Equilibrium fluctuations for the slow boundary exclusion process
aa r X i v : . [ m a t h . P R ] D ec Equilibrium fluctuations for the slowboundary exclusion process
Tertuliano Franco, Patr´ıcia Gon¸calves and Adriana Neumann
Abstract
We prove that the equilibrium fluctuations of the symmetric simpleexclusion process in contact with slow boundaries is given by an Ornstein-Uhlenbeck process with Dirichlet, Robin or Neumann boundary conditionsdepending on the range of the parameter that rules the slowness of the bound-aries.
The study of nonequilibrium behavior of interacting particle systems is oneof the most challenging problems in the field and it has only been completelysolved in very particular cases. The toy model for the study of a system ina nonequilibrium scenario is the symmetric simple exclusion process (SSEP)whose dynamics is rather simple to explain and it already captures manyfeatures of more complicated systems.The dynamics of this model can be described as follows. We fix a scalingparameter n and we consider the SSEP evolving on the discrete space Σ n = { , · · · , n − } to which we call the bulk. To each pair of bonds { x, x + 1 } with Tertuliano FrancoUFBA, Instituto de Matem´atica, Campus de Ondina, Av. Adhemar de Barros, S/N. CEP40170-110, Salvador, Brazil.e-mail: [email protected]
Patr´ıcia Gon¸calvesCenter for Mathematical Analysis, Geometry and Dynamical Systems, Instituto SuperiorT´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugale-mail: [email protected]
Adriana NeumannUFRGS, Instituto de Matem´atica, Campus do Vale, Av. Bento Gon¸calves, 9500. CEP91509-900, Porto Alegre, Brazil.e-mail: [email protected] x = 1 , · · · , n − N x,x +1 ( t ) of rate 1. Now weartificially add two end points at the bulk, namely, we add the sites x = 0 and x = n and we superpose the exclusion dynamics with a Glauber dynamicswhich has only effect at the boundary points of the bulk, namely at the sites x = 1 and x = n −
1. For that purpose, we add extra Poisson processes at thebonds { , } and { n − , n } . In each one of these bonds there are two Poissonprocesses: N , ( t ) with parameter αn − θ , N , ( t ) with parameter (1 − α ) n − θ , N n − ,n ( t ) with parameter βn − θ and N n,n − ( t ) with parameter (1 − β ) n − θ .All the Poisson processes are independent. Above α, β ∈ (0 ,
1) and θ ≥ η = ( η (1) , · · · , η ( n − x ∈ Σ n , η ( x ) = 1 if there is a particleat the site x and η ( x ) = 0 if the site x is empty. Now, if a clock rings for abond { x, x + 1 } in the bulk, then we exchange the coordinates x and x + 1 of η , that is we exchange η ( x ) with η ( x + 1) at rate 1. If the clock rings for thebond at the boundary as, for example, from the Poisson process N , ( t ) thena particle gets into the bulk through the site 1 at rate αn − θ if and only ifthere is no particle at the site 1, otherwise nothing happens. If the clock ringsfrom the Poisson process N , ( t ) and there is a particle at the site 1, then itexits the bulk from the site n − − α ) n − θ . Note that the higherthe value of θ the slower is the dynamics at the boundaries. For a display ofthe description above, see the figure below. βn θ − βn θ − αn θ αn θ
12 12
The dynamics just described is Markovian and can be completely char-acterized in terms of its infinitesimal generator given below in (1). We notethat the space state of this Markov process is Ω n := { , } Σ n . Observe thatthe bulk dynamics preserves the number of particles and our interest is to quilibrium fluctuations for the slow boundary exclusion process 3 describe the space-time evolution of this conserved quantity as a solution ofsome partial differential equation called the hydrodynamic equation .Note that for the choice α = β = ρ a simple computation shows that theBernoulli product measure of parameter ρ given by: ν nρ ( η ∈ Ω n : η ( x ) =1) = ρ is invariant under the dynamics. For this choice of the parameters theboundary reservoirs have the same intensity and we do not see any inducedcurrent on the system. Nevertheless, in the case α = β , let us say for example α < β , there is a tendency to have more particles entering into the bulk fromthe right reservoir and leaving the system from the left reservoir. This is acurrent which is induced by the difference of the density at the boundaryreservoirs. Note that in the bulk the dynamics is symmetric. In the case α = β , since we have a finite state Markov process, there is only one stationarymeasure that we denote by µ ssn which is no longer a product measure as inthe case α = β . By using the matrix ansazt method developed by [3, 10, 11]and references therein, it is possible to obtain information about this measureand an important problem is to analyze the behavior of the system startingfrom this non-equilibrium stationary state.We note that the hydrodynamic limit of this model was studied in [1] andthe hydrodynamic equations consist in the heat equation with different typesof boundary conditions depending on the range of the parameter θ . Moreprecisely, for 0 ≤ θ < α and β ,respectively. In this case we do not see any difference at the macroscopic levelwith respect to the case θ = 0. Nevertheless, for θ = 1 the boundary dynamicsis slowed enough in such a way that macroscopically the Dirichlet boundaryconditions are replaced by a type of Robin boundary conditions. These Robinboundary conditions state that the rate at which particles are injected intothe system through the boundary points, is given by the difference of thedensity at the bulk and the boundary. Finally for θ >
1, the boundaries aresufficiently slowed so that the Robin boundary conditions are replaced byNeumann boundary conditions stating that macroscopically there is no fluxof particles from the boundary reservoirs.We emphasize that there are many similar models to the one studied inthese notes which we summarize as follows. In [7, 8, 9], the authors considera model where removal of particles can only occur at an interval aroundthe left boundary and the entrance of particles is allowed only at an intervalaround the right boundary. Their model presents a current exchange betweenthe two reservoirs and shows some similarities with our model for the choice θ = 1. Another case already studied in the literature (see [12, 19]) is whenthe boundary is not slowed, that corresponds to our model for the choice θ = 0. As mentioned above, the hydrodynamic equation of this model hasDirichlet boundary conditions, see [12] or the equation (7). A similar model,whose hydrodynamic equation has both Dirichlet boundary conditions andNeumann boundary conditions, was studied in [6]. The main difference, atthe macroscopic level, is that the end points of the boundary conditions vary Tertuliano Franco, Patr´ıcia Gon¸calves and Adriana Neumann with time. The microscopic dynamics there is given by the SSEP evolving on Z with additional births and deaths restricted to a subset of configurationswhere there is a leftmost hole and a rightmost particle. In this situation, ata fixed rate j birth of particles occur at the position of the leftmost hole andat the same rate, independently, the rightmost particle dies. Another modelwhich has a current is considered in [4]. The dynamics evolves on the discretetorus Z /n Z without reservoirs, but has a surprising phenomenon: a “batteryeffect”. This effect produces a current of particles through the system and isdue to a single abnormal bond, where the rates to cross from left to right andfrom right to left are different. Finally, another model which has similaritieswith the model we consider in these notes is the SSEP with a slow bond,which was studied in [13, 14, 15]. The dynamics evolves on the discrete torus Z /n Z , and particles exchange positions between nearest neighbor bonds atrate 1, except at one particular bond, where the exchange occurs at rate n − β .In this case β > slow bond . The similarity between the slowbond model and the slow boundary model considered in these notes is that ifwe “open” the discrete torus exaclty at the position of the slow bond, then theslow bond rives rise to a slow boundary. In [13, 15] different hydrodynamicbehaviors were obtained, depending on the range of the parameter β , moreprecisely, the hydrodynamic equation is, in all cases, the heat equation butthe boundary conditions vary with the value of β , exhibiting three differentregimes as for the slow boundary model, see [1].Our interest in these notes is to go further the hydrodynamical behaviorin order to analyze the fluctuations around the hydrodynamical profile. Toaccomplish this, we restrict ourselves to the case α = β = ρ and startingfrom the stationary measure ν nρ defined above. Our result states that thefluctuations starting from ν nρ are given by an Ornstein-Uhlenbeck processsolution of d Y t = ∆ θ Y t dt + p χ ( ρ ) t ∇ θ d W t , where χ ( ρ ) is the variance of η ( x ) with respect to ν nρ , W t is a space-timewhite noise of unit variance and ∆ θ and ∇ θ are, respectively, the Laplacianand derivative operators defined on a space of test functions with differenttypes of boundary conditions depending on the value of θ . We note that thecase θ = 0 was studied in [19] and the case θ = 1 was studied in [16]. In thosearticles, the nonequilibrium fluctuations were obtained starting from generalinitial measures, which include the equilibrium case ν nρ treated here. We notehowever, that the case θ = 1 is quite difficult to attack at the nonequilibriumscenario since we need to establish a local replacement (see Lemma 3) inorder to close the martingale problem, which we can only prove startingthe system from the equilibrium state. In a future work, we will dedicate toextending this result to the nonequilibrium situation as, for example, startingthe system from the steady state when α = β . quilibrium fluctuations for the slow boundary exclusion process 5 Here follows an outline of these notes. In Section 2 we give the definition ofthe model, we recall from [1] the hydrodynamic limit and we state our mainresult, namely, Theorem 3. In Section 4 we characterize the limit processby means of a martingale problem. Tightness is proved in Section 5 and inSection 6 we prove the Replacement Lemma which is the most technical partof these notes.
For n ≥
1, we denote by Σ n the set { , · · · , n − } to which we call the bulk.The symmetric simple exclusion process with slow boundaries is a Markovprocess { η t : t ≥ } with state space Ω n := { , } Σ n . The slowness of theboundaries is ruled by a parameter that we denote by θ ≥
0. If η is a con-figuration of the state space Ω , then for x ∈ Σ n , the random variable η ( x )can take only two values, namely 0 or 1. If η ( x ) = 0, it means that thesite x is vacant, while η ( x ) = 1 means that the site x is occupied. The dy-namics of this model can be described as follows. In the bulk particles moveaccording to continuous time random walks, but whenever a particle wantsto jump to an occupied site, the jump is suppressed. At the left boundary,particles can be created (resp. removed) at rate αn − θ (resp. (1 − α ) n − θ ). Atthe right boundary, particles can be created (resp. removed) at rate βn − θ (resp. (1 − β ) n − θ ).Fix now a finite time horizon T . The Markov process { η t ( x ) : x ∈ Σ n ; t ∈ [0 , T ] } can be characterized in terms of its infinitesimal generator that wedenote by L θn and is defined as follows. For a function f : Ω n → R , we havethat( L θn f )( η ) = h αn θ (1 − η (1)) + (1 − α ) n θ η (1) i(cid:16) f ( η ) − f ( η ) (cid:17) + h βn θ (1 − η ( n − − β ) n θ η ( n − i(cid:16) f ( η n − ) − f ( η ) (cid:17) + n − X x =1 (cid:16) f ( σ x,x +1 η ) − f ( η ) (cid:17) , (1)where σ x,x +1 η is the configuration obtained from η by exchanging the occu-pation variables η ( x ) and η ( x + 1), that is,( σ x,x +1 η )( y ) = η ( x + 1) , if y = x ,η ( x ) , if y = x + 1 ,η ( y ) , otherwise. (2) Tertuliano Franco, Patr´ıcia Gon¸calves and Adriana Neumann and for x = 1 , n − η x is the configuration obtained from η by flipping theoccupation variable η ( x ):( η x )( y ) = (cid:26) − η ( y ) , if y = x ,η ( y ) , otherwise. (3)Let D ([0 , T ] , Ω n ) be the space of trajectories which are right continuousand with left limits, taking values in Ω n . Denote by P θ,nµ n the probability on D ([0 , T ] , Ω n ) induced by the Markov process with generator n L θn and theinitial measure µ n and denote by E θ,nµ n the expectation with respect to P θ,nµ n . The stationary measure µ ssn for this model when α = β = ρ ∈ (0 ,
1) is theBernoulli product measure given by ν nρ (cid:16) η ∈ Ω n : η ( x ) = 1 (cid:17) = ρ . But in the general case, where α = β , the stationary measure µ ssn does nothave independent marginals, see [10]. What we can say about the stationarybehavior of this model is that the density of particles has a behavior veryclose to a linear profile, which depends on the range of θ in the sense of thefollowing definition: Definition 1.
Let γ : [0 , → [0 ,
1] be a measurable profile. A sequence { µ n } n ∈ N is said to be associated to γ if, for any δ > f : [0 , → R the following limit holds:lim n →∞ µ n η : (cid:12)(cid:12)(cid:12) n n − X x =1 f ( xn ) η ( x ) − Z f ( u ) γ ( u ) du (cid:12)(cid:12)(cid:12) > δ ! = 0 . For µ n equal to the stationary measure µ ssn , the limit above is called the hydrostatic limit . Theorem 1 (Hydrostatic Limit, [1]).
Let µ ssn be the stationary probability measure in Ω n wrt the Markov processwith infinitesimal generator n L θn , defined in (1) . The sequence { µ ssn } n ∈ N isassociated (in the sense of Definition 1) to the profile ρ : [0 , → R given by ρ ( u ) = ( β − α ) u + α, if θ ∈ [0 , , β − α u + α + β − α , if θ = 1 , β + α , if θ ∈ (1 , ∞ ) , (4) for all u ∈ [0 , . quilibrium fluctuations for the slow boundary exclusion process 7 Another feature that we can say about the stationary state of the modelstudied in this paper is that the profiles in (4) are very close to the mean of η ( x ) taken with respect to the stationary measure µ ssn . To state this resultproperly, we start by defining for an initial measure µ n in Ω n , for x ∈ Σ n and for t ≥ ρ nt ( x ) := E θ,nµ n [ η t ( x )] . (5)If in the expression above µ n = µ ssn , then ρ nt ( x ) does not depend on t , so that ρ nt ( x ) = ρ n ( x ). From [1], we have that ρ n ( x ) satisfies the following recurrencerelations: ρ n ( x + 1) − ρ n ( x )] + [ ρ n ( x − − ρ n ( x )] , if x ∈ { , . . . , n − } , ρ n (2) − ρ n (1)] + n − θ [ α − ρ n (1)] , n − θ [ β − ρ n ( n − ρ n ( n − − ρ n ( n − . A simple computation shows that ρ n ( x ) is given by ρ n ( x ) = a n x + b n , for all x ∈ Σ n , where a n = β − α n θ + n − and b n = α + a n ( n θ − . Moreover, we concludethat lim n →∞ (cid:16) max x ∈ Σ n (cid:12)(cid:12) ρ n ( x ) − ρ ( xn ) (cid:12)(cid:12)(cid:17) = 0 . In [1] it was established the hydrodynamic limit of the model for any θ ≥ ρ : [0 , → [0 ,
1] and for each n ∈ N , let µ n be a probability measure on Ω n . Theorem 2 (Hydrodynamic Limit, [1]).
Suppose that the sequence { µ n } n ∈ N is associated to a profile ρ ( · ) in thesense of Definition 1. Then, for each t ∈ [0 , T ] , for any δ > and anycontinuous function f : [0 , → R , lim n → + ∞ P θ,nµ n " η · : (cid:12)(cid:12)(cid:12) n n − X x =1 f ( xn ) η tn ( x ) − Z f ( u ) ρ ( t, u ) du (cid:12)(cid:12)(cid:12) > δ = 0 , where ρ ( t, · ) is the unique weak solution of the heat equation ( ∂ t ρ ( t, u ) = ∂ u ρ ( t, u ) , for t > , u ∈ (0 , ,ρ (0 , u ) = ρ ( u ) , u ∈ [0 , . (6) with boundary conditions that depend on the range of θ , which are given by: Tertuliano Franco, Patr´ıcia Gon¸calves and Adriana Neumann
For θ < , ∂ u ρ ( t,
0) = α and ∂ u ρ ( t,
1) = β, for t > . (7) For θ = 1 , ∂ u ρ ( t,
0) = ρ ( t, − α and ∂ u ρ ( t,
1) = β − ρ ( t, , for t > . (8) For θ > , ∂ u ρ ( t,
0) = ∂ u ρ ( t,
1) = 0 , for t > . (9) Remark 1.
We note that the profiles in (4) are stationary solutions of theheat equation with the corresponding boundary conditions given above.
The space C ∞ ([0 , f : [0 , → R such that f iscontinuous in [0 ,
1] as well as all its derivatives.
Definition 2.
Let S θ denote the set of functions f ∈ C ∞ ([0 , k ∈ N ∪ { } it holds that(1) for θ < ∂ ku f (0) = ∂ ku f (1) = 0 . (2) for θ = 1: ∂ k +1 u f (0) = ∂ ku f (0) and ∂ k +1 u f (1) = − ∂ ku f (1) . (3) for θ > ∂ k +1 u f (0) = ∂ k +1 u f (1) = 0 . Definition 3.
For θ ≥
0, let − ∆ θ be the positive operator, self-adjoint on L [0 , f ∈ S θ by ∆ θ f ( u ) = ∂ u f ( u ) , if u ∈ (0 , ,∂ u f (0 + ) , if u = 0 ,∂ u f (1 − ) , if u = 1 . (10)Above, ∂ u f ( a ± ) denotes the side limits at the point a . Analogously, let ∇ θ : S θ → C ∞ ([0 , ∇ θ f ( u ) = ∂ u f ( u ) , if u ∈ (0 , ,∂ u f (0 + ) , if u = 0 ,∂ u f (1 − ) , if u = 1 . (11) Definition 4.
Let T θt : S θ → S θ be the semigroup associated to (6) with thecorresponding boundary conditions for the case α = β = 0. That is, given f ∈ S θ , by T θt f we mean the solution of the homogeneous version of (6) withinitial condition f . Definition 5.
Let S ′ θ be the topological dual of S θ with respect to the topol-ogy generated by the seminorms k f k k = sup u ∈ [0 , | ∂ ku f ( u ) | , (12) quilibrium fluctuations for the slow boundary exclusion process 9 where k ∈ N ∪ { } . In other words, S ′ θ consists of all linear functionals f : S θ → R which are continuous with respect to all the seminorms k · k k .Let D ([0 , T ] , S ′ θ ) (resp. C ([0 , T ] , S ′ θ )) be the space of trajectories which areright continuous and with left limits (resp. continuous), taking values in S ′ θ .The expression for T θt , θ ≥
0, is presented in the next proposition:
Proposition 1.
Let θ ≥ . Suppose that ρ ∈ L [0 , . Then ( T θt ρ )( u ) := ∞ X n =1 a n e − λ n t Ψ n ( u ) , (13) where { Ψ n } n ∈ N is an orthonormal basis of L [0 , constituted by eigenfunc-tions of the associated Regular Sturm-Liouville Problem (concerning the op-erator ∆ θ ) and a n are the Fourier coefficients of ρ in the basis { Ψ n } n ∈ N . • For θ < , the corresponding orthonormal basis of L [0 , is ( Ψ n ( u ) = √ nπu ) , for n ∈ N ,Ψ ( u ) ≡ . The eigenvalues of the associated Regular Sturm-Liouville Problem (con-cerning the operator ∆ θ ) are given by λ n = n π . • For θ = 1 , the corresponding orthornormal basis of L [0 , is a linearcombination of sines and cosines, namely, Ψ n ( u ) = A n sin( p λ n u ) + A n p λ n cos( p λ n u ) , for n ∈ N ∪ { } , where A n is a normalizing constant. The eigenvalues λ n do not have anexplicit formula, but it can verified that λ n ∼ n π . • For θ > , the corresponding orthonormal basis of L [0 , is ( Ψ n ( u ) = √ nπu ) , for n ∈ N ,Ψ ( u ) ≡ . The eigenvalues of the associated Regular Sturm-Liouville Problem (concern-ing the operator ∆ θ ) are given by λ n = n π .Proof. For θ = 1 the expression for T θt has been obtained in [16]. For the case θ = 1, as in [16], we state the associated Regular Sturm-Liouville Problem (for details on this subject we refer to [2], for instance):For θ < ( Ψ ′′ ( u ) + λΨ ( u ) = 0 , u ∈ (0 , ,Ψ (0) = 0 , Ψ (1) = 0 ; For θ ≥ ( Ψ ′′ ( u ) + λΨ ( u ) = 0 , u ∈ (0 , ,Ψ ′ (0) = 0 , Ψ ′ (1) = 0 . The solution of each one of the problems above (the eigenvalues λ n and theeigenfunctions Ψ n ) can be found in Chapter 10 of [5].As a consequence, the series (13) converges exponentially fast, implying that( T θt ρ )( u ) is smooth in space and time for any t >
0. This observation impliesa property of T θt : S θ → S θ stated in the next corollary. Corollary 1. If f ∈ S θ , then for any t > , T θt f ∈ S θ and ∆ θ T θt f ∈ S θ . We observe that the previous result is needed in the proof of uniqueness ofthe corresponding Ornstein-Uhlenbeck process (which is defined in the nextsection). Its proof is a consequence of the formula (13), see [16] for moredetails.
Fix ρ ∈ (0 , d Y t = ∆ θ Y t dt + p χ ( ρ ) t ∇ θ d W t , (14)where W t is a space-time white noise of unit variance and χ ( ρ ) = R ( η ( x ) − ρ ) dν nρ = ρ (1 − ρ ), in terms of a martingale problem. We will see below thatthis process governs the equilibrium fluctuations of the density of particlesof our model. In spite of having a dependence of Y t on θ , we do not index onit to not overload notation. Denote by Q θρ the distribution of Y · and E θρ theexpectation with respect to Q θρ .Define the inner product between the functions f, g : [0 , → R by h f, g i L ,θρ = 2 χ ( ρ ) " Z f ( u ) g ( u ) du + (cid:16) f (0) g (0) + f (1) g (1) (cid:17) θ =1 , where · is the indicator function. Then, L ,θρ ([0 , f : [0 , → R with k f k L ,θρ < ∞ , where k f k L ,θρ = h f, f i L ,θρ . (15) Proposition 2.
There exists an unique random element Y taking values inthe space C ([0 , T ] , S ′ θ ) such that:i) For every function f ∈ S θ , M t ( f ) and N t ( f ) given by quilibrium fluctuations for the slow boundary exclusion process 11 M t ( f ) = Y t ( f ) − Y ( f ) − Z t Y s ( ∆ θ f ) ds , N t ( f ) = (cid:0) M t ( f ) (cid:1) − χ ( ρ ) t k∇ θ f k L ,θρ (16) are F t -martingales, where for each t ∈ [0 , T ] , F t := σ ( Y s ( f ); s ≤ t, f ∈ S θ ) .ii) Y is a Gaussian field of mean zero and covariance given on f, g ∈ S θ by E θρ (cid:2) Y ( f ) Y ( g ) (cid:3) = h f, g i L ,θρ (17) Moreover, for each f ∈ S θ , the stochastic process { Y t ( f ) ; t ≥ } is gaus-sian, being the distribution of Y t ( f ) conditionally to F s , for s < t , normal ofmean Y s ( T θt − s f ) and variance R t − s k∇ θ T θr f k L ,θρ dr , where T θt was given inDefinition 4. The random element Y · is called the generalized Ornstein-Uhlenbeck processof characteristics ∆ θ and ∇ θ . From the second equation in (16) and L´evy’sTheorem on the martingale characterization of Brownian motion, for each f ∈ S θ , the process M t ( f ) (cid:16) χ ( ρ ) t k∇ θ f k L ,θρ (cid:17) − / (18)is a standard Brownian motion. Therefore, in view of Proposition 2, it makessense to say that Y · is the formal solution of (14). We define the density fluctuation field Y n · as time-trajectory of the linearfunctional acting on functions f ∈ S θ as Y nt ( f ) = 1 √ n n − X x =1 f (cid:16) xn (cid:17)(cid:16) η tn ( x ) − ρ nt ( x ) (cid:17) , for all t ≥ , (19)where ρ nt was defined in (5). Our results are given for the case α = β = ρ and for µ n being equal to ν nρ , that is, the Bernoulli product measure withparameter ρ ∈ (0 , ρ nt ( x ) = ρ, for all x ∈ Σ n and t ≥
0. Let Q θ,nρ bethe probability measure on D ([0 , T ] , S ′ θ ) induced by the density fluctuationfield Y n · . We denote E θ,nρ the expectation with respect to Q θ,nρ . Moreover,since we will consider only the initial measure µ n as ν nρ , we will simplify thenotations P θ,nν nρ and E θ,nν nρ as P θ,nρ and E θ,nρ , respectively.Our main result is the following theorem. Theorem 3 (Ornstein-Uhlenbeck limit).
For α = β = ρ ∈ (0 , , if we take the initial measure to be ν nρ , namely,the Bernoulli product measure with parameter ρ , then, the sequence { Q θ,nρ } n ∈ N converges, as n → ∞ , to a generalized Ornstein-Uhlenbeck (O.U.) process,which is the formal solution of equation (14) . As a consequence, the varianceof the limit field Y t is given on f ∈ S θ by E θρ [ Y t ( f ) Y s ( f )] = χ ( ρ ) Z ( f ( u )) du + Z s k T θt − r f k L ,θρ dr , (20) where k · k L ,θρ was defined in (15) . Fix a test function f . By Dynkin’s formula, we have that M nt ( f ) = Y nt ( f ) − Y ( f ) − Z t ( ∂ s + n L θn ) Y ns ( f ) ds, (21) N nt ( f ) = ( M nt ( f )) − Z t n L θn Y ns ( f ) − Y ns ( f ) n L θn Y ns ( f ) ds (22)are martingales with respect to the natural filtration F t := σ ( η s : s ≤ t ).To simplify notation we denote Γ ns ( f ) := ( ∂ s + n L θn ) Y ns ( f ). A long butelementary computation shows that Γ ns ( f ) = 1 √ n n − X x =1 ∆ n f (cid:16) xn (cid:17) ( η s ( x ) − ρ )+ √ n ∇ + n f (0)( η s (1) − ρ ) − √ n ∇ − n f ( n )( η s ( n − − ρ ) − n / n θ f (cid:16) n (cid:17) ( η s (1) − ρ ) − n / n θ f (cid:16) n − n (cid:17) ( η s ( n − − ρ ) . (23)Above ∆ n f ( x ) := n h f (cid:16) x +1 n (cid:17) + f (cid:16) x − n (cid:17) − f (cid:16) x − n (cid:17)i , ∇ + n f ( x ) := n h f (cid:16) x +1 n (cid:17) − f (cid:16) xn (cid:17)i and ∇ − n f ( x ) := n h f (cid:16) xn (cid:17) − f (cid:16) x − n (cid:17)i . We note that for the choice θ = 0, using the fact that f (0) = 0 = f (1),the expression (23) reduces to quilibrium fluctuations for the slow boundary exclusion process 13 Γ ns ( f ) = 1 √ n n − X x =1 ∆ n f (cid:16) xn (cid:17) ( η s ( x ) − ρ ) , (24)which is Y ns ( ∆ n f ) . Now, we close the equation (23) for each regime of θ . The goal is two showthat we can rewrite (23) as (24) plus a term which vanishes as n → ∞ . • The case θ <
1: we note that since f ∈ S θ we can write Γ ns ( f ) as Y ns ( ∆ n f ) + √ n (1 − n − θ ) n ∇ + n f (0)( η s (1) − ρ ) − ∇ − n f ( n )( η s ( n − − ρ ) o . In order to close the equation for the martingale we need to show that:lim n →∞ E θ,nρ (cid:2)(cid:16) Z t √ n (cid:16) η s ( x ) − ρ (cid:17) ds (cid:17) i = 0 , for x = 1 , n − , (25)which is a consequence of Lemma 3, see Remark 2. • The case θ = 1: we can write Γ ns ( f ) as Y ns ( ∆ n f ) + √ n (cid:16) ∂ u f (0) − f (0) (cid:17) ( η s (1) − ρ )+ √ n (cid:16) ∂ u f (1) + f (1) (cid:17) ( η s ( n − − ρ ) + O (cid:16) √ n (cid:17) . Since f ∈ S θ the last expression equals to Y ns ( ∆ n f ) . • The case θ >
1: we can repeat the computations above and since f ∈ S θ ,in order to close the equation for the martingale term we need to show thatlim n →∞ E θ,nρ h(cid:16) Z t n / n θ ( η s ( x ) − ρ ) ds (cid:17) i = 0 , for x = 1 , n − , (26)which is a consequence of Lemma 3, see Remark 2.From the previous observations, for each regime of θ we can rewrite (23)as (24) plus a negligible term. Lemma 1.
For all θ ≥ , t > and f ∈ S θ it holds that lim n →∞ E θ,nρ [ | M nt ( f ) | ] = t k∇ θ f k L ,θρ , where the norm above was defined in (15) .Proof. A simple computation shows that the integral part of the martingale N nt ( f ) can be written as n L θn Y ns ( f ) − Y ns ( f ) n L θn Y ns ( f ) = 1 n n − X x =1 (cid:16) ∇ + n f (cid:16) xn (cid:17)(cid:17) (cid:16) η s ( x ) − η s ( x + 1) (cid:17) + nn θ (cid:16) f (cid:0) n (cid:1)(cid:17) (cid:16) ρ − ρη s (1) + η s (1) (cid:17) + nn θ (cid:16) f (cid:0) n − n (cid:1)(cid:17) (cid:16) ρ − ρη s ( n −
1) + η s ( n − (cid:17) , from where we get that E θ,nρ (cid:2) | M nt ( f ) | (cid:3) = 2 χ ( ρ ) t ( n n − X x =1 (cid:16) ∇ + n f (cid:16) xn (cid:17)(cid:17) + nn θ (cid:16) f (cid:0) n (cid:1)(cid:17) + (cid:16) f (cid:0) n − n (cid:1)(cid:17) !) . (27)Let f ∈ S θ . The first term at the right hand side of the previous expressionconverges to 2 χ ( ρ ) R (cid:16) ∇ θ f ( u ) (cid:17) du , for all θ ≥
0. The second term at theright and side of last expression has to be analyze for each case of θ separately: • The case θ <
1: since f (0) = 0 = f (1), the second term at the righthand side of (27) can be rewritten as 2 χ ( ρ ) t times nn θ (cid:16) f (cid:0) n (cid:1)(cid:17) + (cid:16) f (cid:0) n − n (cid:1)(cid:17) ! = 1 n θ (cid:16) ∇ + n f (0) (cid:17) + (cid:16) ∇ − n f ( n ) (cid:17) ! , which goes to zero as n → ∞ . • The case θ = 1: the second term at the right hand side of (27) converges,as n → ∞ , to 2 χ ( ρ ) (cid:16) f (cid:0) (cid:1) + f (cid:0) (cid:1)(cid:17) . Recalling that f (0) = ∂ u f (0) and f (1) = − ∂ u f (1), the proof ends. • The case θ >
1: since f ∈ S θ and nn θ →
0, as n → ∞ , the second termat the right hand side of (27) converges to zero when n → ∞ .We have just proved that the quadratic variation of the martingale convergesin mean. In the next Lemma below we state the stronger convergence of themartingales to a Brownian motion. Lemma 2.
For f ∈ S θ , the sequence of martingales { M nt ( f ); t ∈ [0 , T ] } n ∈ N converges in the topology of D ([0 , T ] , R ) , as n → ∞ , towards a Brownianmotion W t ( f ) of quadratic variation given by t k∇ θ f k L ,θρ where k · k L ,θρ wasdefined in (15) .Proof. We can repeat here the same proof of [14, page 4170], which is basedon Lemma 1 and the fact that a limit in distribution of a uniformly integrable quilibrium fluctuations for the slow boundary exclusion process 15 sequence of martingales is a martingale. We leave the details to the interestedreader.
Proposition 3.
The sequence { Y n } n ∈ N converges in distribution to Y , where Y is a Gaussian field with mean zero and covariance given by (17) .Proof. We first claim that, for every f ∈ S θ and every t > n → + ∞ log E θ,nρ h exp { iλ Y n ( f ) } i = − λ χ ( ρ ) Z f ( u ) du . Since ν nρ is a Bernoulli product measure,log E θ,nρ [exp { iλ Y n ( f ) } ] = log E θρ h exp n iλ √ n X x ∈ Σ n ( η ( x ) − ρ ) f (cid:16) xn (cid:17)oi = X x ∈ Σ n log E θρ h exp n iλ √ n ( η ( x ) − ρ ) f (cid:16) xn (cid:17)oi . Since f is smooth and using Taylor’s expansion, the right hand side of lastexpression is equal to − λ n X x ∈ Σ n f (cid:16) xn (cid:17) χ ( ρ ) + O (cid:16) √ n (cid:17) . Taking the limit as n → + ∞ and using the continuity of f , the proof of theclaim ends. Replacing f by a linear combination of functions and recallingthe Cr´amer-Wold device, the proof finishes. Now we prove that the sequence of processes { Y nt ; t ∈ [0 , T ] } n ∈ N is tight.Recall that we have defined the density fluctuation field on test functions f ∈ S θ . Since we want to use Mitoma’s criterium [20] for tightness, we needthe following property from the space S θ . Proposition 4.
The space S θ endowed with the semi-norms given in (12) isa Fr´echet space.Proof. The definition of a Fr´echet space can be found, for instance, in [21].Since C ∞ ([0 , a closed subspace of a Fr´echet space is also a Fr´echet space, it is enough toshow that S θ is a closed subspace of C ∞ ([0 , S ′ θ valued processes { Y nt ; t ∈ [0 , T ] } n ∈ N follows from tightnessof the sequence of real-valued processes { Y nt ( f ); t ∈ [0 , T ] } n ∈ N , for f ∈ S θ . Proposition 5 (Mitoma’s criterium, [20]).
A sequence of processes { x t ; t ∈ [0 , T ] } n ∈ N in D ([0 , T ] , S θ ′ ) is tight with respect to the Skorohod topol-ogy if, and only if, the sequence { x t ( f ); t ∈ [0 , T ] } n ∈ N of real-valued processesis tight with respect to the Skorohod topology of D ([0 , T ] , R ) , for any f ∈ S θ . Now, to show tightness of the real-valued process we use the Aldous’ cri-terium:
Proposition 6.
A sequence { x t ; t ∈ [0 , T ] } n ∈ N of real-valued processes istight with respect to the Skorohod topology of D ([0 , T ] , R ) if:i) lim A → + ∞ lim sup n → + ∞ P µ n (cid:16) sup ≤ t ≤ T | x t | > A (cid:17) = 0 , ii) for any ε > , lim δ → lim sup n → + ∞ sup λ ≤ δ sup τ ∈ T T P µ n ( | x τ + λ − x τ | > ε ) = 0 , where T T is the set of stopping times bounded by T . Fix f ∈ S θ . By (21), it is enough to prove tightness of { Y n ( f ) } n ∈ N , { R t Γ ns ( f ) ds ; t ∈ [0 , T ] } n ∈ N , and { M nt ( f ); t ∈ [0 , T ] } n ∈ N . This follows from Proposition 3.
By Lemma 2, the sequence of martingales converges. In particular, it is tight.
The first claim of Aldous’ criterium can be easily checked for the integralterm R t Γ ns ( f ) ds , where the expression for Γ ns ( f ) can be found in (23). Let f ∈ S θ . quilibrium fluctuations for the slow boundary exclusion process 17 • The case θ <
1: by Young’s inequality and Cauchy-Schwarz’s inequalitywe have that E θ,nρ h sup t ≤ T (cid:16) Z t Γ ns ( f ) ds (cid:17) i ≤ CT Z T E θ,nρ h(cid:16) √ n n − X x =1 ∆ n f ( xn )( η sn ( x ) − ρ ) (cid:17) i ds + C ( ∇ + n f (0)) T Z T E θ,nρ h(cid:16) √ n ( η sn (1) − ρ ) (cid:17) i ds + C ( ∇ − n f (1)) T Z T E θ,nρ h(cid:16) √ n ( η sn ( n − − ρ ) (cid:17) i ds. Since f ∈ S θ and by (25), the second and third terms at the right handside of the previous expression go to zero, as n → ∞ . Then there exists C > CT . Thefirst term at the right hand side of last expression is bounded from aboveby T times 1 n n − X x =1 (cid:0) ∆ n f ( xn ) (cid:17) χ ( ρ ) . (28)Now, since f ∈ S θ last expression is bounded from above by a constant.Now we need to check the second claim of Aldous’ criterium. For thatpurpose, fix a stopping time τ ∈ T T . By Chebychev’s inequality togetherwith (28), we get that P θ,nρ (cid:16)(cid:12)(cid:12)(cid:12) Z τ + λτ Γ ns ( f ) ds (cid:12)(cid:12)(cid:12) > ε (cid:17) ≤ ε E θ,nρ h(cid:16) Z τ + λτ Γ ns ( f ) ds (cid:17) i ≤ δCε , which vanishes as δ → • The case θ = 1: we note that it was treated in [16]. • The case θ >
1: as in the case θ <
1, we have that E θ,nρ h sup t ≤ T (cid:16) Z t Γ ns ( f ) ds (cid:17) i ≤ CT Z T E θ,nρ h(cid:16) √ n n − X x =1 ∆ n f ( xn )( η sn ( x ) − ρ ) (cid:17) i ds + C f (cid:16) n (cid:17) T Z T E θ,nρ h(cid:16) n / n θ ( η sn (1) − ρ ) (cid:17) i ds + C f (cid:16) n − n (cid:17) T Z T E θ,nρ h(cid:16) n / n θ ( η sn ( n − − ρ ) (cid:17) i ds , plus a term of order √ n . To bound the first term at the right hand sideof the previous inequality we repeat the same computations as in the case θ <
1. In order to bound the second and the third terms at the right handside of the previous inequality, we use (26) and the proof follows as in thecase θ < This section is devoted to estimate the expectations (25) and (26). In orderto do this we start introducing some notations. Let µ be an initial measure.For x = 0 , , . . . , n −
1, define I x,x +1 ( f, µ ) := Z r x,x +1 ( η ) (cid:0) f ( σ x,x +1 η ) − f ( η ) (cid:1) dµ , where σ x,x +1 η was defined in (2), for x = 1 , ..., n − σ , η := η , σ n − ,n η := η n − (the configurations η and η n − were defined in (3)), and the rates aregiven by r , ( η ) := r α ( η ) := αn θ (1 − η (1)) + 1 − αn θ η (1) ,r n − ,n ( η ) := r β ( η ) := βn θ (1 − η ( n − − βn θ η ( n − ,r x,x +1 ( η ) := 1 , if x = 1 , · · · , n − . Define the quantity: D n ( f, µ ) := n − X x =0 I x,x +1 ( f, µ ) = n − X x =0 Z r x,x +1 ( η ) (cid:0) f ( σ x,x +1 η ) − f ( η ) (cid:1) dµ . (29)The Dirichlet form is defined by h− L θn f, f i µ , where we can rewrite forshort the infinitesimal generator as L θn f ( η ) := n − X x =0 L x,x +1 f ( η ) := n − X x =0 r x,x +1 ( η )( f ( σ x,x +1 η ) − f ( η )) . Now, we recall that we consider the case α = β = ρ ∈ (0 , ν nρ (the Bernoulli product measure) is invariant for this process andit satisfies r x,x +1 ( η ) ν nρ ( η ) = r x,x +1 ( σ x,x +1 η ) ν nρ ( σ x,x +1 η ) , (30)for all x ∈ { , , . . . , n − } . Let us check this equality in the case x = 0, thecase x = n − quilibrium fluctuations for the slow boundary exclusion process 19 r , ( σ , η ) ν nρ ( σ , η ) ν nρ ( η ) = h ρn θ (1 − η (1)) + 1 − ρn θ η (1) i ν nρ ( η ) ν nρ ( η ) . (31)Since ν nρ ( η ) ν nρ ( η ) = η (1)=1 − ρρ + η (1)=0 ρ − ρ , (32)then (31) becomes η (1)=1 h ρn θ i − ρρ + η (1)=0 h − ρn θ i ρ − ρ = r , ( η ) . Thus, using (30), we get h− L θn f, f i ν nρ = 12 D n ( f, ν nρ ) . (33) Lemma 3 (Replacement Lemma).
Let x = 1 , n − and t > fixed. Then E θ,nρ h(cid:16) Z t c n (cid:0) η s ( x ) − ρ (cid:1) ds (cid:17) i ≤ C c n n θ n . Remark 2.
Recall that for θ < c n = √ n , so that the errorabove becomes n θ /n , which vanishes as n → ∞ . Recall that for θ > c n = n / /n θ , so that the error above becomes n/n θ , which vanishesas n → ∞ . Proof.
The proof follows by a classical argument combining both the Kipnis-Varadhan’s inequality (see [18, page 33, Lemma 6.1]) with Young’s inequality.For that purpose let x = 1 (the other case is completely analogous) and notethat the expectation in the statement of the lemma can be bounded fromabove by sup f ∈ L νnρ n Z c n ( η (1) − ρ ) f ( η ) dν nρ + n h L θn f, f i ν nρ o , (34)where L ν nρ is the space of functions f such that R f ( η ) dν nρ < + ∞ . We startby writing the integral R c n ( η (1) − ρ ) f ( η ) dν nρ as twice its half and in one ofthe terms we make the exchange η → η to have12 Z c n ( η (1) − ρ ) f ( η ) dν nρ + 12 Z c n (1 − η (1) − ρ ) f ( η ) ν nρ ( η ) ν nρ ( η ) dν nρ , see (32) to get the expression of ν nρ ( η ) ν nρ ( η ) . A simple computation shows that theintegral at the right hand side of last expression is equal to − Z c n ( η (1) − ρ ) f ( η ) dν nρ , so that the display above is equal to12 Z c n ( η (1) − ρ )( f ( η ) − f ( η )) dν nρ . By Young’s inequality we can bound the previous expression by B Z c n ( η (1) − ρ ) dν nρ + 14 B Z ( f ( η ) − f ( η )) dν nρ . Now, remember the notation η = σ , η and multiply and divide by r , ( η )the integrand function inside the second integral above. We can do it, becausethat there exists ˜ C ρ such that ˜ C ρ n θ ≤ r , ( η ) ≤ C ρ n θ . Then we can bound theprevious expression from above by B Z c n ( η (1) − ρ ) dν nρ + n θ B ˜ C ρ Z r , ( η ) ( f ( σ , η ) − f ( η )) dν nρ . Using (29) the second integral in the last expression is bounded from aboveby D n ( f, ν nρ ). Recalling (33), we get Z c n ( η (1) − ρ ) f ( η ) dν nρ ≤ B c n Z ( η (1) − ρ ) dν nρ + n θ B ˜ C ρ h− L θn f, f i ν nρ . Putting this inequality in (34) and choosing B = n θ − / C ρ , the term at theright hand side of lthe last expression cancels with n h L θn f, f i ν nρ . Therefore,the expectation appearing in the statement of the lemma is bounded fromabove by c n n θ C ρ n Z ( η (1) − ρ ) dν nρ . Since η is bounded the proof ends. Acknowledgements
A. N. was supported through a grant “L’OR´EAL - ABC - UNESCO ParaMulheres na Ciˆencia”. P. G. thanks FCT/Portugal for support through theproject UID/MAT/04459/2013. T. F. was supported by FAPESB throughthe project Jovem Cientista-9922/2015. quilibrium fluctuations for the slow boundary exclusion process 21
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