ R ad u Λ, we deote fu + := lim v u v Λ fv ad fu := lim v u fv, 2.5 v Λ that is, fu + is the limit of fv as v approaches u Λ through the complemet of Λ, while fu is the limit of fv as v approaches u Λ through Λ. Let A be the idicator fuctio of a set A, that is, A a = if a A ad zero otherwise. Deote by ζu the ormal uitary exterior vector to the regio Λ at the poit u Λ ad by / ζ the directioal derivative with respect to ζu. Below, by u, v we deote the caoical ier product of two vectors u ad v i R d, which shall ot be misuderstood with the ier product i L 2 T d as defied i 2.3. By ds we idicate a surface itegral. Defiitio 2. A bouded fuctio ρ :, T T d R is said to be a weak solutio of the followig heat equatio with Robi boudary coditios t ρt, u = ρt, u, t, u T d, ρt, u + ζu = ρt, u d ζu = α ρt, u + ρt, u ζu, e j, t, u Λ, j= ρ, u = ρ u, u T d. 2.6 if ρ L 2, T, H T d \ Λ ad, for all fuctios H = h Λ + h 2 Λ with h, h 2 C 2 T d ad for all t, T, the followig the itegral equatio holds: ρ t, H ρ, H + + t t Λ Λ ρ s u t ρ s, H ds t Λ ρ s u + d uj Hu ζu, e j dsuds j= d uj Hu + ζu, e j dsuds d α ρ s u ρ s u + Hu + Hu ζu, e j dsuds =. The reader should ote that the fuctio H is possibly discotiuous at the boudary Λ. ote also that the expressio d j= u j Hu ± ζu, e j appearig i the itegral equatio above is othig but Hu ± / ζ due to liearity of the directioal derivative. j= j= 8

22 Defiitio 3. A bouded fuctio ρ :, T T d R is said to be a weak solutio of the heat equatio with euma boudary coditios t ρt, u = ρt, u, t, u T d, ρt, u + ζu = ρt, u ζu =, t, u Λ, 2.7 ρ, u = ρ u, u T d, if ρ L 2, T, H T d \ Λ ad, for all fuctios H = h Λ + h 2 Λ with h, h 2 C 2 T d ad for all t, T, the followig itegral equatio holds: t t d ρ t, H ρ, H ρ s, H ds ρ s u + uj Hu + ζu, e j dsuds + t Λ ρ s u Λ j= d uj Hu ζu, e j dsuds =. j= Sice i Defiitios 2 ad 3 we impose ρ L 2, T, H T d \ Λ, the itegrals above are well-defied o the boudary due to the otio of trace i Sobolev spaces, see 5 o the subject. We clarify that the otio of weak solutios above have bee defied i the stadard way of Aalysis: the reader ca check that a strog solutio of 2.4, 2.6 or 2.7 is ideed a weak solutio of the respective PDE. Fix a measurable desity profile ρ : T d,. For each, let µ be a probability measure o Ω. A sequece of probability measures {µ : } is said to be associated to a profile ρ : T d, if, for every δ > ad every cotiuous fuctio H : T d R the followig limit holds: { lim µ } Hx/ηx Huρ d udu > δ =. 2.8 x T d Below, we establish the mai result of this paper, the hydrodyamic limit for the exclusio process with slow bods, which depeds o the regime of β. Theorem Fix β,. Cosider the exclusio process with slow bods over Λ with rate α β at each oe of these slow bods. Fix a Borel measurable iitial profile ρ : T d, ad cosider a sequece of probability measures {µ } o Ω associated to ρ i the sese of 2.8. The, for each t, T, lim P µ η : d x T d Hx/ η t x Hu ρt, udu T > δ d for every δ > ad every fuctio H CT d where: If β,, the ρ is the uique weak solutio of 2.4. If β =, the ρ is the uique weak solutio of 2.6. If β,, the ρ is the uique weak solutio of 2.7. =, The assumptio that Λ is simple ad coected may be dropped, beig imposed oly for the sake of clarity. Otherwise, otatio would be highly overloaded. 9

23 2.3 Scalig Limit ad Proof s Structure Let M be the space of positive Rado measures o T d with total mass bouded by oe, edowed with the weak topology. Let πt M the empirical measure at time t associated to η t, it is a measure o T d obtaied rescalig space by : π t du = π t η t, du := d x T d η t xδ x/ du, where δ u deotes the Dirac measure cocetrated o u T d. For a measurable fuctio H : T d R which is π-itegrable, deote by πt, H the itegral of H with respect to πt : πt, H = H x ηt x. d x T d ote that this otatio, is also used as the ier product of L 2 T d. Fix oce ad for all a time horizo T >. Let D, T, M be the space of M-valued càdlàg trajectories π :, T M edowed with the Skorohod topology. The, the M- valued process {πt : t } is a radom elemet of D, T, M determied by {η t : t }. For each probability measure µ o Ω, deote by Q β, µ the distributio of {πt : t } o the path space D, T, M, whe η has distributio µ. Fix a cotiuous Borel measurable profile ρ : T d, ad cosider a sequece {µ : } of measures o Ω associated to ρ. Let Q β be the probability measure o D, T, M cocetrated o the determiistic path πt, du = ρt, udu, where: if β,, the ρ is the uique weak solutio of 2.4, if β =, the ρ is the uique weak solutio of 2.6, if β,, the ρ is the uique weak solutio of 2.7. Propositio For ay β,, the sequece of probability measures Q β, µ coverges weakly to Q β as goes to ifiity. The proof of this result is divided ito three parts. I the ext sectio, we show that tightess of the sequece {Q β, µ : }. I Sectio 3.3, we prove a suitable Replacemet Lemma for each regime of β, which will be crucial i the task of characterizig limit poits. I Sectio 2.6 we characterize the limit poits of the sequece for each regime of the parameter β. Fially, the uiqueess of weak solutios is preseted i Sectio 2.7 ad this implies the uiqueess of limit poits of the sequece {Q β, µ : }. Fially, we ote that Theorem 2.2. is a cosequece of Propositio Actually, sice Q β, µ weakly coverges to Q β for all cotiuous fuctios H : T d R, it follows that the path { πt, H : t T } coverges i distributio to { π t, H :

24 t T }. Sice { π t, H : t T } is a determiistic path, covergece i distributio is equivalet to covergece i probability. Therefore, { } lim P µ Hx/ η d t x Huρt, udu T > δ d x T d = lim Qβ, µ { π t, H π t, H > δ } =, for all δ > ad t T. This gives the strategy of proof for the hydrodyamic limit. ext, we make some geeral observatios. Sice particles i the exclusio process evolve idepedetly as a earest eighbor radom walk, except for exclusio rule, the exclusio process with slow bods over Λ is related to the radom walk o T d that describes the evolutio of the system with a sigle particle. To be used throughout the paper we itroduce the geerator of the radom walk described above, which is L H x d = {ξ x,x+ej j= H x+e j H x + ξ x,x e j H x e j H x } 2.9 for every H : T d R ad every x Td. Above, it is uderstood that ξ x±e j,x = ξ x,x±ej. By Dyki s formula see A..5. i 8, M t H = π t, H π, H t 2 L π s, H ds is a martigale with respect to the atural filtratio F t := σηs : s t. By some elemetary calculatios, 2 L πs, H = x η d 2 s xl H = πs, 2 L H, x T d hece the martigale ca be rewritte as M t H = π t, H π, H t π s, 2 L H ds. 2. ote that this observatio stads for ay jump rates. The particular form of jump rates for the SSEP with slow bods over Λ will play a role whe characterizig limit poits ad provig replacemet lemmas. 2.4 Tightess This sectio deals with the issue of tightess for the sequece {Q β, µ : } of probability measures o D, T, M. Propositio For ay fixed β,, the sequece of measures {Q β, µ : } is tight i the Skorohod topology of D, T, M.

25 Proof. I order to prove tightess of {πt : t T }, it is eough to show tightess of the real-valued process { πt, H : t T } for H CT d. I fact, cf. Propositio.7, chapter 4 of 8 it is eough to show tightess of { πt, H : t T } i D, T, R for a dese set of fuctios i CT d with respect to the uiform topology. For that purpose, fix H C 2 T d. Sice the sum of tight processes is tight, i order to prove tightess of { πt, H : }, it is eough to assure tightess of each term i 2.. The quadratic variatio of Mt H is give by t M H t = implyig that d j= x T d ξx,x+e j η 2d 2 s x η s x + e j H x+e j 2ds, H x 2. M H t αt d d uj H 2, 2.2 where H := sup u T d Hu, hece Mt coverges to zero as i L 2 P β µ. Therefore, by Doob s iequality, for every δ >, lim sup Mt H > δ =, 2.3 P µ t T which implies tightess of the sequece of martigales {Mt H : }. ext, we will prove tightess for the itegral term i 2.. Let Γ be the set of vertices i T d havig some icidet edge with exchage rate ot equal to oe, that is, Γ = {x T d : for some j =,..., d, ξ x,x+ej = α or β ξx,x ej = α }. 2.4 β The term π s, 2 L H appearig iside the time itegral i 2. ca be the writte as d d η s x 2 x/ Γ j= + d d η s x x Γ j= H x+e j j= + H x e j 2H x ξ x,x+e j H x+e j H x +ξ x,x e j H x e j H x sice ξ x,x+ej = ξ x+ej,x = for every x / Γ. By a Taylor expasio o H C 2 T d, the absolute value of the summad i the first double sum above is bouded by H. Sice there are O d elemets i Γ, ad ξ x,x+ej α, the absolute value of summad i secod double sum above is bouded by d j= α u j H. Therefore, there exists C >, depedig oly o H, such that 2 L πs, H C, which yields t 2 L πs, H dr C t s. s By 8, Propositio 4..6, last iequality implies tightess of the itegral term, cocludig the proof of the propositio. 2

26 2.5 Replacemet Lemma ad Eergy Estimates This sectio gives a fudametal result that allow us to replace a mea occupatio of a site by the mea desity of particles i a small macroscopic box aroud this site. We start by itroducig some tools to be used i the sequel. Deote by H µ ν θ the relative etropy of µ with respect to the ivariat state ν θ. For a precise defiitio ad properties of the etropy, we refer the reader to 8. Assumig < θ <, the formula i 8, Theorem A.8.3 assures the existece a fiite costat κ = κ θ such that H µ ν θ κ d 2.5 for ay probability measure µ o {, } Td. Deote by D the Dirichlet form of the process, which is the fuctioal actig o fuctios f : {, } Td R as D f := f, L f νθ = d j= x T d ξx,x+e j fη x,x+e j fη 2 νθ dη I the sequece, we will make use of the fuctioal D f, where f is a probability desity with respect to ν θ Replacemet Lemma for β, Below, we defie the local desity of particles, which correspods a to the mea occupatio i a box aroud a give site. Abusig of otatio, we deote by ε the iteger part of ε. For β,, we defie the local mea by η ε x = ε d ε j,j 2,...,j d = η x + j e j d e d. 2.7 ote that the sum o the right had side of above may cotai sites i ad out of Λ i the sese that x/ Λ or x/ Λ. By Of we will mea a fuctio bouded i modulus by a costat times f. Lemma Fix β,. Let f be a desity with respect to the ivariat measure ν θ, λ : T d R a fuctio such that λ M < ad γ >. The, γ x Γ λ x { ηx η ε x } fην θ dη γ2 M 2 O d 2 β α + dε + 2 D f. Proof. By the defiitio 2.7 of local mea η ε x, { } λ x ηx η ε x fην θ dη = 3

27 T d y Λ x Λ Figure 2.2: Illustratio i dimesio 2 of a polygoal path joiig the sites x ad y = x + j e + j 2 e 2, with j = j 2 = 3. ote the embeddig i the cotiuous torus T d. = λ x ε d d ε j,...,j d = { } ηx ηx + j e j d e d fην θ dη. 2.8 The ext step is to write ηx ηx + j e + + j d e d as a telescopic sum: ηx ηx + j e j d e d = j + +j d l= ηa l ηa l, where a = x, a j + +j l = x + j e + + j d e d, ad a l a l = for ay l =,..., j + + j d. ote that the path a, a,..., a j + +j l depeds o the iitial poit x ad the fial poit x + j e + + j d e d. See Figure 2.2 for a illustratio ad keep i mid that the legth of this path is bouded by dε. Isertig the previous equality ito 2.8, we get λ x ε d ε j,...,j d = { j + +j d l= } ηa l ηa l fη ν θ dη. Rewritig the expressio above as twice the half ad performig the trasformatio η η a l,a l for which the probability measure νθ is ivariat, expressio above becomes: 2ε d ε j,...,j d = j + +j d l= λ x ηa l ηa l f η a l,a l f η dν θ. Sice ab = ca b c 2 ca2 + b 2, which holds for ay c >, the previous expressio 2 c is smaller or equal tha 2ε d + A 2ξ a l,a l ε j,...,j d = j + +j d l= ξ al,a l 2A f η a l,a l f η 2 dνθ λ 2 x ηa l ηa l 2 f η a l,a l + f η 2dνθ 4.

28 Summig over x Γ, we ca boud the last expressio by 2ε d ε j + +j d ξ al,a l 2A + x Γ j,...,j d = A 2ξ x Γ a l,a l l= f η a l,a l f η 2 dνθ λ 2 x ηa l ηa l 2 f η a l,a l + f η 2dνθ Recallig 2.6, we ca boud the first parcel i the sum above by 2ε d ε j,...,j d = A D f = 2A D f. Sice f is a desity ad λ x M, the secod parcel is bouded by 2ε d ε x Γ j,...,j d = ε Up to here we have achieved that j + +j d l= A 2 4M 2 ξ a l,a l AM 2 O d β ε d α + dε j,...,j d = = AM 2 O d β α + dε. λ x { ηx η ε x } fην θ dη x Γ AM 2 O d β α + dε + 2A D f. We poit out that the quatity of sites o Γ is of order O d, which is a cosequece of the fact that Λ is a smooth surface of dimesio d. The, multiplyig the iequality above by γ gives us γ x Γ λ x { ηx η ε x } fην θ dη AγO d M 2 β α + dε + γ 2A D f. ow choosig A = γ /2 the proof eds. Recall the defiitio of Γ i 2.4. Lemma Replacemet lemma. Fix β,. Let λ : T d R be a sequece of fuctios such that λ M <. The, t lim lim ε Eβ µ d λ x{η ε s x Γ 5 x η s x} ds =..

29 Proof. Usig the variatioal formula for etropy, for ay γ R which will be chose large a posteriori, E β µ t d = γ d Eβ µ γ λ x{η s x η ε s x Γ t x}ds λ x{η s x η ε s x Γ H µ ν θ γ d + γ d log E ν θ exp γ t x}ds λ x{η s x η ε s x Γ x}ds. 2.9 By the estimate 2.5 o the etropy, the first parcel of above is egligible as sice we will choose γ arbitrarily large. Therefore, we ca focus o the secod parcel. Usig that e x e x + e x ad lim { loga d + b = max lim d log a, lim } log b d 2.2 for ay sequeces a, b >, oe ca see that the secod parcel o the right had side of 2.9 is less tha or equal to the sum of lim { t γ log E d νθ exp γ λ x{η s x η ε s x Γ } x}ds 2.2 ad lim { t γ log E d νθ exp γ λ x{η s x η ε s x Γ } x}ds We hadle oly 2.2, beig 2.22 aalogous. By Feyma-Kac s formula, see 8, Appedix, Lemma 7.2, expressio 2.2 is bouded by where Φ = sup f desity lim { γ log exp d t } Φ ds t Φ = lim γ, d { γ } λ x{ηx η ε x}fην θ dη 2 D f. x Γ Applyig Lemma 2.5. fiishes the proof Replacemet Lemma for β, Here, some additioal otatio is required. The idea is actually very simple: the local mea shall be over a regio avoidig slow bods. Let B x, l T d be the discrete box cetered o x T d which edge has size 2l, that is, B x, l = {y T d : 6

30 y x l}, where we have writte for the supremum orm o T d, that is, x,..., x d = max { x x,..., x d x d }. Λ T d Λ x Figure 2.3: Illustratio i dimesio two of C x, 2. The sites i C x, 2 are those layig i the gray regio. Let Λ = {x T d : x Λ} the set of sites i Td belogig to Λ. We defie ow the regio C x, l T d by B x, l Λ if x C x, l := Λ, 2.23 B x, l Λ if x Λ, see Figure 2.3 for a illustratio. For β,, we defie the local desity as the average over C x, l, that is, η ε x := #C x, ε y C x,ε ηy Lemma Fix β,. Let f be a desity with respect to the ivariat measure ν θ, let λ : T d R a fuctio such that λ M < ad γ >. The, the followig iequalities hold: γ λ x { ηx η ε x } fην θ dη 2 γ2 M 2 O d dε + 2 D f 2.25 x Γ ad γ x T d λ x{ηx η ε x}fην θ dη 2 γ2 M 2 O d dε + 2 D f Proof. Let us prove the iequality As commeted i the begiig of this subsectio, the local average η ε is take over C x, ε. Thus, we ca write λ x{ηx η ε x}fην θ dη 7

31 = { λ x #C x, ε y C x,ε ηx ηy } fην θ dη For each y Cx, ε, let γx, y be a polygoal path of miimal legth coectig x to y which does ot crosses Λ. That is, γx, y is a sequece of sites a,..., a M such that x = a, y = a M, a i a i+ = ad ξ a,ai+ = for i =,..., M, ad γx, y has miimal legth, that is, M = Mx, y = x y +. ow we repeat the steps i the proof of Lemma 2.5., observig that i this case the sum will be over T d, obtaiig that 2.27 is bouded from above by Mx,y 2 fη a l,a l fη dνθ 2#C x, ε 2A x T d y C x,ε l= + A 2ηal 2 λ x ηa l 2 fη a l,a l + fη dνθ. 2 We ca boud the first parcel i the sum above by 2A D f ad the secod parcel by 2#C x, ε #C x, ε x T d y C x,ε y C x,ε Mx,y l= 4AM 2 2 AM 2 O d dε = AM 2 O d dε. We hece have λ x { ηx η ε x } fην θ dη AM 2 O d dε + 2A D f. x T d The, multiplyig the iequality above by γ gives us γ λ x { ηx η ε x } fην θ dη AγO d M 2 dε + γ 2A D f. x T d ow choosig A = γ 2 /2 the proof of 2.25 eds. The proof of iequality 2.25 similar to the proof of Lemma 2.5., uder the additioal feature that rates of bods over a path coectig two sites will be always equal to oe, which facilitates the argumet. Lemma Replacemet lemma. Fix β,. Let λ : T d R be a sequece of fuctios such that λ c <. The, t lim lim ε Eβ µ λ d x{η ε s x η s x} ds = x Γ ad t lim lim ε Eβ µ d x T d λ x{η ε s 8 x η s x} ds =.

32 Proof. The proof is similar to the oe of Lemma 2.5.2, beig sufficiet to show that expressios satisfy { Φ 2 := sup γ } λ x{η ε x ηx}fηdν θ 2 D f, f desity x Γ { Φ 3 := sup γ λ x{η ε x ηx}fηdν θ 2 D } f f desity x T d lim tφ 2 tφ 3 = ad lim γ d γ =, d which is a cosequece of Lemma 2.5.3, fiishig the proof Eergy Estimates I this subsectio, cosider β,. Our goal here is to prove that ay limit poit Q β of the sequece {Q β, µ : > } is cocetrated o trajectories ρt, udu with fiite eergy, meaig that ρt, u belogs to a suitable Sobolev space. This result plays a both role i the uiqueess of weak solutios of 2.7 ad i the characterizatio of limit poits. The fact that Q β is cocetrated i trajectories with desity with respect to the Lebesgue measure of the form ρt, udu, with ρ, is a cosequece of maximum of oe particle per site, see 8. The issue here is to prove that the desity ρt, u belogs to the Sobolev space L 2, T ; H T d \ Λ, see Sectio 2.2 for its defiitio. Assume without loss of geerality that the etire sequece {Q β, µ : } weakly coverges to Q β. Let Bu, ε := {r T d : r u < ε} ad Bu, ε Λ if u Λ, Cu, ε := Bu, ε Λ if u Λ, where we have writte for the supremum orm o the cotiuous torus T d =, d, that is, u,..., u d = max { u u,..., u d u d }. See Figure 2.4 for a illustratio. 9

33 Td Λ Λ u Figure 2.4: Illustratio i dimesio two of Cu, ε, which is represeted by the regio i gray, while Bu, ε is represeted by the square delimited by the dashed lie. ote that Cu, ε is the cotiuous couterpart of C x, l defied i We defie a approximatio of the idetity ι ε i the cotiuous torus T d by ι ε u, v := Cu, ε Cu,εv, 2.28 where Cu, ε above deotes the Lebesgue measure of the set Cu, ε. Recall that the covolutio of a measure π with ι ε is defied by π ι ε u = ι ε u, vπdv for ay u T d T d Give a fuctio ρ, the covolutio ρ ι ε shall be uderstood as the covolutio of the measure ρvdv with ι ε. A importat remark ow is the equality πt ι ε x = η ε t x + O ε d, 2.3 where ηt ε has bee defied i 2.24, beig the small error above due to the fact that sites o the boudary of C x, l may or may ot belog to Cu, ε whe takig u = x/ ad l = ε. Give a fuctio H : T d R, let V ε, j, H, η := d x T d H x {ηx ηx + εe j } ε 2 d x T d H x Lemma Cosider H,..., H k fuctios i C,, T T d with compact support cotaied i, T T d \ Λ. Hece, for every ε > ad j =,..., d, lim lim δ Eβ µ max i k { T where κ has bee defied i 2.5. } V ε, j, H i s,, ηs δ ds Proof. Provided by Lemma 2.5.4, it is eough to prove that { t lim Eβ µ max i k } V ε, j, H i s,, η s ds 2 κ, 2.32 κ.

34 By the etropy iequality, for each fixed, the expectatio above is smaller tha Hµ ν θ + { { T }} d log E d ν θ exp max d V ε, j, H i s,, η s ds. i k Usig 2.5, we boud the first parcel above by κ. Sice exp { } max i k a j i k exp{a j} ad by 2.2, we coclude that the limsup as of the secod parcel above is less tha or equal to lim = max log E d ν θ lim i k i k log E d ν θ exp T } exp { d V ε, j, H i s,, η s ds T { d } V ε, j, H i s,, η s ds. Thus, i order to coclude the proof, it is eough to show that the limsup above is o positive for each i =,..., k. By the Feyma-Kac formula see 8, p. 332, Lemma 7.2 for each fixed ad d 2, log E d ν θ T sup f T exp { d } V ε, j, H i s,, η s ds 2.33 { V ε, j, H i s,, ηfηdν θ 2 d D f} ds, 2.34 where the supremum above is take over all probability desities f with respect to ν θ. By assumptio, each of the fuctios {H i : i =,..., k} vaishes i a eighborhood of Λ. Thus, we make followig observatio about the first sum i the RHS of 2.3: for small ε, o-zero summads are such that x/ ad x + εe j lay both i Λ or both i Λ. Heceforth, i such a case, it is possible to fid a path o slow bods coectig x ad x + εe j. Keepig this i mid, we ca repeat the argumets i the proof of Lemma to deduce that d x T d H x 2 d D f + 2 d {ηx ηx + εe j } fηdν θ ε x T d H x 2. Pluggig this iequality ito 2.34 implies that 2.33 has a opositive limsup, showig ad therefore fiishig the proof. Lemma { T E Q β sup uj Hs, uρs, ududs 2 H T d T } Hs, u 2 duds T d κ, where the supremum is carried over all fuctios H C,, T T d with compact support cotaied i, T T d \ Λ. 2

35 Proof. Cosider a sequece {H i : i } dese i the subset of C 2, t T d of fuctios with support cotaied i, T T d \ Λ, beig the desity with respect to the orm H + u H. Recall we are assumig that {Q β, µ : } coverges to Q β. The, by 2.32 ad the Portmateau Theorem, { T lim E δ Q max H β i s, u{ρ δ i k ε su ρ δ su + εe j } duds T d T } 2 H i s, u 2 duds κ, T d where ρ δ su = ρ s ι δ u as defied i Lettig δ, the Lebesgue Differetiatio Theorem assures that ρ δ su coverges almost surely to ρ s. The, performig a chage of variables ad lettig ε, we obtai that E Q β max i k { T T d uj H i s, uρ s u duds 2 T } H i s, u 2 duds κ. T d Sice the maximum icreases to the supremum, we coclude the lemma by applyig the Mootoe Covergece Theorem to {H i : i }, which is a dese sequece i the subset of fuctios C 2, T T d with compact support cotaied i, T T d \ Λ. Propositio The measure Q β such that ρ L 2, T ; H T d \ Λ. is cocetrated o paths πt, u = ρt, udu Proof. Deote by l : C 2, T T d R the liear fuctioal defied by lh = T T d uj Hs, uρs, u du ds. Sice the set of fuctios H C 2, T T d with support cotaied i, T T d \ Λ is dese i L 2, T T d ad sice by Lemma l is a Q β -a.s. bouded fuctioal i C 2, T T d, we ca exted it to a Q β -a.s. bouded fuctioal i L 2, T T d, which is a Hilbert space. The, by the Riesz Represetatio Theorem, there exists a fuctio G L 2, T T d such that cocludig the proof. T lh = Hs, ugs, u du ds, T d 2.6 Characterizatio of limit poits Before goig ito the details of each regime β,, β = or β,, we make some useful cosideratios for all cases. We will prove i this sectio that all limit poits of the sequece {Q β, µ : } are cocetrated o trajectories of measures πt, du = ρt, u du, whose desity 22

36 ρt, u with respect to the Lebesgue measure is the weak solutio of the hydrodyamic equatio 2.4, 2.6 or 2.7 for each correspodig value of β. Provided by tightess, let Q β be a limit poit of the sequece {Q β, µ : } ad assume, without loss of geerality, that {Q β, µ : } coverges to Q β. Sice there is at most oe particle per site, it is easy to show that Q β is cocetrated o trajectories πt, du which are absolutely cotiuous with respect to the Lebesgue measure πt, du = ρt, u du ad whose desity ρt,, is oegative ad bouded by oe. Recall the martigale Mt H i 2.. Lemma If a β, ad H C 2 T d, or b β, ad H C 2 T d \ Λ, the, for all δ >, lim P µ sup t T Mt H > δ = Proof. Item a has bee already proved i 2.3. For item b, recallig 2. ote that M T d 2 H t ξ 2d 2 x,x+e j H x+e j H x j= x T d Sice H C 2 T d \ Λ, H is differetiable with bouded derivative except over Λ. Therefore, if the edge x, x + e j is ot a slow bod, the ξ x,x+e j H x+e j H x 2 2 u j H O the other had, if the edge x, x + e j is a slow bod, the ξ x,x+e j H x+e j H t x 2 4α H 2 β Sice the umber of slow bods is of order O d, pluggig 2.37 ad 2.38 ito 2.36 gives us M H t t O/ d. The, Doob s iequality cocludes the proof Characterizatio of limit poits for β,. Propositio Let H C 2 T d. The, for ay δ >, Q β π. : sup π t, H π, H t T t π s, H ds > δ =. Proof. Sice Q β, µ coverges weakly to Q β, by Portmateau s Theorem see 2, Theorem 2., Q β π. : sup π t, H π, H t T t 23 π s, H ds > δ

37 lim Qβ, µ π. : sup π t, H π, H t T t π s, H ds > δ 2.39 sice the supremum above is a cotiuous fuctio i the Skorohod metric, see Propositio Recall that Q β, µ is the probability measure iduced by P β µ via the empirical measure. With this i mid ad the addig ad subtractig πs, 2 L H, expressio 2.39 ca be bouded from above by lim Pβ µ π. : sup t T + lim Pβ µ π. : sup t T π t, H π, H t t πs, 2 L H ds > δ/2 πs, H 2 L H ds > δ/2. By Lemma 2.6., the first term above is ull. Sice there is at most oe particle per site, the secod term i last expressio is bouded by lim Pβ µ T d + lim Pβ µ x/ Γ sup t T x x H 2 L > δ/4 t d { x x } H 2 L η s x ds > δ/4. x Γ Outside Γ, the operator 2 L coicides with the discrete Laplacia. Sice H C 2 T d, the first probability above vaishes for sufficietly large. Recall that the umber of elemets i Γ is of order d. Applyig the triagular iequality, the secod expressio i the previous sum becomes bouded by the sum of lim Pβ µ O T H > δ/8 2.4 ad lim Pβ µ sup t T t d x L η s x ds > δ/ x Γ For large, the probability i 2.4 vaishes. We deal ow with 2.4. Let x Γ. By defiitio of Γ, some adjacet bod to x is a slow bod. Thus, the opposite vertex to x with respect to this bod is also i Γ, see Figure 2.5. Recall the defiitio of L i 2.9. Wheever {x, x e j } either {x, x + e j } are slow bods, the expressio ξx,x+e j H x+e j H x + ξ x,x e j H x e j H x is of order O 2 due to assumptio H C 2 T d. Therefore, i 2.4 we ca disregard terms of this kid, reducig the proof that 2.4 is ull to prove that lim Pβ µ sup t T t d e={x,x+e j } e is a slow bod Ae ds > δ/6 =,