Scalings limits for the exclusion process with a slow site
SSCALING LIMITS FOR THE EXCLUSIONPROCESS WITH A SLOW SITE
TERTULIANO FRANCO, PATR´ICIA GONC¸ ALVES, AND GUNTER M. SCH ¨UTZA
BSTRACT . We consider the symmetric simple exclusion processes with a slowsite in the discrete torus with n sites. In this model, particles perform nearest-neighbor symmetric random walks with jump rates everywhere equal to one,except at one particular site, the slow site , where the jump rate of enteringthat site is equal to one, but the jump rate of leaving that site is given by aparameter g ( n ) . Two cases are treated, namely g ( n ) = 1 + o (1) , and g ( n ) = αn − β with β > , α > . In the former, both the hydrodynamic behaviorand equilibrium fluctuations are driven by the heat equation (with periodicboundary conditions when in finite volume). In the latter, they are driven bythe heat equation with Neumann boundary conditions. We therefore establishthe existence of a dynamical phase transition. The critical behavior remainsopen. C ONTENTS
1. Introduction 22. Statement of results 52.1. The model 52.2. Hydrodynamics 62.3. Equilibrium density fluctuations 82.4. Ornstein-Uhlenbeck process 93. Hydrodynamics 113.1. Tightness 123.2. Entropy 133.3. Dirichlet form 133.4. Hydrodynamic limit for g ( n ) = 1 + o (1) g ( n ) = αn − β g ( n ) = 1 + o (1) g ( n ) = αn − β k neighboring slow bonds 28Acknowledgements 30References 31 Mathematics Subject Classification.
Key words and phrases.
Exclusion processes, hydrodynamics, fluctuations, phase transition,Ornstein-Uhlenbeck process. a r X i v : . [ m a t h . P R ] A ug TERTULIANO FRANCO, PATR´ICIA GONC¸ ALVES, AND GUNTER M. SCH ¨UTZ
1. I
NTRODUCTION
In the seventies, Dobrushin and Spitzer, see [30] and references therein,initiated the idea of obtaining a mathematically precise understanding of theemergence of macroscopic behavior in gases or fluids from the microscopic in-teraction of a large number of identical particles with stochastic dynamics.This approach has turned out to be extremely fruitful both in probability the-ory and statistical physics (e.g. see the books [32, 19]) and it still raises at-tention nowadays. In this context, recent studies have been made in hydrody-namic limit/fluctuations of interacting particle systems in random/non homo-geneous medium, see for instance [9, 10, 11, 18] and references therein.So far, most of the work done in this field concerns the bulk hydrodynamics,i.e., the derivation of macroscopic partial differential equations arising fromthe bulk interactions of the underlying particle system. To this end, one usu-ally considers an infinite system or a finite torus with periodic boundary con-ditions and then takes the thermodynamic limit. However, in applications tophysical systems one is usually confronted with finite systems, which requiresthe study of a partial differential equation on a finite interval with prescribedboundary conditions. This raises the question from which microscopic bound-ary interactions a given type of boundary condition emerges at the macroscopicscale.This is an important issue both for boundary-driven open systems, whereboundary interactions can induce long-range correlations [31] and bulk phasetransitions due to the absence of particle conservation at the boundaries [3],and for bulk-driven conservative systems on the torus where even a singledefect bond between two neighboring sites can change bulk relaxation behav-ior or lead to macroscopic discontinuities in the hydrostatic density profiles. Given such rich behaviour due to boundary effects in non-conservative or bulk-driven systems it is natural to explore the macroscopic role of a microscopicdefect on a torus in a conservative system in the absence of bulk-driving andto ask whether such a defect can be described on macroscopic scale in terms ofa boundary condition for the PDE describing the bulk hydrodynamics.In this work we address this problem for the symmetric simple exclusionprocess (SSEP) on the discrete torus in the presence of a defect site. The modelcan be described as follows. Each site of the discrete torus with n sites, thatwe denote by T n = Z /n Z , is allowed to have at most one particle. To eachsite is associated a Poisson clock, all of them being independent. If there is aparticle in the associated site, this particle chooses one of its nearest neighborswith equal probability when the clock rings. If the chosen site is empty, theparticle jumps to it. Otherwise nothing happens. All sites have a Poisson clockof parameter two, except the origin, which has a Poisson clock of parameter g ( n ) . If g ( n ) < , the origin behaves as a trap , and (in average) it keeps aparticle there for a longer time than the other sites do. We call this site a slowsite . The main results of the present work are the hydrodynamic limit and theequilibrium fluctuations for the exclusion process with such a slow site. See [16, 29, 1, 4, 2, 25] for numerical, exact and rigorous results for the asymmetric simpleexclusion process and [28, 27] for a review, including experimental applications of interactingparticle systems with boundary interactions in physical and biological systems.
SEP WITH A SLOW SITE 3 F IGURE
1. Exclusion process with a slow site .Specifically, for g ( n ) = 1 + o (1) it is shown here that the limit for the timetrajectory of the spatial density of particles is given by the solution of the heatequation with periodic boundary conditions, namely: (cid:40) ∂ t ρ ( t, u ) = ∂ u ρ ( t, u ) , t ≥ , u ∈ T ,ρ (0 , u ) = ρ ( u ) , u ∈ T , (1.1)where T is the one-dimensional continuous torus.Moreover, considering the same particle system evolving on Z , we provethat the equilibrium fluctuations of the system are driven by a generalizedOrnstein-Uhlenbeck process Y t which is the solution of d Y t = ∆ Y t dt + (cid:112) χ ( p ) ∇ d W t where W t is a Brownian motion on the space S (cid:48) ( R ) of tempered distributionsand χ ( p ) is the compressibility, which is a coefficient related to the invariantmeasure of the system. Both results are true irrespective of whether o (1) ispositive or negative, i.e., of whether the origin is a fast site or a slow site.On the other hand, if g ( n ) = αn − β , α > , β > , the limit for the timetrajectory of the spatial density of particles is given by the solution of the heatequation with Neumann boundary conditions, namely: ∂ t ρ ( t, u ) = ∂ u ρ ( t, u ) , t ≥ , u ∈ (0 , ,∂ u ρ ( t, + ) = ∂ u ρ ( t, − ) = 0 , t ≥ ,ρ (0 , u ) = ρ ( u ) , u ∈ (0 , , (1.2)where + and − denotes right and left side limits, respectively. This repre-sents no passage of particles in the continuum limit.Again considering the same particle system evolving on Z , we prove thatthe equilibrium fluctuations of the system when g ( n ) = αn − β , α > , β > aredriven by the solution of d Y t = ∆ Neu Y t dt + (cid:112) χ ( p ) ∇ Neu d W t , which is essentially a version of the previous generalized Ornstein-Uhlenbeckprocess associated to the PDE (1.2), in the same setting of [11]. These Ornstein-Uhlenbeck processes are precisely stated in Section 2.We point out that a similar model with conservative dynamics on the torushas been considered in [10, 11, 9], which consists in the SSEP with a slowbond of intensity g ( n ) = αn − β , α > , β ≥ . In that model particles performnearest-neighbor symmetric random walks, whose jump rate is equal to one atall bonds, except at a particular bond , where it is equal to g ( n ) . From [10, 11, 9] TERTULIANO FRANCO, PATR´ICIA GONC¸ ALVES, AND GUNTER M. SCH ¨UTZ it is known that hydrodynamic limit/fluctuations for exclusion processes witha slow bond have three different behaviors depending on the regime of β .For the SSEP with a slow site that we treat here the methods used for theslow bond problem cannot be adapted in any straightforward fashion as theasymmetry at the slow site gives rise to novel difficulties in the study of itshydrodynamic behavior and fluctuations. The model is reversible, as is theSSEP with a slow bond, but it is not self-dual, in contrast to the SSEP witha slow bond. Moreover, the invariant measures for the SSEP with a slow siteare not translation invariant, as happens for the SSEP with a slow bond. Asa consequence, the proof of the hydrodynamic limit for the SSEP with a slowsite requires different approaches from the ones of [10, 11, 9] and one cannotnaively extend the results obtained for the slow bond to the case of the slowsite.As described above, in this paper we are able to characterize the hydrody-namic limit and the equilibrium fluctuations for g ( n ) = αn − β when α > , β > . The case ≤ β ≤ remains open. However, we present and moti-vate a conjecture on the behavior of the system in that case. Moreover, sincewe present also the hydrodynamic limit and the equilibrium fluctuations for g ( n ) close to one, namely g ( n ) = 1 + o (1) , the existence of a dynamical phasetransition in the behaviour of the system from periodic boundary conditions toNeumann boundary conditions at a critical value of β in the range ≤ β ≤ isestablished.In order to put our results into a broader perspective conservative particlesystems with defects we point out that for a single slow bond the SSEP treated[10, 11, 9] exhibits the same hydrodynamic behaviour as non-interacting ran-dom walks with a single slow bond. On the other hand, with a single slow siteboth hydrodynamic limit and fluctuations of non-interacting particles wouldbe driven by a disconnect behavior for any β > , leading to Dirichlet boundaryconditions with boundary densities 0. Thus the similarity between the SSEPand non-interacting particles that one finds for a slow bond breaks down for aslow site, adding further motivation for a detailed investigation of the SSEPwith a slow site.It is also worthwhile to compare our result with a result derived in [18] fora related problem. The model considered there is called the Bouchaud trapmodel. In that model, particles perform independent random walks in a ran-dom environment with traps given by i.i.d. alpha-stable random variables. In[18] it was proved that the hydrodynamic limit for such model is given by ageneralized partial differential equation depending on an alpha-stable subor-dinator. The present paper suggests that a trap model of exclusion type shouldnot have the same limit as obtained in [18] for a trap model of independentrandom walks. For the SSEP the asymmetry at the slow site yields a limitthat has some properties in common with a slow bond and therefore a behav-ior completely different from the one observed in [18].Here follows the outline of this paper. In Section 2 we give notations, pre-cise definitions and statements of the results. In Section 3 we present thehydrodynamic limit of the model. In Section 4 we present the equilibrium fluc-tuations (in infinite volume). In Section 5 we state a conjecture on what shouldbe the complete scenario for exclusion processes with a slow site. In Section 6
SEP WITH A SLOW SITE 5 we present an extra result on the hydrodynamic behavior of the SSEP with k neighboring slow bonds, which we use as an argument to sustain our conjec-ture in Section 5. 2. S TATEMENT OF RESULTS
The model.
A particle system can be constructed through its generatoror via Poisson processes. In this work we will make use of both.Let T n = Z /n Z = { , , . . . , n − } be the one-dimensional discrete torus with n points. The simple symmetric exclusion process (SSEP) with a slow site isthe Markov process with state space { , } T n and with generator L n acting onfunctions f : { , } T n → R as L n f ( η ) = (cid:88) x,y ∈ T n | x − y |≤ ξ nx η ( x )(1 − η ( y )) [ f ( η x,y ) − f ( η )] , (2.1)where the jump rates ξ nx are given by ξ nx = (cid:40) g ( n ) , if x = 0 , , if x ∈ T n \{ } , where g ( n ) > and η x,x +1 is the configuration obtained from η by exchangingthe occupation variables η ( x ) and η ( x + 1) . Formally, ( η x,x +1 )( y ) = η ( x + 1) , if y = x ,η ( x ) , if y = x + 1 ,η ( y ) , otherwise. (2.2)Its dynamics can be described as follows. To each site we attach two Poissonprocesses, one corresponding to jumps from x to x +1 and the other correspond-ing to jumps from x to x − . If the site x is occupied and the site x + 1 is empty,the particle moves from site x to site x + 1 at a time arrival of the Poissonprocess associated to { x, x + 1 } , and analogously for sites { x, x − } . The jumprates corresponding to those transitions are shown in Figure 1.For fixed n , let { η τ : τ ≥ } be the Markov process with generator L n . Noticethat η τ depends on g ( n ) , but we do not display this dependence in the notation.We denote by { η t : t ≥ } the Markov process with generator n L n . Thistime factor n is the so-called diffusive time scaling . We observe that this isequivalent to define η t := η n τ .Next we establish a family of invariant measures (in fact, reversible) for thedynamics introduced above. Proposition 2.1.
For any p ∈ [0 , , the Bernoulli product measure ν p on thespace { , } T n with marginals given by ν p { η ; η ( x ) = 1 } = m p ( x ) = pg ( n ) (1 − p ) + pg ( n ) , if x = 0 ,p , if x ∈ T n \{ } , (2.3) is reversible for the Markov process { η τ : τ ≥ } . TERTULIANO FRANCO, PATR´ICIA GONC¸ ALVES, AND GUNTER M. SCH ¨UTZ
The proof of this proposition consists only in checking the detailed balanceequation, which is straightforward and for that reason it will be omitted.Above and in what follows, a sub-index in a function means a variable, not aderivative . Denote by T the one-dimensional continuous torus R / Z = [0 , andby (cid:104)· , ·(cid:105) the inner product in L [0 , .2.2. Hydrodynamics.Definition 1.
Let ρ : T → [0 , be a measurable function. We say that ρ is aweak solution of the heat equation with periodic boundary conditions given by (cid:26) ∂ t ρ ( t, u ) = ∂ u ρ ( t, u ) , t ≥ , u ∈ T ,ρ (0 , u ) = ρ ( u ) , u ∈ T , (2.4) if, for all t ∈ [0 , T ] and for all H ∈ C ( T ) , (cid:104) ρ t , H t (cid:105)−(cid:104) ρ , H (cid:105)− (cid:90) t (cid:10) ρ s , ∂ u H s (cid:11) ds = 0 . (2.5)Next, we define what we mean by weak solutions of the heat equation withNeumann boundary conditions, as given in (1.2). We introduce first some tech-nical background on Sobolev spaces. Definition 2.
Let H be the set of all L functions ζ : [0 , → R such that thereexists a function ∂ u ζ ∈ L [0 , satisfying (cid:104) ∂ u G, ζ (cid:105) = −(cid:104) G, ∂ u ζ (cid:105) , for all G ∈ C ∞ [0 , with compact support contained in (0 , . For ζ ∈ H , wedefine the norm (cid:107) ζ (cid:107) H := (cid:16) (cid:107) ζ (cid:107) L [0 , + (cid:107) ∂ u ζ (cid:107) L [0 , (cid:17) / . Let L (0 , T ; H ) be the space of all measurable functions ξ : [0 , T ] → H suchthat (cid:107) ξ (cid:107) L (0 ,T ; H ) := (cid:90) T (cid:107) ξ t (cid:107) H dt < ∞ . Abusing notation slightly, we denote by C [0 , the set of functions H : T → R that are continuously twice differentiable in T \{ } and have a C -extensionto the closed interval [0 , . Definition 3.
Let ρ : T → [0 , be a measurable function. We say that ρ is aweak solution of the heat equation with Neumann boundary conditions ∂ t ρ ( t, u ) = ∂ u ρ ( t, u ) , t ≥ , u ∈ (0 , ,∂ u ρ ( t, + ) = ∂ u ρ ( t, − ) = 0 , t ≥ ,ρ (0 , u ) = ρ ( u ) , u ∈ (0 , , (2.6) if ρ belongs to L (0 , T ; H ) and for all t ∈ [0 , T ] and for all H ∈ C [0 , , (cid:104) ρ t , H (cid:105) − (cid:104) ρ , H (cid:105) − (cid:90) t (cid:10) ρ s , ∂ u H (cid:11) ds − (cid:90) t (cid:0) ρ s (0 + ) ∂ u H (0 + ) − ρ s (0 − ) ∂ u H (0 − ) (cid:1) ds = 0 . SEP WITH A SLOW SITE 7
Let D ( R + , { , } T n ) be the path space of c `adl `ag trajectories with values in { , } T n . For a measure µ n on { , } T n , denote by P µ n the probability measureon D ( R + , { , } T n ) induced by the initial state µ n and the Markov process { η t : t ≥ } . Notice that in fact P µ n = P g ( n ) ,nµ n but we will not carry the dependenceon n nor g in order to not overload notation. By E µ n we mean the expectationwith respect to P µ n .The notation η · is reserved to represent elements of the Skorohod space D ( R + , { , } T n ) , i.e., time trajectories of the exclusion process with a slow site.This notation η · should not be confused with the notation η for elements of { , } T n .From now on we fix a profile γ : T → [0 , , representing the initial densityof particles. To avoid uninteresting technical complications, we assume that γ is continuous at all x ∈ T \{ } and bounded from below by a positive constant: ζ := inf x ∈ T γ ( x ) > . (2.7) Theorem 2.2.
For each n ∈ N , let µ n be a Bernoulli product measure on { , } T n with marginal distributions given by µ n { η ; η ( x ) = 1 } = γ ( xn ) . (2.8) Then, for any t > , for every δ > and every H ∈ C ( T ) , it holds that lim n →∞ P µ n (cid:110) η · : (cid:12)(cid:12)(cid:12) n (cid:88) x ∈ T n H ( xn ) η t ( x ) − (cid:90) T H ( u ) ρ ( t, u ) du (cid:12)(cid:12)(cid:12) > δ (cid:111) = 0 , (2.9) where • for g ( n ) = 1 + o (1) , ρ is the unique weak solution of (2.4) ; • for g ( n ) = αn − β , α > , β > , ρ is the unique weak solution of (2.6) ;and where, in both cases, the initial condition of the corresponding partial dif-ferential equation is given by ρ = γ . Remark 2.3.
About the constants: To avoid repetitions along the paper, wefix, once and for all, the assumptions α > and β > . Remark 2.4.
About the statement of the Theorem: If at the initial time thedensity of particles converges to the profile γ ( · ) , then, in the future time t , thedensity of particles converges to a profile ρ ( t, · ) which is the weak solution ofthe heat equation with the corresponding boundary conditions and with initialcondition ρ = γ . Remark 2.5.
About the scaling: In the claim of the Theorem 2.2 one can seethat the space is rescaled by n − (space between sites) and time is rescaled by n , since the “future time” is indeed tn . This is the diffusive time scaling. Remark 2.6.
About the initial measure: We can weaken the hypothesis on µ n by dropping the condition of being a product measure, and assuming that { µ n } n ∈ N is associated to γ ( · ) , see [19]. In that case the statement of Theorem2.2 remains in force. However, this hypothesis would complicate the attrac-tiveness tools at Subsection 3.5 and for this reason we assume (2.8). From the French, “continuous from the right with limits from the left”.
TERTULIANO FRANCO, PATR´ICIA GONC¸ ALVES, AND GUNTER M. SCH ¨UTZ
Remark 2.7.
About the weak solution: The weak solution of (2.6) is a func-tion defined on the interval [0 , , not on the torus. But, as already explained,Lebesgue almost sure, it is the same. Thus it makes sense to integrate ρ in thetorus T , as it appears in the equation (2.9).2.3. Equilibrium density fluctuations.
In this section we consider η t evolv-ing on the one-dimensional lattice Z and starting from the invariant state ν p ,with p ∈ (0 , . Therefore, the generator of the process is given by (2.1) with T n replaced by Z , namely L n f ( η ) = (cid:88) x,y ∈ Z | x − y |≤ ξ nx η ( x )(1 − η ( y )) [ f ( η x,y ) − f ( η )] , for local functions f : { , } Z → R .From now on we fix p ∈ (0 , . In order to establish the central limit theorem(C.L.T.) for the density under the invariant state ν p , we need to introduce thefluctuation field as the linear functional acting on test functions H as Y nt ( H ) = 1 √ n (cid:88) x ∈ Z H (cid:16) xn (cid:17) ( η t ( x ) − m p ( x )) , (2.10)where m p ( x ) is the mean of η t ( x ) with respect to ν p introduced in (2.3). We em-phasize that { η t : t ≥ } is the Markov process with generator n L n . Denote by P p the probability measure on the Skorohod path space D ( R + , { , } Z ) inducedby the initial state ν p and the Markov process { η t : t ≥ } and we denote by E p the expectation with respect to P p .Now we introduce the space of test functions. Since the hydrodynamics isgoverned by different partial differential equations, the state space for H de-pends on the jump rate g ( n ) that we defined at the slow site. Definition 4.
Let S ( R ) be the usual Schwartz space of functions H : R → R such that H ∈ C ∞ ( R ) and (cid:107) H (cid:107) k,(cid:96) := sup x ∈ R | (1 + | x | (cid:96) ) d k Hdx k ( x ) | < ∞ , for all integers k, (cid:96) ≥ . We define S Neu ( R ) as the space composed of functions H : R → R such that(1) Except possibly at x = 0 , the function H is continuous and infinitelydifferentiable,(2) The function H is continuous from the right at zero,(3) For all integers k, (cid:96) ≥ , (cid:107) H (cid:107) k,(cid:96), + := sup x> | (1 + | x | (cid:96) ) d k dx k H ( x ) | < ∞ , and (cid:107) H (cid:107) k,(cid:96), − := sup x< | (1 + | x | (cid:96) ) d k Hdx k ( x ) | < ∞ , (4) For any integer k ≥ , lim x → + d k +1 Hdx k +1 ( x ) = lim x → − d k +1 Hdx k +1 ( x ) = 0 . SEP WITH A SLOW SITE 9
Notice that it is not required that H is continuous at x = 0 . Intuitively,this space S Neu ( R ) corresponds to two independent Schwartz spaces in eachhalf line. The chosen notation comes from the expression Neumann boundaryconditions .Both spaces S ( R ) and S Neu ( R ) are Fr´echet spaces. The proof that S ( R ) isFr´echet can be found in [26], for instance. The proof that S Neu ( R ) is Fr´echet isquite similar and will be omitted.The set of continuous linear functions f : S ( R ) → R and f : S Neu ( R ) → R with respect to the topology generated by the corresponding semi-norms willbe denoted by S (cid:48) ( R ) and S (cid:48) Neu ( R ) , respectively.The notation ∇ and ∆ mean the first and second space derivatives. In thecase of S Neu ( R ) , we will make use of the following definition: Definition 5.
We define the operators ∇ Neu : S Neu ( R ) → S Neu ( R ) and ∆ Neu : S Neu ( R ) → S Neu ( R ) by ∇ Neu H ( u ) = (cid:40) dHdu ( u ) , if u (cid:54) = 0 , lim u → + dHdu ( u ) , if u = 0 , ∆ Neu H ( u ) = (cid:40) d Hdu ( u ) , if u (cid:54) = 0 , lim u → + d Hdu ( u ) , if u = 0 , Notice that these operators are essentially the first and second space deriva-tives, but defined in specific domains, which changes the meaning of the opera-tor. Roughly speaking, the operator ∆ Neu is the operator associated to a systemblocked at the origin.2.4.
Ornstein-Uhlenbeck process.
Denote by χ ( p ) = p (1 − p ) the so-called static compressibility of the system. Based on [15, 19], we have a characteriza-tion of the generalized Ornstein-Uhlenbeck process, which is a solution of d Y t = ∆ Y t dt + (cid:112) χ ( p ) ∇ d W t , (2.11)where d W t is a space-time white noise of unit variance, in terms of a mar-tingale problem. We will see later that this process, which take values on S (cid:48) ( R ) , governs the equilibrium fluctuations of the density of particles whenthe strength of the slow site is given by g ( n ) = 1 + o (1) .On the other hand, when the strength is given by g ( n ) = αn − β , the corre-sponding Ornstein-Uhlenbeck process will be the solution of d Y t = ∆ Neu Y t dt + (cid:112) χ ( p ) ∇ Neu d W t , (2.12)and taking values on S (cid:48) Neu ( R ) .In what follows D ([0 , T ] , S (cid:48) ( R )) (resp. C ([0 , T ] , S (cid:48) ( R )) ) is the space of c `adl `ag(resp. continuous) S (cid:48) ( R ) valued functions endowed with the Skohorod topology.Analogous definitions hold for D ([0 , T ] , S (cid:48) Neu ( R )) and C ([0 , T ] , S (cid:48) Neu ( R )) .The rigorous meaning of equations (2.11) and (2.12) is given in terms of thetwo next propositions. Denote by T t : S ( R ) → S ( R ) the semi-group of the heatequation in the line (see [12] for instance). It is well known that Proposition 2.8.
There exists an unique random element Y · taking values inthe space C ([0 , T ] , S (cid:48) ( R )) such that: i) For every function H ∈ S ( R ) , M t ( H ) and N t ( H ) , given by M t ( H ) = Y t ( H ) − Y ( H ) − (cid:90) t Y s (∆ H ) ds , N t ( H ) = (cid:0) M t ( H ) (cid:1) − χ ( p ) t (cid:107)∇ H (cid:107) L ( R ) , (2.13) are F t -martingales, where for each t ∈ [0 , T ] , F t := σ ( Y s ( H ); s ≤ t, H ∈S ( R )) . ii) Y is a Gaussian field of mean zero and covariance given on G, H ∈ S ( R ) by E p (cid:2) Y ( G ) Y ( H ) (cid:3) = χ ( p ) (cid:90) R G ( u ) H ( u ) du . (2.14) Moreover, for each H ∈ S ( R ) , the stochastic process {Y t ( H ) : t ≥ } is Gaussian,being the distribution of Y t ( H ) conditionally to F s , for s < t , normal of mean Y s ( T t − s H ) and variance (cid:82) t − s (cid:107)∇ T r H (cid:107) L ( R ) dr . We call the random element Y · the generalized Ornstein-Uhlenbeck processof characteristics ∇ and ∆ . From the second equation in (2.13) and L´evy’sTheorem on the martingale characterization of Brownian motion, the process (cid:0) χ ( p ) (cid:107)∇ H (cid:107) L ( R ) (cid:1) − / M t ( H ) (2.15)is a standard Brownian motion. Therefore, in view of Proposition 2.8, it makessense to say that Y · is the formal solution of (2.11).Now, let T Neu t : S Neu ( R ) → S Neu ( R ) be the semi-group associated to the fol-lowing partial differential equation with Neumann boundary conditions: ∂ t u ( t, x ) = ∂ x u ( t, x ) , t ≥ , x ∈ R \{ } ∂ x u ( t, + ) = ∂ x u ( t, − ) = 0 t ≥ u (0 , x ) = H ( x ) , x ∈ R . (2.16)See [12] for an explicit expression of T Neu t . In a similar way, we have Proposition 2.9 (See [12]) . There exists an unique random element Y · takingvalues in the space C ([0 , T ] , S (cid:48) Neu ( R )) such that: i) For every function H ∈ S Neu ( R ) , M t ( H ) and N t ( H ) given by M t ( H ) = Y t ( H ) − Y ( H ) − (cid:90) t Y s (∆ Neu H ) ds , N t ( H ) = (cid:0) M t ( H ) (cid:1) − χ ( p ) t (cid:107)∇ Neu H (cid:107) L ( R ) (2.17) are F t -martingales, where for each t ∈ [0 , T ] , F t := σ ( Y s ( H ); s ≤ t, H ∈S Neu ( R )) . ii) Y is a Gaussian field of mean zero and covariance given on G, H ∈S Neu ( R ) by (2.14) .Moreover, for each H ∈ S Neu ( R ) , the stochastic process {Y t ( H ) ; t ≥ } is Gauss-ian, being the distribution of Y t ( H ) conditionally to F s , for s < t , normal ofmean Y s ( T Neu t − s H ) and variance (cid:82) t − s (cid:107)∇ T Neu r H (cid:107) L ( R ) dr . We call the random element Y · the generalized Ornstein-Uhlenbeck processof characteristics ∇ Neu and ∆ Neu . We are in position to state our result for thefluctuations of the density of particles.
SEP WITH A SLOW SITE 11
Theorem 2.10 (C.L.T. for the density of particles) . Consider the Markov process { η t : t ≥ } starting from the invariant state ν p under the assumption g ( n ) = 1 + o (1) . Then, the sequence of processes {Y nt } n ∈ N converges in distribution, as n → + ∞ , with respect to the Skorohod topology of D ([0 , T ] , S (cid:48) ( R )) to Y t in C ([0 , T ] , S (cid:48) ( R )) , the generalized Ornstein-Uhlenbeck pro-cess of characteristics ∆ , ∇ which is the formal solution of the equation (2.11) .On the other hand, if we consider g ( n ) = αn − β , α > and β > , then {Y nt } n ∈ N converges in distribution, as n → + ∞ , with respect to the Skoro-hod topology of D ([0 , T ] , S (cid:48) Neu ( R )) to Y t in C ([0 , T ] , S (cid:48) Neu ( R )) , the generalizedOrnstein-Uhlenbeck process of characteristics ∆ Neu , ∇ Neu which is the formalsolution of the equation (2.12) .
3. H
YDRODYNAMICS
We proceed to define the spatial density of particles of the exclusion process,where we embed the discrete torus T n in the continuous torus T .Let M be the space of positive measures on T with total mass bounded byone, endowed with the weak topology. Let π nt ∈ M be the measure on T ob-tained by rescaling time by n , rescaling space by n − , and assigning mass n − to each particle, i.e., π nt ( η, du ) = n (cid:88) x ∈ T n η t ( x ) δ x/n ( du ) , (3.1)where δ u is the Dirac measure concentrated on u . The usual name for π nt ( η, du ) is empirical measure . For an integrable function H : T → R , the expression (cid:104) π nt , H (cid:105) stands for the integral of H with respect to π nt : (cid:104) π nt , H (cid:105) = n (cid:88) x ∈ T n H ( xn ) η t ( x ) . This notation is not to be mistaken with the inner product in L ( T ) . Also, when π t has a density ρ , namely when π ( t, du ) = ρ ( t, u ) du , we sometimes write (cid:104) ρ t , H (cid:105) for (cid:104) π t , H (cid:105) .To avoid unwanted topological issues, in the entire paper a time horizon T > is fixed. Let D ([0 , T ] , M ) be the space of M -valued c `adl `ag trajectories π : [0 , T ] → M endowed with the Skorohod topology. For each probability mea-sure µ n on { , } T n , denote by Q nµ n the measure on the path space D ([0 , T ] , M ) induced by the measure µ n and the process π nt introduced in (3.1).Recall the profile γ : T → [0 , and the sequence { µ n } n ∈ N of measures on { , } T n defined through (2.8). Let Q be the probability measure on the space D ([0 , T ] , M ) concentrated on the deterministic path π ( t, du ) = ρ ( t, u ) du , where • if g ( n ) = 1 + o (1) , the function ρ is the unique weak solution of (2.4); • if g ( n ) = αn β , the function ρ is the unique weak solution of (2.6). Proposition 3.1.
Considering the two possibilities above for the function g , thesequence of probability measures { Q nµ n } n ∈ N converges weakly to Q as n → ∞ . Since Theorem 2.2 is an immediate corollary of the previous proposition, ourgoal is to prove Proposition 3.1.
The proof is divided in several parts. In Subsection 3.1, we show that thesequence { Q nµ n } n ∈ N is tight. In Sections 3.4, 3.6, we show that, for each caseof g , Q is the only possible limit along subsequences of { Q nµ n } n ∈ N . This assuresthat the sequence { Q nµ n } n ∈ N converges weakly to Q , as n → ∞ .3.1. Tightness.
In order to prove tightness of { π nt : 0 ≤ t ≤ T } n ∈ N it is enoughto show tightness of the real-valued processes {(cid:104) π nt , H (cid:105) : 0 ≤ t ≤ T } n ∈ N for aset of functions H ∈ C ( T ) , provided this set of functions is dense in C ( T ) withrespect to the uniform topology (see [19, page 54, Proposition 1.7]). For thatpurpose, let H ∈ C ( T ) . By Dynkin’s formula, M nt ( H ) := (cid:104) π nt , H (cid:105) − (cid:104) π n , H (cid:105) − (cid:90) t n L n (cid:104) π ns , H (cid:105) ds (3.2)is a martingale with respect to the natural filtration F t := σ ( η s : s ≤ t ) . More-over, (cid:0) M nt ( H ) (cid:1) − (cid:90) t (cid:16) n L n [ (cid:104) π ns , H (cid:105) ] − (cid:104) π ns , H (cid:105) n L n (cid:104) π ns , H (cid:105) (cid:17) ds (3.3)is also a martingale with respect to the same filtration, see [19]. In order toprove tightness of { π nt ( H ) : 0 ≤ t ≤ T } n ∈ N , we shall prove tightness of eachterm in the formula above and then we invoke the fact that a sequence of afinite sum of tight processes is again tight.Since (3.3) is a martingale, doing elementary calculations we obtain the qua-dratic variation of M nt ( H ) at time T as (cid:104) M n ( H ) (cid:105) T = (cid:90) T (cid:88) x ∈ T n \{ } (cid:16) ( η s ( x ) − η s ( x + 1))( H ( x +1 n ) − H ( xn )) (cid:17) ds + (cid:90) T (cid:16) η s (1)(1 − η s (0)) + g ( n ) η s (0)(1 − η s (1)) (cid:17) ( H ( n ) − H ( n )) ds + (cid:90) T (cid:16) η s ( − − η s (0)) + g ( n ) η s (0)(1 − η s ( − (cid:17) ( H ( − n ) − H ( n )) ds . (3.4)The smoothness of H implies that lim n →∞ E µ n (cid:2) (cid:104) M n ( H ) (cid:105) T (cid:3) = 0 . Hence M nT ( H ) converges to zero in L ( P µ n ) as n → ∞ and, by Doob’s inequality, for every δ > , lim n →∞ P µ n (cid:26) η · : sup ≤ t ≤ T | M nt ( H ) | > δ (cid:27) = 0 . (3.5)In particular, this yields tightness of the sequence of martingales { M nt ( H ) :0 ≤ t ≤ T } n ∈ N .A long computation, albeit completely elementary, shows us that the term n L n (cid:104) π ns , H (cid:105) appearing inside the time integral in (3.2) can be rewritten as n (cid:88) x ∈ T n \{ } (cid:16) H ( x +1 n ) + H ( x − n ) − H ( xn ) (cid:17) η s ( x )+ ng ( n ) (cid:16) H ( n ) + H ( − n ) − H ( n ) (cid:17) η s (0)+ n (1 − g ( n )) (cid:104) ( H ( − n ) − H ( n )) η s (0) η s ( −
1) + ( H ( n ) − H ( n )) η s (0) η s (1) (cid:105) . (3.6) SEP WITH A SLOW SITE 13
We note that the first term above corresponds to the discrete Laplacian lead-ing to the heat equation, while the other two terms arise from the boundaryconditions.By the smoothness of H again, there exists a constant c H > such that | n L n (cid:104) π ns , H (cid:105)| ≤ c H , which in turn gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) tr n L n (cid:104) π ns , H (cid:105) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ c H | t − r | . By the Arzel `a-Ascoli Theorem the sequence of these integral terms is a rela-tively compact set, with respect to the uniform topology, therefore it is tight.The term (cid:104) π n , H (cid:105) is constant in time and bounded, thus is tight as well. Thisconcludes the proof that the set of measures { Q nµ n } n ∈ N is tight.3.2. Entropy.
Denote by H ( µ | ν p ) the entropy of a probability measure µ withrespect to the invariant state ν p . For a precise definition and properties of theentropy, we refer the reader to [19]. Proposition 3.2.
There exists a finite constant K := K ( p ) , such that H ( µ | ν p ) ≤ K n , for any probability measure µ on { , } T n .Proof. Recall the definition of ν p and notice that H ( µ | ν p ) = (cid:88) η ∈{ , } T n µ ( η ) log (cid:16) µ ( η ) ν p ( η ) (cid:17) ≤ (cid:88) η ∈{ , } T n µ ( η ) log (cid:16) ν p ( η ) (cid:17) . Recall (2.3). By the assumption < p < and the inequality ν p ( η ) ≥ ( p ∧ (1 − p )) n − ( m p (0) ∧ (1 − m p (0))) we conclude that H ( µ | ν p ) ≤ K n for some K > depending only on p . (cid:3) A remark: In particular, the estimate H ( µ n | ν p ) ≤ K n holds for the mea-sures µ n defined in (2.8).3.3. Dirichlet form.
Let f be any density with respect to the invariant mea-sure ν p . In others words, f is a non negative function f : { , } T n → R satisfying (cid:82) f ( η ) ν p ( dη ) = 1 . The Dirichlet form D n is the convex and lower semicontinu-ous functional defined through D n ( (cid:112) f ) = − (cid:90) (cid:112) f ( η ) L n (cid:112) f ( η ) ν p ( dη ) . Invoking a general result [19, Appendix 1, Prop. 10.1] we can write D n as D n ( (cid:112) f ) = g ( n )2 (cid:90) η (0)(1 − η (1)) (cid:110)(cid:112) f ( η , ) − (cid:112) f ( η ) (cid:111) ν p ( dη )+ 12 (cid:90) η (1)(1 − η (0)) (cid:110)(cid:112) f ( η , ) − (cid:112) f ( η ) (cid:111) ν p ( dη )+ 12 (cid:90) η ( − − η (0)) (cid:110)(cid:112) f ( η , − ) − (cid:112) f ( η ) (cid:111) ν p ( dη )+ g ( n )2 (cid:90) η (0)(1 − η ( − (cid:110)(cid:112) f ( η , − ) − (cid:112) f ( η ) (cid:111) ν p ( dη )+ 12 (cid:88) x ∈ T nx (cid:54) =0 , − (cid:90) (cid:110)(cid:112) f ( η x,x +1 ) − (cid:112) f ( η ) (cid:111) ν p ( dη ) , (3.7)where η x,x +1 has been defined in (2.2). Proposition 3.3.
Let Q ∗ be a limit of a subsequence of the sequence of probabil-ities measures { Q nµ n } n ∈ N . Then Q ∗ is concentrated on trajectories π ( t, du ) with adensity with respect to the Lebesgue measure, i.e., of the form π ( t, du ) = ρ ( t, u ) du .Moreover, the density ρ ( t, u ) belongs to the space L (0 , T ; H ) , see Definition 2. The proof of the proposition above can be adapted from [10, Proposition 5.6]and for this reason it will be omitted. For the interested reader, we brieflyindicate some steps of this adaptation.We begin by observing that [10, Proposition 5.6] is in fact a consequence of[10, Lemma 5.8]. Thus we just describe how to prove, in our case, the state-ment in [10, Lemma 5.8].There are two basic ingredients in the proof of [10, Lemma 5.8]. The first oneis that the entropy (with respect to the invariant measure) of any probabilitymeasure on the state space of the process, namely, { , } T n does not grow morethan linearly. In our case, this result is proved in Proposition 3.2.The second ingredient in the proof of [10, Lemma 5.8] is the fact that, exceptat the defect, the Dirichlet form of the considered process coincides with theDirichlet form of the homogeneous exclusion process. This fact indeed holdsfor the exclusion process with a slow site and therefore the same proof of [10,Lemma 5.8] applies here.3.4. Hydrodynamic limit for g ( n ) = 1 + o (1) . Proof of Proposition 3.1 for g ( n ) = 1 + o (1) . Let Q ∗ be the weak limit of someconvergent subsequence { Q n j µ nj } j ∈ N of the sequence { Q nµ n } n ∈ N . In order notto overburden the notation, denote this subsequence just by { Q nµ n } n ∈ N . ByProposition 3.3, the probability measure Q ∗ is concentrated on trajectories π ( t, du ) = ρ ( t, u ) du such that ρ ( t, u ) ∈ L (0 , T ; H ) . Our goal is to concludethat ρ is a weak solution of the partial differential equation (2.4).Let H ∈ C ( T ) . We claim that Q ∗ (cid:110) π : (cid:104) π t , H (cid:105) − (cid:104) π , H (cid:105) − (cid:90) t (cid:104) π s , ∂ u H (cid:105) ds = 0 , ∀ t ∈ [0 , T ] (cid:111) = 1 . (3.8) SEP WITH A SLOW SITE 15
To prove this, it suffices to show that Q ∗ (cid:110) π : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) π t , H (cid:105) − (cid:104) π , H (cid:105) − (cid:90) t (cid:104) π s , ∂ u H (cid:105) ds (cid:12)(cid:12)(cid:12) > δ (cid:111) = 0 , for every δ > . Since the supremum is a continuous function in the Skorohodmetric, by Portmanteau’s Theorem, the probability above is smaller or equalthan lim inf n →∞ Q nµ n (cid:110) π : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) π t , H (cid:105) − (cid:104) π , H (cid:105) − (cid:90) t (cid:104) π s , ∂ u H (cid:105) ds (cid:12)(cid:12)(cid:12) > δ (cid:111) . Since Q nµ n is the measure on the space D ([0 , T ] , M ) induced by P nµ n via theempirical measure, we can rewrite the expression above as lim inf n →∞ P µ n (cid:110) η · : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) π nt , H (cid:105) − (cid:104) π n , H (cid:105) − (cid:90) t (cid:104) π ns , ∂ u H (cid:105) ds (cid:12)(cid:12)(cid:12) > δ (cid:111) . Adding and subtracting n L n (cid:104) π ns , H (cid:105) to the integral term above, we can seethat the previous expression is bounded from above by the sum of lim sup n →∞ P µ n (cid:110) sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) π nt , H (cid:105) − (cid:104) π n , H (cid:105) − (cid:90) t n L n (cid:104) π ns , H (cid:105) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) (3.9)and lim sup n →∞ P µ n (cid:110) sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:90) t (cid:0) n L n (cid:104) π ns , H (cid:105) − (cid:104) π ns , ∂ u H (cid:105) (cid:1) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) . (3.10)As already verified in Subsection 3.1, the quadratic variation of the martingale M nt ( H ) given in (3.2) goes to zero, as n → ∞ . Therefore, by Doob’s inequality,expression (3.9) is null.It remains to show that (3.10) also vanishes. Recall (3.6) for n L n (cid:104) π ns , H (cid:105) .Let us examine its terms.The first term in the sum (3.6) is n (cid:88) x ∈ T n \{ } (cid:16) H ( x +1 n ) + H ( x − n ) − H ( xn ) (cid:17) η s ( x ) , from which one can subtract (cid:104) π ns , ∂ u H (cid:105) = n (cid:88) x ∈ T n ∂ u H ( xn ) η s ( x ) , this difference being bounded (in modulus) by c H /n , again because H ∈ C ( T ) ,where c H > is a constant depending only on H .The second term in (3.6) is n (cid:16) o (1) (cid:17)(cid:16) H ( n ) + H ( − n ) − H ( n ) (cid:17) η s (0) , which converges to zero as n → ∞ , because H ∈ C ( T ) .The last term in (3.6) is n (1 − g ( n )) (cid:104) ( H ( − n ) − H ( n )) η s (0) η s ( −
1) + ( H ( n ) − H ( n )) η s (0) η s (1) (cid:105) , which goes to zero, as n goes to infinity, because H is smooth and g ( n ) = 1+ o (1) .By the facts above we conclude that (3.10) is zero, proving the claim.Now, let { H i } i ≥ be a countable dense set of functions in C ( T ) , with respectto the norm (cid:107) H (cid:107) ∞ + (cid:107) ∂ u H (cid:107) ∞ + (cid:107) ∂ u H (cid:107) ∞ . Intersecting a countable number of sets of probability one, (3.8) can be extended for all functions H ∈ C ( T ) simultaneously, proving that Q ∗ is concentrated on weak solutions of (2.4).Since there exists only one weak solution of (2.4), it means that Q ∗ is equalto the aforementioned probability measure Q . Invoking tightness proved inSubsection 3.1, we conclude that the entire sequence { Q nµ n } n ∈ N converges to Q as n → ∞ . (cid:3) Law of large numbers for the occupation at the origin.
Recall thedefinition of µ n given in (2.8) . Proposition 3.4.
Consider g ( n ) = αn − β . Let γ : T → [0 , be a continuousprofile, except possibly at x = 0 and satisfying (2.7) . Then, for all t > and ε > , lim n →∞ P µ n (cid:110) η · : nt (cid:90) t (1 − η s (0)) ds > ε (cid:111) = 0 . (3.11)This statement says that the site x = 0 remains empty a fraction of timesmaller than /n . A simple heuristics for this statement is the following. Thetime a particles takes to escape the slow site is, at least, an exponential randomvariable of parameter g ( n ) . If a random variable has exponential distributionof parameter λ , its expectation is /λ . Hence, the time average that a trappedparticle takes to escape from the slow site is at least n β /α (if its neighboringsites are occupied, the trapped particle can spend even more time there). Astime goes by, the slow site will remain empty a fraction of time at most g ( n ) = αn − β . Since β > , this would lead to (3.11).Despite the simplicity of the heuristics above, it is not straightforward totransform it into a rigorous argument. In order to prove Proposition 3.4, wewill make use of attractiveness and the knowledge on the invariant measures.For that purpose, in { , } T n we introduce the natural order between configura-tions: η ≤ ζ if and only if η ( x ) ≤ ζ ( x ) for all x ∈ T n . A function f : { , } T n → R is said to be monotone if f ( η ) ≤ f ( ζ ) whenever η ≤ ζ . This partial order isnaturally extended to the space of measures.We write µ ≤ st µ if, and only if, (cid:90) f dµ ≤ (cid:90) f dµ for all monotone functions f . In this case, we say that µ is stochasticallydominated from above by µ . The next result is well known and can found in[22] for instance. Theorem 3.5.
Let µ and µ be two probability measures on { , } T n . Thestatements below are equivalent:(1) µ ≤ st µ ;(2) There exists a probability measure ¯ µ on { , } T n × { , } T n such that itsfirst and second marginals are µ and µ , respectively, and ¯ µ is “concen-trated above the diagonal”, which means ¯ µ (cid:8) ( η, ζ ) : η ≤ ζ (cid:9) = 1 . Next, we construct such a measure ¯ µ , with the aforementioned property, bymeans of the so-called graphical construction . SEP WITH A SLOW SITE 17
Fix n ∈ N . For each site x of T n , we associate two Poisson point processes N n, − x and N n, + x , all of them being independent. The parameters of those Pois-son process agree with the Figure 1. In other words, the parameter of N n, − x and of N n, + x is one for all x ∈ T n , except for x = 0 , for which the parameter of N n, − and N n, +0 is equal to g ( n ) .Given an initial configuration of particles η ∈ { , } T n and the “toss” of thosePoisson processes, the dynamics will be the following. At a time arrival of somePoisson process, let us say, a time arrival of N n, − x , if there is a particle at thesite x , and there is no particle at the site x − , the particle at x moves to x − . The analogous happens with respect to a Poisson process of type N n, + x ,in which the movement (if possible) is from x to x + 1 . This construction yieldsthe same Markov process previously defined via the generator given in (2.1).Consider now two probability measures µ and µ in { , } T n such that µ ≤ st µ . By Theorem 3.5, there exists a measure ¯ µ on { , } T n ×{ , } T n concentratedabove the diagonal, as it is stated there. Evolving a configuration ( η , η ) ,chosen by ¯ µ , by the same set of Poisson point processes described above, weare lead to η t ≤ η t , for any future time t > . A stochastic process enjoying thisproperty of preserving the partial order is said to be attractive . See Liggett’sbook [22] for more details on the subject.Notice that the specific value of g ( n ) does not play any role in the argumentabove. In resume, we can say that the defect does not destroy attractiveness.Having established attractiveness of the exclusion process with a slow sitewe make the following observation. Once the Theorem 2.2 is true for initialmeasures µ n conditioned to have a particle at the origin , the statement willremain in force for initial measures µ n . This is explained as follows.By attractiveness, we can construct both processes (the one starting from µ n and the one starting from µ n conditioned to have a particle at the origin)in such a way that these processes will differ at most at one site, for any latertime t . Therefore, the empirical measures (3.1) for each process will have thesame limit in distribution.Without loss of generality, we assume henceforth that there is a particle atthe origin at the initial time. Proof of Proposition 3.4.
Since we have assumed γ ( x ) ≥ ζ > , for all x ∈ T ,since there is a particle at the origin and since µ n is a product measure, we canfind p > small enough such that µ n ≥ st ν p , for any n ∈ N .Fix ε > . By attractiveness, P µ n (cid:110) η · : nt (cid:90) t η s (0) ds > ε (cid:111) ≥ P p (cid:110) η · : nt (cid:90) t η s (0) ds > ε (cid:111) , which in turn implies P µ n (cid:110) η · : nt (cid:90) t (1 − η s (0)) ds > ε (cid:111) ≤ P p (cid:110) η · : nt (cid:90) t (1 − η s (0)) ds > ε (cid:111) . By Chebyshev’s inequality and Fubini’s Theorem, P p (cid:110) η · : nt (cid:90) t (1 − η s (0)) ds > ε (cid:111) ≤ nε E p (cid:104) t (cid:90) t (1 − η s (0)) ds (cid:105) = nε (cid:16) − t (cid:90) t E p [ η s (0)] ds (cid:17) . Since ν p { η ; η (0) = 1 } = pg ( n ) (1 − p ) + pg ( n ) , we obtain that P µ n (cid:110) η · : nt (cid:90) t (1 − η s (0)) ds > ε (cid:111) ≤ nε (cid:16) − pg ( n ) (1 − p ) + pg ( n ) (cid:17) , finishing the proof since g ( n ) = αn − β , β > . (cid:3) Hydrodynamic limit for g ( n ) = αn − β . Recall that we have denoted (cid:98) εn (cid:99) , the integer part of εn , simply by εn . Define the right average at ∈ T n ,and the left average at − ∈ T n by η εn, R (1) := εn εn (cid:88) y =1 η ( y ) and η εn, L ( −
1) := εn n − (cid:88) y = n − εn η ( y ) , (3.12)respectively. Notice that none of these sums involve the occupation at the slowsite ∈ T n . Proposition 3.6.
For any t > , lim sup ε ↓ lim sup n →∞ E µ n (cid:104) (cid:12)(cid:12)(cid:12) (cid:90) t (cid:16) η s (1) − η εn, R s (1) (cid:17) ds (cid:12)(cid:12)(cid:12) (cid:105) = 0 and lim sup ε ↓ lim sup n →∞ E µ n (cid:104) (cid:12)(cid:12)(cid:12) (cid:90) t (cid:16) η s ( − − η εn, L s ( − (cid:17) ds (cid:12)(cid:12)(cid:12) (cid:105) = 0 . The last result says that we can replace the occupation at the neighboringsites of the slow site by their averages in closed boxes, provided that theseboxes do not cross the slow site. This kind of argument appears often in theliterature and can be found, for example, in [20].
Proof.
We treat the case x = +1 , the case x = − being analogous. FromJensen’s inequality and the definition of the entropy, for any N > , the expec-tation appearing in the statement of this proposition is bounded from aboveby H ( µ n | ν p ) N n + 1
N n log E p (cid:104) exp (cid:110) N n (cid:12)(cid:12)(cid:12) (cid:90) t { η s (1) − η εn, R s (1) } ds (cid:12)(cid:12)(cid:12)(cid:111)(cid:105) . (3.13)By Proposition 3.2, H ( µ n | ν p ) ≤ K n , hence the term on the left hand side oflast expression is bounded from above by K /N . Now, we bound the remainingterm. Since e | x | ≤ e x + e − x and lim sup n n log( a n + b n ) = max (cid:110) lim sup n n log( a n ) , lim sup n n log( b n ) (cid:111) , (3.14)we can remove the modulus inside the exponential. Moreover, by the Feynman-Kac formula the term on the right hand side of (3.13) is less than or equal to t sup f density (cid:110) (cid:90) { η (1) − η εn, R s (1) } f ( η ) ν p ( dη ) − n D n ( (cid:112) f ) (cid:111) ds . See, for example, Lemma A1.7.2 of [19].
SEP WITH A SLOW SITE 19
Notice that the expression above does not depend on N . We claim now that,for any density f , (cid:90) { η (1) − η εn, R s (1) } f ( η ) ν p ( dη ) ≤ ε + n D n ( (cid:112) f ) Since N is arbitrary large, once we prove this claim the proof will be finished.By the definition in (3.12), (cid:90) { η (1) − η εn, R s (1) } f ( η ) ν p ( dη ) = (cid:90) (cid:110) εn εn (cid:88) y =1 ( η (1) − η ( y )) (cid:111) f ( η ) ν p ( dη ) . Writing η ( x ) − η ( y ) as a telescopic sum, the right hand side of above can berewritten as (cid:90) (cid:110) εn εn (cid:88) y =1 y − (cid:88) z =1 ( η ( z ) − η ( z + 1)) (cid:111) f ( η ) ν p ( dη ) . Rewriting the expression above as twice the half and making the transforma-tion η (cid:55)→ η z,z +1 (for which the probability ν p is invariant) it becomes εn εn (cid:88) y =1 y − (cid:88) z =1 (cid:90) { η ( z ) − η ( z + 1) } ( f ( η ) − f ( η z,z +1 )) ν p ( dη ) . By ( a − b ) = ( √ a − √ b )( √ a + √ b ) and Cauchy-Schwarz’s inequality, we boundthe previous expression from above by εn εn (cid:88) y =1 y − (cid:88) z =1 A (cid:90) { η ( z ) − η ( z + 1) } (cid:16)(cid:112) f ( η ) + (cid:112) f ( η z,z +1 ) (cid:17) ν p ( dη )+ εn εn (cid:88) y =1 y − (cid:88) z =1 1 A (cid:90) (cid:16)(cid:112) f ( η ) − (cid:112) f ( η z,z +1 ) (cid:17) ν p ( dη ) , for any A > . Since f is a density and recalling (3.7), the expression aboveis bounded by A − D n ( f ) + 2 Aεn . Choosing A = 1 /n we achieve the claim,concluding the proof. (cid:3) Proof of Proposition 3.1 for g ( n ) = αn − β , α > , β > . Again, let Q ∗ be the weaklimit of some convergent subsequence { Q n j µ nj } j ∈ N of the sequence { Q nµ n } n ∈ N andto keep notation simple, denote this subsequence by { Q nµ n } n ∈ N . Recall thatby Proposition 3.3, the probability measure Q ∗ is concentrated on trajectories π ( t, du ) = ρ ( t, u ) du such that ρ ( t, u ) ∈ L (0 , T ; H ) . By the notion of trace inSobolev spaces, the integrals (cid:90) t ρ ( s, ds and (cid:90) t ρ ( s, ds (3.15)are well defined and are finite. See [6] for the properties of Sobolev spaces.More than that, since the Sobolev space in one dimension is composed of func-tions which are absolutely continuous, ρ has indeed left and right limits atzero.Our goal here is to conclude that ρ is a weak solution of the partial differen-tial equation (2.6). For this purpose, let H ∈ C [0 , . Notice that, if H is seen as a function inthe torus, H is possibly discontinuous at zero. We impose that H (0) = 0 . Weclaim that Q ∗ (cid:110) π : (cid:104) ρ t , H (cid:105) − (cid:104) ρ , H (cid:105) − (cid:90) t (cid:104) ρ s , ∂ u H (cid:105) ds − (cid:90) t (cid:0) ρ s (0) ∂ u H (0) − ρ s (1) ∂ u H (1) (cid:1) ds = 0 , ∀ t ∈ [0 , T ] (cid:111) = 1 . (3.16)To prove this, it suffices to show that Q ∗ (cid:110) π : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) ρ t , H (cid:105) − (cid:104) ρ , H (cid:105) − (cid:90) t (cid:104) ρ s , ∂ u H (cid:105) ds − (cid:90) t (cid:0) ρ s (0) ∂ u H (0) − ρ s (1) ∂ u H (1) (cid:1) ds (cid:12)(cid:12)(cid:12) > δ (cid:111) = 0 , for any δ > . Since the integrals in (3.15) are not defined in the whole Skoro-hod space D ([0 , T ] , M ) , we cannot apply Portmanteau’s Theorem yet.For that purpose, let ι ε ( u ) = ε − (0 ,ε ] ( u ) . Adding and subtracting the convo-lution of ρ ( t, u ) with ι ε at the boundaries, we bound the previous probability bythe sum of Q ∗ (cid:110) π : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) ρ t , H (cid:105) − (cid:104) ρ , H (cid:105) − (cid:90) t (cid:104) ρ s , ∂ u H (cid:105) ds − (cid:90) t (cid:0) ρ s ∗ ι ε )(0) ∂ u H (0) ds + (cid:90) t ( ρ s ∗ ι ε )(1 − ε ) ∂ u H (1) (cid:1) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) (3.17)and Q ∗ (cid:110) π : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:90) t (cid:0) ρ s ∗ ι ε )(0) ∂ u H (0) ds − (cid:90) t ( ρ s ∗ ι ε )(1 − ε ) ∂ u H (1) (cid:1) ds − (cid:90) t (cid:0) ρ s (0) ∂ u H (0) ds + (cid:90) t ρ s (1) ∂ u H (1) (cid:1) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) . Since ρ has left and right side limits, taking ε small, the previous probabilitygoes to zero, as n → ∞ . It remains to bound (3.17). By Portmanteau’s Theoremand since there is at most one particle per site, (3.17) is bounded from aboveby lim sup n →∞ Q nµ n (cid:110) π : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) π nt , H (cid:105) − (cid:104) π n , H (cid:105) − (cid:90) t (cid:104) π ns , ∂ u H (cid:105) ds − (cid:90) t (cid:104) π nt , ε − (0 ,ε ] (cid:105) ∂ u H (0) ds + (cid:90) t (cid:104) π nt , ε − (1 − ε, (cid:105) ∂ u H (1) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) . (3.18)Noticing the identities η εn, R s (1) = (cid:104) π ns , ε − (0 ,ε ] (cid:105) and η εn, L s ( −
1) = (cid:104) π ns , ε − (1 − ε, (cid:105) , SEP WITH A SLOW SITE 21 we can rewrite (3.18) as lim sup n →∞ P µ n (cid:110) η : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) π nt , H (cid:105) − (cid:104) π n , H (cid:105) − (cid:90) t (cid:104) π ns , ∂ u H (cid:105) ds − (cid:90) t η εn, R s (1) ∂ u H (0) ds + (cid:90) t η εn, L s ( − ∂ u H (1) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) . Recalling Proposition 3.6, in order to prove that the limit above is equal to zero,it is enough to show that the limit below is null: lim sup n →∞ P µ n (cid:110) η : sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) π nt , H (cid:105) − (cid:104) π n , H (cid:105) − (cid:90) t (cid:104) π ns , ∂ u H (cid:105) ds − (cid:90) t η s (1) ∂ u H (0) ds + (cid:90) t η s ( − ∂ u H (1) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) . Adding and subtracting n L n (cid:104) π ns , H (cid:105) , the previous expression is bounded fromabove by the sum of lim sup n →∞ P µ n (cid:110) sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:104) π nt , H (cid:105) − (cid:104) π n , H (cid:105) − (cid:90) t n L n (cid:104) π ns , H (cid:105) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) (3.19)and lim sup n →∞ P µ n (cid:110) sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:90) t n L n (cid:104) π ns , H (cid:105) ds − (cid:90) t (cid:104) π ns , ∂ u H (cid:105) ds − (cid:90) t η s (1) ∂ u H (0) ds + (cid:90) t η s ( − ∂ u H (1) ds (cid:12)(cid:12)(cid:12) > δ/ (cid:111) . (3.20)As can be easily verified, by the imposed conditions on the test function H ,the quadratic variation of the martingale M nt ( H ) given in (3.2) goes to zero, as n → ∞ . Therefore, by Doob’s inequality, (3.19) is null.It remains to show that (3.20) also vanishes. Recall (3.6) for n L n (cid:104) π ns , H (cid:105) .Let us examine its terms. The first term in the sum (3.6) is n (cid:88) x ∈ T n \{ } (cid:16) H ( x +1 n ) + H ( x − n ) − H ( xn ) (cid:17) η s ( x ) , which we split into the sum of n (cid:88) x ∈ T n \{− , , } (cid:16) H ( x +1 n ) + H ( x − n ) − H ( xn ) (cid:17) η s ( x ) (3.21)and n (cid:16) H ( n ) − H ( n ) (cid:17) η s (1) + n (cid:16) H ( − n ) − H ( − n ) (cid:17) η s ( − . (3.22)The difference between (3.21) and (cid:104) π ns , ∂ u H (cid:105) = n (cid:88) x ∈ T n ∂ u H ( xn ) η s ( x ) in bounded (in modulus) by c H /n , because H ∈ C [0 , , where c H > is aconstant depending only on H . The second term in the sum (3.6) is n − β (cid:16) H ( n ) + H ( − n ) − H ( n ) (cid:17) η s (0) which converges to zero as n → ∞ , because β > . The last term in (3.6) is n (1 − g ( n )) (cid:104) ( H ( − n ) − H ( n )) η s (0) η s ( −
1) + ( H ( n ) − H ( n )) η s (0) η s (1) (cid:105) , which can be rewritten as the sum of n (1 − g ( n )) (cid:104) ( H ( − n ) − H ( n )) η s ( −
1) + ( H ( n ) − H ( n )) η s (1) (cid:105) (3.23)and n ( η s (0) − − g ( n )) (cid:104) ( H ( − n ) − H ( n )) η s ( −
1) + ( H ( n ) − H ( n )) η s (1) (cid:105) . By Proposition 3.4, the time integral of the last term in the previous expressionconverges to zero in probability, as n → ∞ . Since H ( n ) = 0 and β > , theexpression (3.22) plus the expression (3.23) is equal to n (cid:16) H ( n ) − H ( n ) (cid:17) η s (1) + n (cid:16) H ( − n ) − H ( − n ) (cid:17) η s ( − , plus an error of order n − β . The expression above is, asymptotically in n , thesame as η s (1) ∂ u H (0) − η s ( − ∂ u H (1) , whose time integral cancels with the remaining time integrals of (3.20) andtherefore proves that (3.20) vanishes. This concludes the claim (3.16).Now, let { H i } i ≥ be a countable dense set of functions on C [0 , , with re-spect to the norm (cid:107) H (cid:107) ∞ + (cid:107) ∂ u H (cid:107) ∞ + (cid:107) ∂ u H (cid:107) ∞ . Since (3.16) is true for eachone of these functions H i , we can extend (3.16) for all functions H ∈ C [0 , simultaneously by intersecting a countable number of sets of probability one.This proves that Q ∗ is concentrated on weak solutions of (2.6). Since thereexists only one weak solution of (2.6), it means that Q ∗ is equal to the afore-mentioned probability measure Q . Invoking the tightness that we have provedin Subsection 3.1, we conclude that the entire sequence { Q nµ n } n ∈ N converges to Q , as n → ∞ . (cid:3)
4. E
QUILIBRIUM DENSITY FLUCTUATIONS
In this section we prove Theorem 2.10. Recall that here we take the processevolving on Z and recall also (2.10). By Dynkin’s formula, M nt ( H ) := Y nt ( H ) − Y n ( H ) − (cid:90) t n L n Y ns ( H ) ds , (4.1)is a martingale with respect to the natural filtration F t := σ ( η s : s ≤ t ) . Besidesthat, (cid:16) M nt ( H ) (cid:17) − (cid:90) t (cid:16) n L n (cid:0) Y nt ( H ) (cid:1) − Y ns ( H ) n L n Y ns ( H ) (cid:17) ds , (4.2)is also a martingale with respect to the same filtration. By I nt ( H ) = (cid:90) t n L n Y ns ( H ) ds we denote the integral part in (4.1). First we observe that the expression (cid:88) x ∈ Z (cid:16) H ( x +1 n ) + H ( x − n ) − H ( xn ) (cid:17) p SEP WITH A SLOW SITE 23 is well defined since H decays fast and is equal to zero. Analogously to (3.6),some direct calculations then yield n L n Y ns ( H ) = n / (cid:88) x (cid:54) = − , , (cid:104) H ( x +1 n ) + H ( x − n ) − H ( xn ) (cid:105) ¯ η s ( x )+ n / (cid:104) H ( n ) + H ( n ) − H ( n ) (cid:105) ¯ η s (1) + n / (cid:104) H ( − n ) + H ( n ) − H ( − n ) (cid:105) ¯ η s ( − n / (1 − g ( n )) (cid:104) ( H ( n ) − H ( n )) η s (0) η s (1) + ( H ( − n ) − H ( n )) η s (0) η s ( − (cid:105) + n / g ( n ) (cid:104) H ( n ) + H ( − n ) − H ( n ) (cid:105) η s (0) + Θ( n, p, H ) , (4.3)where ¯ η s ( x ) = η ( x ) − m p ( x ) is the centered occupation variable, as in (2.10) and Θ( n, p, H ) = − n / (cid:104) H ( n ) + H ( − n ) − H ( n ) (cid:105) p. The next two propositions are direct calculations very similar to [11], and forthis reason their proofs are omitted.
Proposition 4.1.
Consider g ( n ) = 1 + o (1) and H ∈ S ( R ) . In this case, lim n →∞ E p (cid:104)(cid:0) M nt ( H ) (cid:1) (cid:105) = 2 t χ ( p ) (cid:107)∇ H (cid:107) L ( R ) . and lim sup n →∞ E p (cid:104)(cid:0) I nt ( H ) (cid:1) (cid:105) ≤ t χ ( p ) (cid:107)∇ H (cid:107) L ( R ) Proposition 4.2.
Consider g ( n ) = αn − β , α > , β > , and H ∈ S Neu ( R ) . Inthis case, lim n →∞ E p (cid:104)(cid:16) M nt ( H ) (cid:17) (cid:105) = 2 t χ ( p ) (cid:107)∇ Neu H (cid:107) L ( R ) and lim sup n →∞ E p (cid:104)(cid:0) I nt ( H ) (cid:1) (cid:105) ≤ t χ ( p ) (cid:107)∇ Neu H (cid:107) L ( R ) . The next result is concerned with convergence at initial time, for either caseof the function g ( n ) . Proposition 4.3. {Y n } n ∈ N converges in distribution to Y , as n → ∞ , where Y is a Gaussian field with mean zero and covariance given by (2.14) .Mutatis mutandis , the same proof of [11, Prop. 3.2] applies and is sup-pressed here.4.1. Tightness.
Here we prove tightness of the process {Y nt ; t ∈ [0 , T ] } n ∈ N inboth cases of g ( n ) . First we notice that by Mitoma’s criterion [23] and the factthat S ( R ) and S Neu ( R ) are Fr´echet spaces, it is enough to prove tightness of thesequence of real-valued processes {Y nt ( H ); t ∈ [0 , T ] } n ∈ N , where H ∈ S ( R ) if weconsider g ( n ) = 1 + o (1) , and H ∈ S Neu ( R ) if we consider g ( n ) = αn − β , α > , β > .In order to prove tightness of {Y nt ( H ) : 0 ≤ t ≤ T } n ∈ N , we shall prove tight-ness of each term in the formula (4.1).Fix a test function H belonging to the respective space for each case of g ( n ) .By (4.1), it is enough to prove tightness of the stochastic processes {Y n ( H ) } n ∈ N , {I nt ( H ); t ∈ [0 , T ] } n ∈ N , and {M nt ( H ); t ∈ [0 , T ] } n ∈ N . By Proposition 4.3 we have convergence at initial time, hence {Y n ( H ) } n ∈ N is obviously tight.To show tightness of the remaining real-valued processes we use the Aldouscriterion: Proposition 4.4 (Aldous’ criterion) . A sequence { x nt ; t ∈ [0 , T ] } n ∈ N of real-valued processes is tight with respect to the Skorohod topology of D ([0 , T ] , R ) if: (i) lim A → + ∞ lim sup n → + ∞ P (cid:16) sup ≤ t ≤ T | x nt | > A (cid:17) = 0 , (ii) for any ε > , lim δ → lim sup n → + ∞ sup λ ≤ δ sup τ ∈T T P ( | x nτ + λ − x nτ | > ε ) = 0 , where T T is the set of stopping times bounded by T . For the martingale term, the claim (i) of Aldous’ criterion is achieved by anapplication of Doob’s inequality together with Proposition 4.1 or Proposition4.2 (depending on the chosen g ).By Proposition 4.1 or Proposition 4.2, the claim (i) of Aldous’ criterion canbe easily checked for the integral term. It remains to check (ii). Fix a stoppingtime τ ∈ T T and suppose that g ( n ) = 1 + o (1) . By Chebychev’s inequality, P p (cid:0)(cid:12)(cid:12) M nτ + λ ( H ) − M nτ ( H ) (cid:12)(cid:12) > ε (cid:1) ≤ ε E p (cid:2)(cid:0) M nτ + λ ( H ) − M nτ ( H ) (cid:1) (cid:3) . Thus, by Proposition 4.1, lim sup n →∞ P p (cid:0)(cid:12)(cid:12) M nτ + λ ( H ) − M nτ ( H ) (cid:12)(cid:12) > ε (cid:1) ≤ ε χ ( p ) λ (cid:107)∇ H (cid:107) L ( R ) ≤ ε χ ( p ) δ (cid:107)∇ H (cid:107) L ( R ) , which vanishes as δ → . Similarly, P p (cid:0)(cid:12)(cid:12) I nτ + λ ( H ) − I nτ ( H ) (cid:12)(cid:12) > ε (cid:1) ≤ ε E p (cid:2)(cid:0) I nτ + λ ( H ) − I nτ ( H ) (cid:1) (cid:3) . Again by Proposition 4.1, we obtain lim sup n →∞ P p (cid:0)(cid:12)(cid:12) I nτ + λ ( H ) − I nτ ( H ) (cid:12)(cid:12) > ε (cid:1) ≤ tε δ χ ( p ) (cid:107)∇ H (cid:107) L ( R ) , which vanishes as δ → .The proof in the case g ( n ) = αn − β is analogous (invoking is this case Propo-sition 4.2) and for this reason will be omitted. This finishes the proof of tight-ness.4.2. Characterization of limit points for g ( n ) = 1 + o (1) . We shall provethat any limit of {Y nt ( H ) } n ∈ N is concentrated on solutions of the martingaleproblem described in Proposition 2.8, with H ∈ S ( R ) . Suppose that {Y nt } n ∈ N converges along a subsequence to Y t . In slight abuse of notation, we denotethis convergent subsequence also by {Y nt } n ∈ N .In this case H ∈ S ( R ) is smooth, hence we have (cid:12)(cid:12)(cid:12) n (cid:104) H ( x +1 n ) + H ( x − n ) − H ( xn ) (cid:105) − ∆ H ( x ) (cid:12)(cid:12)(cid:12) ≤ c H n . SEP WITH A SLOW SITE 25
A similar analysis to the one presented in (4.3) for the hydrodynamic limitimplies that M nt ( H ) := Y nt ( H ) − Y n ( H ) − (cid:90) t Y ns (∆ H ) ds + e ( n ) , where the error function e ( n ) is bounded, in modulus, by cn − / . Since we aresupposing that {Y nt } n ∈ N converges, we conclude that {M nt ( H ) } n ∈ N converges.By similar arguments to those presented in [11], we know that the sequenceof martingales {M nt ( H ) } n ∈ N is uniformly integrable. This implies that thelimit of {M nt ( H ) } n ∈ N , which we denote by M t ( H ) , is again a martingale. ByProposition 4.1, its quadratic variation is χ ( p ) t (cid:107)∇ H (cid:107) L ( R ) . Now, Proposition2.8 finishes the characterization of limit points in this case.4.3. Characterization of limit points for g ( n ) = αn − β . We shall prove inthis case that any limit of {Y nt ( H ) } n ∈ N with H ∈ S Neu ( R ) is a solution of themartingale problem described in Proposition 2.9.In this situation, there is no analogous result of Proposition 3.4. The keyingredient here will be the following tricky lemma: Lemma 4.5.
Let g ( n ) = αn − β and x = ± . For some constant C > notdepending on n the estimates E p (cid:34)(cid:16) (cid:90) t (cid:16) n / η s ( x )(1 − η s (0)) − αn − β η s (0)(1 − η s ( x )) (cid:17) ds (cid:17) (cid:35) ≤ Cn − β , hold.Proof. We prove only the inequality for x = 1 , the case x = − is completelyanalogous. By the Kipnis-Varadhan inequality (see [19, Proposition A1.6.1]),the expectation E p (cid:34)(cid:16) (cid:90) t (cid:16) n / η s (1)(1 − η s (0)) − αn − β η s (0)(1 − η s (1)) (cid:17) ds (cid:17) (cid:35) is less or equal than t sup f ∈ L ( ν p ) (cid:110) (cid:90) n / η (1)(1 − η (0)) f ( η ) ν p ( dη ) − αn − β (cid:90) η (0)(1 − η (1)) f ( η ) ν p ( dη ) − n D n ( f ) (cid:111) . where D n is the Dirichlet form given in (3.7). In the first integral inside thesupremum above we perform the change of variables η (cid:55)→ η , . Thus, the ex-pression above can be rewritten as t sup f ∈ L ( ν p ) (cid:110) (cid:90) n / η (0)(1 − η (1)) f ( η , ) ν p ( dη , ) − αn − β (cid:90) η (0)(1 − η (1)) f ( η ) ν p ( dη ) − n D n ( f ) (cid:111) . (4.4) Notice that here the Dirichlet form is evaluated at f instead of √ f as in Subsection 3.3. Now, from (2.3) we have that ν p ( { η , } ) ν p ( { η } ) = (cid:16) m p (0) p (cid:17) η (1) (cid:16) pm p (0) (cid:17) η (0) (cid:16) − p − m p (0) (cid:17) − η (0) (cid:16) − m p (0)1 − p (cid:17) − η (1) from where we get (cid:90) n / η (0)(1 − η (1)) f ( η , ) ν p ( dη , ) = (cid:90) αn − β η (0)(1 − η (1)) f ( η , ) ν p ( dη ) , and therefore (4.4) is the same as t sup f ∈ L ( ν p ) (cid:110) αn − β (cid:90) η (0)(1 − η (1)) (cid:16) f ( η , ) − f ( η ) (cid:17) ν p ( dη ) − n D n ( f ) (cid:111) . By the inequality xy ≤ Ax + y A , ∀ A > , the expression above is smaller than t sup f ∈ L ( ν p ) (cid:110) Aαn − β (cid:90) η (0)(1 − η (1)) (cid:16) f ( η , ) − f ( η ) (cid:17) ν p ( dη )+ αn − β A (cid:90) η (0)(1 − η (1)) ν p ( dη ) − n D n ( f ) (cid:111) for any A > . Picking A = √ n the last expression becomes equal to t sup f ∈ L ( ν p ) (cid:110) αn − β (cid:90) η (0)(1 − η (1)) (cid:16) f ( η , ) − f ( η ) (cid:17) ν p ( dη )+ αn − β (cid:90) η (0)(1 − η (1)) ν p ( dη ) − n D n ( f ) (cid:111) . Since the Dirichlet form (3.7) is a sum of positive terms, and since the firstterm above is exactly the first term in n D n ( f ) , we conclude that the expressionabove is less or equal than tαn − β (cid:90) η (0)(1 − η (1)) ν p ( dη ) = Cn − β , for C > not depending on n . This concludes the proof. (cid:3) Let us proceed to the characterization of limit points in this case. We beginwith an observation that will strongly simplify the analysis.First of all, we notice that M nt ( H ) defined in (4.1) is a martingale where H ∈ S Neu ( R ) does not play any special role, except the decay at infinity tomake the sum well defined. In fact, we can take H = H n depending on n . Wewill do that in the following way. For each n ∈ N , we impose H n ( xn ) = H ( xn ) forall x (cid:54) = 0 while for x = 0 we impose H n ( n ) = (cid:16) H ( n ) + H ( − n ) (cid:17) . (4.5)In this way, we obtain H n ( n ) + H n ( − n ) − H n ( n ) = 0 , ∀ n ∈ N (4.6)which cancels two parcels in (4.3). To simplify notation we will write H insteadof H n , keeping in mind (4.6).We examine carefully all the terms in (4.3). By the discussion above, both Θ( n, p, H ) and n / g ( n ) (cid:104) H ( n ) + H ( − n ) − H ( n ) (cid:105) η (0) (4.7) SEP WITH A SLOW SITE 27 vanish. Let us see the remaining terms. The first sum on the right hand sideof (4.3) is equal to √ n (cid:88) x (cid:54) = − , , ∆ Neu H ( xn ) ¯ η s ( x ) plus an error of order O ( n − / ) . Since the side derivatives of H ∈ S Neu ( R ) atzero vanish, we also have that the second and third terms in (4.3) are equal to n / (cid:104) H ( n ) − H ( n ) (cid:105) ¯ η s (1) + n / (cid:104) H ( n ) − H ( − n ) (cid:105) ¯ η s ( − plus another error of order O ( n − / ) . By (4.5), the expression above can berewritten as n / (cid:104) H ( − n ) − H ( n ) (cid:105) η (1) + n / (cid:104) H ( n ) − H ( − n ) (cid:105) η ( − . Last expression together with the remaining two parcels in (4.3) gives us thesum of n / (cid:104) H ( − n ) − H ( n ) (cid:105) η (1) + n / − g ( n )) (cid:104) H ( n ) − H ( − n ) (cid:105) η s (0) η s (1) (4.8)and n / (cid:104) H ( n ) − H ( − n ) (cid:105) η ( −
1) + n / − g ( n )) (cid:104) H ( − n ) − H ( n ) (cid:105) η s (0) η s ( − . (4.9)At this point, we will use (4.7). Regardless of the fact that (4.7) is null, we cansplit it in two parts, namely n / g ( n ) (cid:104) H ( n ) − H ( n ) (cid:105) η (0) and n / g ( n ) (cid:104) H ( − n ) − H ( n ) (cid:105) η (0) . The first one we add to (4.8) and the second one to (4.9). Recalling(4.5), it gives us the sum of n / (cid:16) H ( − n ) − H ( n ) (cid:17)(cid:104) η s (1)(1 − η s (0)) − g ( n ) η s (0)(1 − η s (1)) (cid:105) (4.10)and n / (cid:16) H ( n ) − H ( − n ) (cid:17)(cid:104) η s ( − − η s (0)) − g ( n ) η s (0)(1 − η s ( − (cid:105) . (4.11)Lemma 4.5 asserts that the time integrals of expressions (4.10) and (4.11) areasymptotically negligible in L .For H ∈ S Neu ( R ) , the sequence of martingales {M nt ( H ) } n ∈ N presented in(4.1) is uniformly integrable. This implies that the L -limit of {M nt ( H ) } n ∈ N ,denoted by M t ( H ) , is again a martingale which quadratic variation given χ ( p ) t (cid:107)∇ Neu H (cid:107) L ( R ) , assured by Proposition 4.1.The entire previous discussion on the integral part of M nt ( H ) lead us toconclude that M t ( H ) satisfies M t ( H ) := Y t ( H ) − Y ( H ) − (cid:90) t Y s (∆ Neu H ) ds , which concludes the characterization of limit points by Proposition 2.9.
5. O
PEN QUESTIONS AND CONJECTURES
As presented in this paper, the critical defect strength and behaviour at thecritical point remains open in sense that it is not clear what should be the limitfor ≤ β ≤ when g ( n ) = αn − β , α > . One conceivable scenario is that forany β > , both hydrodynamic limit and fluctuations would be driven by adisconnect behavior corresponding to Neumann boundary conditions, meaningthat the critical point would be β c = 0 .Our guess instead is that the correct critical point should be achieved at β = 1 , much more close to the scenario of [10] where a slow bond is consideredinstead of a slow site . The physical intuition behind the dynamical phase tran-sition taking place at β = 1 is the fact that, in a large but finite system and atlarge but finite times, the particle current, which is of diffusive origin, will beof order /n everywhere (with some space-dependent amplitude that dependson the initial state) before equilibrium is reached. However, a weak site with β > cannot allow such a current to flow and hence it acts like a total blockagecorresponding to Neumann boundary conditions. On the other hand, a defectrate with β < does not make a current of order /n impossible, correspondingto a macroscopically irrelevant local perturbation of the particle system.Specifically we conjecture that the behavior for β = 1 should be described bythe partial differential equation ∂ t ρ ( t, u ) = ∂ u ρ ( t, u ) , t ≥ , u ∈ (0 , ,∂ u ρ ( t, + ) = ∂ u ρ ( t, − ) = α ( ρ ( t, + ) − ρ ( t, − )) , t ≥ ,ρ (0 , u ) = ρ ( u ) , u ∈ (0 , , (5.1)where + and − denotes right and left side limits, respectively. This conjectureis motivated by the observation that the equation above is the hydrodynamicequation of two neighbouring slow bonds at β = 1 , a claim made precise in nextsection. Thus our conjecture in the slow site setting is the following. • For α > and ≤ β < , the hydrodynamic limit and the equilibriumfluctuations for g ( n ) = αn − β should be driven by the heat equation pe-riodic boundary conditions, achieving the same limits we have obtainedfor g ( n ) = 1 + o ( n ) . • For α > and β = 1 , the hydrodynamic limit and the equilibriumfluctuations for g ( n ) = αn − β should be driven by (5.1) in the same senseof [10, 11, 12] with the correction of / in the boundary condition.6. H YDRODYNAMICS OF THE
SSEP
WITH k NEIGHBORING SLOW BONDS
Here we characterize the hydrodynamic behavior of the SSEP with k neigh-boring slow bonds. This is an additional result we append in order to supportthe conjecture presented in Section 5. We point out that, for the regime β = 1 ,the result presented here is not a corollary of [10, 11], since here we consider k neighboring slow bonds, while the mentioned references considered macro-scopically separated slow bonds.The notation and topology issues will be the same as those we have consid-ered in this paper. Fix k a positive integer. The SSEP with k neighboring slow SEP WITH A SLOW SITE 29 bonds is the Markov process on { , } T n defined through the generator L n f ( η ) = k − (cid:88) x =0 αn β [ f ( η x,x +1 ) − f ( η )] + (cid:88) x ∈ T nx/ ∈{ ,...,k − } [ f ( η x,x +1 ) − f ( η )] acting on functions f : { , } T n → R .F IGURE
2. Exclusion process with three neighboring slow bonds . Definition 6.
Let ρ : T → [0 , be a measurable function. We say that ρ is aweak solution of the heat equation with Robin’s boundary conditions given by ∂ t ρ ( t, u ) = ∂ u ρ ( t, u ) , t ≥ , u ∈ (0 , ,∂ u ρ ( t, + ) = ∂ u ρ ( t, − ) = αk ( ρ ( t, + ) − ρ ( t, − )) , t ≥ ,ρ (0 , u ) = γ ( u ) , u ∈ (0 , , (6.1) if ρ belongs to L (0 , T ; H ) and for all t ∈ [0 , T ] and for all H ∈ C [0 , , suchthat ∂ u H (0 + ) = ∂ u H (0 − ) = αk ( H (0 + ) − H (0 − )) , (6.2) holds that (cid:104) ρ t , H (cid:105) − (cid:104) γ , H (cid:105) − (cid:90) t (cid:10) ρ s , ∂ u H (cid:11) ds = 0 . Proposition 6.1.
For each n ∈ N , let µ n be a Bernoulli product measure on { , } T n as in (2.8) . Then, for any t > , for every δ > and every H ∈ C ( T ) , itholds that lim n →∞ P µ n (cid:110) η · : (cid:12)(cid:12)(cid:12) n (cid:88) x ∈ T n H ( xn ) η t ( x ) − (cid:90) T H ( u ) ρ ( t, u ) du (cid:12)(cid:12)(cid:12) > δ (cid:111) = 0 , (6.3) where • if ≤ β < , the function ρ is the unique weak solution of (2.4) ; • if β = 1 , the function ρ is the unique weak solution of (6.1) ; • if β > , the function ρ is the unique weak solution of (2.6) .Proof. As usual, the proof consists in proving tightness of the process inducedby the empirical measure, plus the uniqueness of the limit points.Tightness can be handled in same way as we have done in Subsection 3.1.Characterization of the limit points for the cases β ∈ [0 , and β ∈ (1 , ∞ ) follows closely the steps of [10, 11]. The case β = 1 is tricky and described inmore detail below. Let G n : { n , n , . . . , n − n } → R be some function depending on n . Performingelementary computations, n L n (cid:104) π ns , G n (cid:105) can be rewritten as the sum of (cid:104) n (cid:16) G n ( k +1 n ) − G n ( kn ) (cid:17) + αn − β (cid:16) G n ( k − n ) − G n ( kn ) (cid:17)(cid:105) η s ( k )+ k − (cid:88) j =1 (cid:104) αn − β (cid:16) G n ( j +1 n ) − G n ( jn ) (cid:17) + αn − β (cid:16) G n ( j − n ) − G n ( jn ) (cid:17)(cid:105) η s ( j )+ (cid:104) αn − β (cid:16) G n ( n ) − G n ( n ) (cid:17) + n (cid:16) G n ( − n ) − G n ( n ) (cid:17)(cid:105) η s (0) (6.4)and n (cid:88) x ∈ T nx/ ∈{ ,...,k } (cid:16) G n ( x +1 n ) + G n ( x − n ) − G n ( xn ) (cid:17) η s ( x ) . (6.5)Let H ∈ C [0 , satisfying (6.2). We define G n by G n ( xn ) = (cid:40) H ( xn ) , if x ∈ { k + 1 , . . . , n − } ,H (0 − ) + xk ( H (0 + ) − H (0 − )) , if x ∈ { , . . . , k } . (6.6)In other words, the function G n is equal to H outside the region where theslow bonds are contained. At the sites { , , . . . , k } , the function G n is a linearinterpolation of H (0 + ) and H (0 − ) .Since H ∈ C [0 , , (6.5) is close to (cid:104) π ns , ∂ u H (cid:105) . We claim now that (6.4) con-verges to zero, as n → ∞ . First, notice that (6.6) tells us that k − (cid:88) j =1 (cid:104) αn − β (cid:16) G n ( j +1 n ) − G n ( jn ) (cid:17) + αn − β (cid:16) G n ( j − n ) − G n ( jn ) (cid:17)(cid:105) η s ( j ) is null. Let us analyze the remaining terms in (6.4). Since β = 1 , the term n (cid:16) G n ( k +1 n ) − G n ( kn ) (cid:17) + αn − β (cid:16) G n ( k − n ) − G n ( kn ) (cid:17) converges to ∂ u H (0 + ) + αk (cid:16) H (0 − ) − H (0 + ) (cid:17) , which vanishes by (6.2). The same analysis assures that αn − β (cid:16) G n ( n ) − G n ( n ) (cid:17) + n (cid:16) G n ( − n ) − G n ( n ) (cid:17) converges to zero as n → ∞ . Provided by this claim and similar arguments ofthose in Section 5 one can conclude the proof. (cid:3) A CKNOWLEDGEMENTS
TF was supported through a project PRODOC/UFBA and a project JCB1708/2013-FAPESB.PG thanks CNPq (Brazil) for support through the research project “Addi-tive functionals of particle systems”, Universal n. 480431/2013-2, also thanksFAPERJ “Jovem Cientista do Nosso Estado” for the grant E-25/203.407/2014and the Research Centre of Mathematics of the University of Minho, for thefinancial support provided by “FEDER” through the “Programa OperacionalFactores de Competitividade COMPETE” and by FCT through the researchproject PEst-C/MAT/UI0013/ 2011.GMS acknowledges financial support by DFG.
SEP WITH A SLOW SITE 31
The authors thank CMAT at the University of Minho, where this work wasinitiated, and PUC-Rio, where this work was finished, for the warm hospitality.R
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