A stochastic Burgers equation from a class of microscopic interactions
aa r X i v : . [ m a t h . P R ] J a n The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2015
A STOCHASTIC BURGERS EQUATION FROM A CLASS OFMICROSCOPIC INTERACTIONS
By Patr´ıcia Gonc¸alves , Milton Jara and Sunder Sethuraman PUC-RIO and Universidade do Minho, IMPA and University of Arizona
We consider a class of nearest-neighbor weakly asymmetric massconservative particle systems evolving on Z , which includes zero-range and types of exclusion processes, starting from a perturba-tion of a stationary state. When the weak asymmetry is of order O ( n − γ ) for 1 / < γ ≤
1, we show that the scaling limit of the fluc-tuation field, as seen across process characteristics, is a generalizedOrnstein–Uhlenbeck process. However, at the critical weak asymme-try when γ = 1 /
2, we show that all limit points satisfy a martingaleformulation which may be interpreted in terms of a stochastic Burg-ers equation derived from taking the gradient of the KPZ equation.The proofs make use of a sharp “Boltzmann–Gibbs” estimate whichimproves on earlier bounds.
1. Introduction.
There has been much recent work on the classificationof fluctuations of certain interfaces and currents, corresponding to mass con-servative particle dynamics in one-dimensional nearest-neighbor interactingparticle systems such as simple exclusion and its variants, with respect toso-called Edwards–Wilkinson (EW) and Kardar–Parisi–Zhang (KPZ) classes(cf. [19] for a review and references). Following recent sensibilities, a d = 1particle system is in the EW class if the standard deviation of the associated“height” function h t ( x ) of the interface at time t and space point x , or theintegrated current at time t ≥ x ∈ R , is of order t / , and also spatial correlations are nontrivial at range t / . Examples inthis class are independent random walk systems, random averaging and re-versible simple exclusion processes starting from a stationary state or evenin nonstationary states [9, 22, 32, 37, 54]. Received September 2012; revised July 2013. Supported by FCT research project “Nonequilibrium Statistical Physics” PTDC/MAT/109844/2009 and PEst-C/MAT/UI0013/2011. Supported in part by NSF DMS-11-59026.
AMS 2000 subject classifications.
Primary 60K35; secondary 82C22.
Key words and phrases.
KPZ equation, Burgers, weakly asymetric, zero-range, kineti-cally constrained, speed-change, fluctuations.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2015, Vol. 43, No. 1, 286–338. This reprint differs from the original in paginationand typographic detail. 1
P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
On the other hand, a system is in the KPZ class if its “height” functionand integrated current have standard deviation of order t / , and nontrivialspatial correlations at range t / . A well-studied particle system model in thisclass is the asymmetric simple exclusion process starting from deterministicinitial configurations such as step profile and alternating conditions, or froma stationary state (cf. [6, 7, 10, 11, 16, 17, 24, 39, 46, 49, 58] and referencestherein).These two classes can be seen in the study of the famous KPZ stochasticpartial differential equation first mentioned in [33]: ∂ t h t ( x ) = D ∆ h t ( x ) + a ( ∇ h t ( x )) + σ ˙ W t ( x ) , (1.1)where ˙ W t ( x ) is a space–time white noise with unit variance. When a = 0and D, σ >
0, then h t ( x ) is a generalized Ornstein–Uhlenbeck process inEW class. However, when a = 0 and D, σ >
0, a physical argument indicatesthat h t ( x ) is in the KPZ class (cf. [33, 59]). Also, in another sense, it hasbeen shown that the “Cole–Hopf” solution of the KPZ equation, startingfrom certain initial conditions, interpolates between the two classes whenthe centered solution is examined in different asymptotic scaling regimes,that is, when normalized by t / as t ↑ ∞ or when normalized by t / as t ↓
0, nontrivial limits are obtained (cf. [1, 8]).Moreover, it is believed that in many “critical” weakly asymmetric, d = 1particle systems, that is, when the weak asymmetry is scaled at a criticallevel, the diffusively scaled “height” function or integrated current shouldconverge to the solution of the KPZ equation with parameters dependingon the structure of particle interactions and initial conditions. Recently,much progress has been made in making clear this convergence. Part of thedifficulty is that, since “solutions” to the KPZ equation are expected tobe distribution-valued, the nonlinear term in the equation does not makesense, and so the equation is ill-posed. Hence, what does it mean to solvethe KPZ equation and also, when properly interpreted, how to derive theKPZ equation from microscopic particle interactions?One way to approach these questions is to observe that the Cole–Hopftransformation z t ( x ) := exp { ( a/D ) h t ( x ) } linearizes the KPZ equation to astochastic heat equation ∂ t z t ( x ) = D ∆ z t ( x ) + ( aσ/D ) z t ( x ) ˙ W t ( x ) , (1.2)which can be solved uniquely starting from a class of initial conditions and isalso strictly positive for times t > h t ( x ) := log z t ( x ). In [14], starting from near stationary mea-sures in a certain weakly asymmetric simple exclusion process observed indiffusive scale, this sentiment was made rigorous. Namely, it was proved thatthe microscopic Cole–Hopf transform of the microscopic height function, us-ing a clever device in [25] which linearizes the simple exclusion dynamics TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS to a more manageable system, converges to the Cole–Hopf transform of theKPZ equation, the solution to the stochastic heat equation (1.2). More re-cently, in [1, 52] this notion of solution further gained traction in that theresult in [14] was nontrivially generalized to step profile deterministic initialconfigurations. At the same time, in [30], it has been shown that log z t ( x )is the unique solution of a well-posed equation on a torus, derived from a“rough paths” approximation of (1.1), so that it is clear what sort of KPZequation the “Cole–Hopf” solution actually solves.In this article, another approach is considered which allows to general-ize the types of microscopic particle interactions considered, given that thedevice in [25] seems limited to simple exclusion and a few variants such as q -TASEP dynamics [15]. At the microscopic level, the height function H t ( x ),evaluated for t ≥ x ∈ Z , takes form H t ( x ) = J ( t ) − x − X y =0 η t ( y ) , for x ≥ ,J ( t ) , for x = 0 ,J ( t ) + − X y = x η t ( y ) , for x ≤ − , (1.3)where J y ( t ) is the current across bond ( y − , y ) and η t ( y ) is the particlenumber at y ∈ Z at time t ≥
0. Then the discrete gradients of the microscopicheight function are the particle numbers, H t ( x + 1) − H t ( x ) = η t ( x ), and thecorresponding fluctuation field examined in diffusive scale, that is, whentime is scaled in terms of n and space is scaled by n , is the particle densityfluctuation field Y nt . The guiding idea is that Y nt should converge to Y t = ∇ h t in some sense.Formally, by carrying through the “ ∇ ” operation, Y t satisfies a type ofstochastic Burgers equation, ∂ t Y t ( x ) = D ∆ Y t ( x ) + a ∇ ( Y t ( x )) + σ ∇ ˙ W t ( x ) , (1.4)which again for the same reasons as for the original KPZ equation is ill-posedwhen a = 0. If a = 0, however, it is a type of Ornstein–Uhlenbeck equationwhich possesses a unique solution when starting from a large class of initialdistributions (cf. [13, 61]).A main contribution of the article is to understand the derived stochasticBurgers equation (1.4) in the context of a general class of nearest-neighborweakly asymmetric interacting particle systems on Z , starting from pertur-bations of the invariant measure ν ρ . This class is composed of systems with“gradient” dynamics, not necessarily product invariant measures, sufficientspectral gap and “equivalence of ensembles” estimates among other tech-nical conditions (cf. Section 2.1), which include in particular the already P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN studied simple exclusion process, and also zero-range and exclusion modelswith kinetically constrained or speed-change interactions, which have vary-ing and sometimes slow mixing behaviors. The initial distributions consist of“bounded entropy” perturbations of the invariant measure ν ρ (cf. Section 2.1for a precise statement).Our results describe the limit points of the fluctuation field Y n,γt in diffu-sive scale, in a reference frame moving with a process characteristic velocity υ n ( t ) ∼ n − ⌊ n ( p n − q n ) υt ⌋ . Here, p n − q n is the difference of the single par-ticle jump rates which identifies the strength of the weak asymmetry consid-ered, and υ is a homogenized velocity parameter depending on the particledynamics. Given the size of p n − q n , a dichotomy emerges in the form of thelimits derived. Namely, for p n − q n = O ( n − γ ), when 1 / < γ ≤
1, we showa “crossover result” (Theorem 2.2) that Y n,γt converges to an Ornstein–Uhlenbeck field with certain homogenized parameters. When γ = 1, con-vergence of Y n,γt to the same Ornstein–Uhlenbeck field has been known formany particle systems since the work [18]. For discussions of “crossover”results with respect to simple exclusion, see [28, 53].However, when γ = 1 /
2, a critical value, we prove (Theorem 2.3) thatlimit points of Y n,γt satisfy a martingale formulation, which we dub as an“energy” formulation (cf. Theorem 2.3), also with homogenized constants,which interprets the stochastic Burgers equation: Namely, the nonlinear termin (1.4) is understood in terms of a certain Cauchy limit of a function ofthe fluctuation field acting on an approximation of a point mass as theapproximation becomes more refined. We remark, however, with respectto simple exclusion processes, convergence of Y n,γt to a unique limit when γ = 1 / ν ρ (cf. point 2 ofRemark 2.4). Also, we note another martingale formulation was given withrespect to the Burgers equation in [4].In our general framework, convergence of Y n,γt to a unique limit when γ = 1 / H n,γt ( x ) := n − / H n t ( nx − nυ n ( t )), via (1.3)given subsequential convergence of Y n,γt . Although this is not the purpose ofthis paper, we indicate how this might be accomplished to be more complete.Indeed, by (1.3) and J ( t ) − J x ( t ) = P x − y =0 ( η t ( y ) − η ( y )), one has H n,γt ( x ) = n − / J nx − nυ n ( t ) ( n t ) − n − / P nx − nυ n ( t ) − y =0 η ( y ), say for x > υ n ( t ). To writethe current in terms of the fluctuation field, formally, n − / J nx − nυ n ( t ) ( n t ) = Y n,γt (1 [ x, ∞ ) ) − Y n,γ (1 [ x, ∞ ) ) + o (1), although as there are an infinite number TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS of particles and 1 [ x, ∞ ) is not a compactly supported function some sortof truncation is needed to make a rigorous argument. Using the methodin [51] and [32], one can approximate n − / J nx − nυ n ( t ) ( n t ) by Y n,γt ( G k,x ) −Y n,γ ( G k,x ) for large k where G k,x ( z ) = (1 − ( z − x ) + /k ) + , and so it is possibleto take subsequential limits of H n,γt .Finally, we remark if uniqueness of solution for the γ = 1 / Y n,γt , namely d Y n,γt = ( ∂ t Y n,γt + L n Y n,γt ) dt + d M n,γt , where L n is the system infinitesimal generator and M n,γt is a martingale.We note, because the reference frame moves with velocity υ n ( t ), the term ∂ t Y n,γt does not vanish. Beginning in a perturbed invariant measure, themartingale term can be handled by an ergodic theorem. However, to writethe drift term ∂ t Y n,γt + L n Y n,γt , in terms of the fluctuation field itself and,therefore, “close” the equation, is a more difficult task, and requires whathas been known as a “Boltzmann–Gibbs” principle. Such a principle wasfirst proved in [18] when γ = 1. In our context, we would like to recover asecond-order term, and the principle would replace Z t n γ − / X x ∈ Z ∇ G ( x/n ) τ x V ( η n s ) ds with ϕ ′′ V ( ρ )2 Z t n γ +1 / X x ∈ Z ∇ G ( x/n ) × (cid:26) Y n,γs (cid:18) ε [ x − ε,x + ε ] (cid:19) − E ν ρ (cid:20) Y n,γs (cid:18) ε [ x − ε,x + ε ] (cid:19) (cid:21)(cid:27) ds in L ( P ν ρ ) as n ↑ ∞ and ε ↓
0. Here, G is a function in the Schwarz class, τ x is the x -shift operator, V is a mean-zero function with the property thatthe derivative of its “tilted mean” ϕ V ( z ) vanishes at z = ρ [cf. definitionnear (2.4)]. Given such a replacement principle (cf. Section 3.2 for precisestatements), one can prove the sequence Y n,γt is tight and derive martingaleformulations of limit points as desired. P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
The case γ = 1 / / < γ ≤ γ = 1 / H − renormalization scheme to bound er-rors in the replacement. Such a scheme makes good use of three assumedingredients [cf. precise statements (R), (G), (EE) in Section 2.1]: First, themeasure ν ρ is invariant with respect to all asymmetric and symmetric ver-sions of the process, the main reason for the “gradient dynamics” condition.Second, a spectral gap lower bound for the symmetric process localized ona interval Λ ℓ with width ℓ and P x ∈ Λ ℓ η ( x ) particles which, after averagingwith respect to ν ρ , is of order O ( ℓ − ). Also, third, an “equivalence of en-sembles” estimate holds with respect to canonical ν ρ ( ·| P | x |≤ ℓ η ( x ) = k ) andgrand canonical ν ρ measures.We note the current article is an evolution of the arXiv paper [27], encom-passing the work there on a type of exclusion model starting from a Bernoulliproduct invariant measure and a model specific Boltzmann–Gibbs principle.See also [3] for a different type of resolvent method specific to simple ex-clusion. In this context, the current article is a nontrivial generalization tomore diverse models, starting from perturbations of the stationary state,using a more general H − renormalization scheme. We remark that part ofthis improvement, of its own interest, is that the Boltzmann–Gibbs principle(Theorem 3.2) shown here does not rely on the independence structure ofa product measure, or on a sharp spectral gap estimate, or on a process“duality.” Finally, we note elements of our H − renormalization scheme goback to [26] and [56] in different contexts.We now give the structure of the article. In Section 2, the general class ofmodels studied, results and specific systems satisfying the class assumptionsare discussed. Then, in Section 3, we outline the proof of the main results,Theorems 2.2 and 2.3, stating the form of Boltzmann–Gibbs principle used.In Section 4, this principle is proved. Finally, in Section 5, we prove for aclass of systems, including the specific processes discussed in Section 2, the“equivalence of ensembles” estimate assumed for the proofs in Section 3.
2. Abstract framework, results and models.
We now discuss the abstractframework we work with in Section 2.1, and state results in this frameworkin Section 2.2. This framework covers a wide class of models such as zero-range models and different types of exclusion processes which we detail inSections 2.3–2.5. A reader focusing on one of these models, might skip toits subsection while referring to Section 2.1, and then proceed to results inSection 2.2.
TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Notation and assumptions.
We consider a sequence of “weakly asym-metric” nearest-neighbor “mass conservative” particle systems { η nt : t ≥ } on the state space Ω = N Z where N = { , , , . . . } . The configuration of thesystem η t = { η t ( x ) : x ∈ Z } is a collection of occupation numbers η t ( x ) whichcounts the numbers of particles at sites x ∈ Z at time t ≥
0. In some of theexamples we will consider, the occupation number is bounded by 1, in whichcase the effective state space reduces to { , } Z . “Gradient” dynamics. The dynamics will be of “gradient” type. That is,we suppose there are functions { b R,nx } n ≥ and { b L,nx } n ≥ satisfying the follow-ing conditions (R1) and (R2). Let τ x be the shift operator where ( τ x η )( z ) = η ( x + z ) and τ x f ( η ) = f ( τ x η ) for x ∈ Z . Let also Λ k = { j : | j | ≤ k } ⊂ Z for k ≥ n ≥ b R,nx = τ x b R,n and b L,nx = τ x b L,n are nonnegative finite-range functions on Ω such that b R,n and b L,n are supported on { η ( y ) : y ∈ Λ R } for some R ≥
1. We suppose uniformly in n that | b R,n ( η ) | + | b L,n ( η ) | ≤ C P y ∈ Λ R η ( y ). Moreover, there are nonnegative functions c nx = τ x c n on Ω,supported on { η ( x ) : x ∈ Λ R } such that b R,nx ( η ) − b L,nx ( η ) = c nx ( η ) − c nx +1 ( η ) . In addition, suppose there are fixed functions b R , b L and c such thatconfiguration-wiselim n ↑∞ b R,n ( η ) = b R ( η ) , lim n ↑∞ b L,n ( η ) = b L ( η ) and lim n ↑∞ c n ( η ) = c ( η ) . In some of the models considered, such as zero-range processes in Sec-tion 2.3, the functions b R,n = b R , b L,n = b L and c n = c are fixed and donot depend on the parameter n . However, for the kinetically constrainedexclusion models in Section 2.4, the rates do depend on n .(R2) With respect to a fixed measure ν ρ on Ω, for all n ≥
1, we have b R,nx ( η x +1 ,x ) dν x +1 ,xρ dν ρ ( η ) = b L,nx ( η ) , where ν x +1 ,xρ is the measure of the variable ζ = η x +1 ,x under ν ρ .We also define b nx ( η ) = b R,nx ( η ) + b L,nx ( η ), b n ( η ) = b n ( η ) and c n ( η ) = c n ( η ) tosimplify notation.We now specify the process generator. For a ∈ R and γ >
0, let p n = 12 + a n γ and q n = 1 − p n = 12 − a n γ . Let also n be such that 0 ≤ p n , q n ≤
1, and
T >
P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN (M) Suppose, for each a ∈ R , γ > n ≥ n , that { η nt : t ∈ [0 , T ] } isa L ( ν ρ ) Markov process with strongly continuous Markov semigroup P nt and Markov generator L n (cf. Chapter I; Section IV.4 of [40]) with a corecomposed of local L ( ν ρ ) functions on which L n f ( η ) = n X x ∈ Z { b R,nx ( η ) p n ∇ x,x +1 f ( η ) + b L,nx ( η ) q n ∇ x +1 ,x f ( η ) } , (2.1)where ∇ x,y f ( η ) = f ( η x,y ) − f ( η ), and η x,y is the configuration obtained from η by moving a particle from x to y : η x,y ( z ) = η ( y ) + 1 , when z = y,η ( x ) − , when z = x,η ( z ) , otherwise.The role of a ∈ R and γ > Invariant measure ν ρ . We now specify some technical properties which ν ρ should satisfy. Define for a probability measure κ , the path measure P κ gov-erning the process { η nt : t ∈ [0 , T ] } with initial configurations η distributedaccording to κ . Let then E κ and E κ denote expectations with respect to κ and P κ , respectively.(IM1) Suppose ν ρ is a translation-invariant measure which is “spatiallymixing.” That is, for local L ( ν ρ ) functions f and h ,lim | x |↑∞ E ν ρ [ f ( η ) τ x h ( η )] = E ν ρ [ f ] E ν ρ [ h ] . In addition, suppose the mean E ν ρ [ η (0)] = ρ , and moment-generating func-tion E ν ρ [ e λη (0) ] < ∞ for | λ | ≤ λ ∗ for a λ ∗ > ν ρ are considered in most of the examples, wenote, in Section 2.5, a nonproduct measure ν ρ corresponding to an exponen-tially mixing ergodic Markov chain is used.Now, the measure ν ρ , by (IM1) and the “gradient dynamics” conditions(R1) and (R2), is an invariant measure with respect to L n for all a ∈ R and γ >
0. Indeed, let φ be a local L ( ν ρ ) function supported with respect tosites in Λ k . Then, for ℓ > k , we have E ν ρ [ L n φ ] = − n E ν ρ (cid:20) X | x |≤ ℓ ( p n − q n ) φ ( η )[ c nx ( η ) − c nx +1 ( η )] (cid:21) = − n ( p n − q n ) E ν ρ [ φ ( η )( c n − ℓ ( η ) − c nℓ +1 ( η ))] . The limit as ℓ ↑ ∞ vanishes, by translation-invariance and the spatial mixingassumption in (IM1). TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS One can also compute that the L ( ν ρ ) adjoint L ∗ n is the generator withparameter − a , that is, when the jump probability is reversed. Define S n =( L n + L ∗ n ) /
2. Then the Dirichlet form D ν ρ ,n ( f ) := E ν ρ [ f ( − L n f )] = E ν ρ [ f ( − S n f )] on local L ( ν ρ ) functions is given by D ν ρ ( f ) = 12 X x ∈ Z E ν ρ [ b R,nx ( η )( ∇ x,x +1 f ( η )) ] . (2.2)Moreover, when a = 0, S n is the generator of the associated process and ν ρ is a reversible measure.Consider now the empirical measure Y n = 1 √ n X x ∈ Z ( η ( x ) − ρ ) δ x/n and its covariance under measure κ , on compactly supported functions, C nκ ( G, H ) = E κ [( Y n ( G ) − E κ [ Y n ( G )])( Y n ( H ) − E κ [ Y n ( H )])] . (IM2) We assume, starting from ν ρ , that Y n converges weakly to a spa-tial Gaussian process with covariance C ν ρ ( G, H ) := lim n ↑∞ C nν ρ ( G, H ) suchthat C ν ρ ( G, G ) ≤ C ( ρ ) k G k L ( R ) . Also, suppose the moment bound holdssup ℓ ≥ k ( √ ℓ P ℓx =1 ( η ( x ) − ρ )) k L ( ν ρ ) < ∞ .It will be convenient to define the variances σ n ( ρ ) := C nν ρ ( H, H ) = E ν ρ (cid:20)(cid:18) √ n + 1 X x ∈ Λ n ( η ( x ) − ρ ) (cid:19) (cid:21) and σ ( ρ ) = C ν ρ ( H, H ) = lim n ↑∞ σ n ( ρ ) when H ( x ) = 1 [ − , ( x ).When ν ρ is sufficiently mixing, the case of our examples, (IM2) holds with C ν ρ ( G, H ) = σ ( ρ ) h G, H i L ( R ) .Now, for λ ∈ ( − λ ∗ , λ ∗ ), consider the tilted measure ν λρ with “tilt” or“chemical potential” λ given by its finite-dimensional projections dν λρ dν ρ ( η ( x ) = e ( x ) , x ∈ Λ ℓ | η ( x ) = ξ ( x ) , x / ∈ Λ ℓ ) = e λ P x ∈ Λ ℓ ( e ( x ) − ρ ) Z ( λ, ℓ, ξ ) , (2.3)where e, ξ ∈ Ω and Z ( λ, ℓ, ξ ) is the normalization.(D1) We will assume the measures { ν λρ : | λ | < λ ∗ } are well defined on Ω,that is a limit of (2.3) as Λ ℓ ր Z can be taken, not depending on ξ . Also,we assume that the measures can be indexed by density, that is, E ν λρ [ η (0)]is strictly increasing in λ for | λ | ≤ λ ∗ . P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
These assumptions hold when ν ρ is a nontrivial product measure satisfying(IM1): ddλ E ν λρ [ η (0)] = E ν λρ [( η (0) − E ν λρ [ η (0)]) ] >
0. They also hold when ν ρ corresponds to the ergodic Markov chain in the case for the exclusion withspeed-change model (cf. details in Section 2.5).The measures { ν λρ : | λ | < λ ∗ } are translation-invariant since ν ρ is assumedtranslation-invariant (IM1). Also, given exponential moments of ν ρ (IM1), E ν λρ [ η (0)] is continuous in λ for | λ | < λ ∗ . Hence, by the strict increasingassumption in (D1), one can reparameterize { ν λρ } in terms of density: Let z ∈ ( ρ ∗ , ρ ∗ ) where ρ ∗ = lim λ ↓− λ ∗ E ν λρ [ η (0)] and ρ ∗ = lim λ ↑ λ ∗ E ν λρ [ η (0)]. Let λ ( z ) ∈ ( − λ ∗ , λ ∗ ) be the parameter such that E ν λ ( z ) ρ [ η (0)] = z . Then we willdefine ν z = ν λ ( z ) ρ .Define also, for a local L ( ν ρ ) function f , the “tilted mean” function ϕ f ( z ) : ( ρ ∗ , ρ ∗ ) → R where ϕ f ( z ) = E ν z [ f ( η )] . We consider the derivatives of ϕ f ( z ) as the formal limits of the derivativesof E ν z [ f ( η ) | η ( x ) = ξ ( x ) , x ∈ Λ ℓ ] as ℓ ↑ ∞ . Define ϕ ′ f ( z ) := λ ′ ( z ) E ν z (cid:20) ( f ( η ) − E ν z [ f ]) (cid:18)X x ∈ Z ( η ( x ) − z ) (cid:19)(cid:21) ,ϕ ′′ f ( z ) := ( λ ′ ( z )) E ν z (cid:20) ( f ( η ) − E ν z [ f ]) (cid:18)X x ∈ Z ( η ( x ) − z ) (cid:19) (cid:21) (2.4) + λ ′′ ( z ) E ν z (cid:20) ( f ( η ) − E ν z [ f ]) (cid:18)X x ∈ Z ( η ( x ) − z ) (cid:19)(cid:21) . For the 0th derivative, we set ϕ (0) f ( z ) := E ν z [ f ].(D2) For local L ( ν ρ ) functions f , suppose the limits (2.4) are well de-fined and | ϕ ′ f ( ρ ) | , | ϕ ′′ f ( ρ ) | ≤ C ( ρ ) k f k L ( ν ρ ) ; already, | ϕ f ( ρ ) | ≤ k f k L ( ν ρ ) . Also,suppose lim n ↑∞ ϕ ′ f n ( ρ ) = ϕ ′ f ( ρ ) and lim n ↑∞ ϕ ′′ f n ( ρ ) = ϕ ′′ f ( ρ )when { f n } and f are local functions such that lim n ↑∞ f n ( η ) = f ( η ) and f n ( η ) ≤ ˆ f ( η ) configuration-wise for each n where ˆ f ∈ L ( ν ρ ).When { ν x } are product or rapidly mixing Markov measures, again the casefor our examples, this condition also holds by calculation with (2.3). Spectral gap . We now give a “spectral gap” condition. For ℓ ≥
1, recallΛ ℓ is the box of size 2 ℓ + 1, namely Λ ℓ := { x ∈ Z : | x | ≤ ℓ } . Let also, for TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS k ≥ ξ ∈ Ω, G k,ℓ,ξ := { η : P x ∈ Λ ℓ η ( x ) = k, η ( y ) = ξ ( y ) for y / ∈ Λ ℓ } be thehyperplane of configurations on Λ ℓ with k particles which equal ξ outsideΛ ℓ . Denote by ν k,ℓ,ξ the canonical measure on G k,ℓ,ξ , namely ν k,ℓ,ξ ( · ) := ν ρ (cid:18) · (cid:12)(cid:12)(cid:12) X x ∈ Λ ℓ η ( x ) = k, η ( y ) = ξ ( y ) for y / ∈ Λ ℓ (cid:19) . Consider now the process, restricted to the hyperplane G k,ℓ,ξ with gener-ator S n, G k,ℓ,ξ f ( η ) = 12 X | x − y | =1 x,y ∈ Λ ℓ b nx ( η ) ∇ x,y f ( η ) . This is a finite-state Markov process with reversible invariant measure ν k,ℓ,ξ .Denote by λ k,ℓ,ξ,n the spectral gap, that is the second largest eigenvalue of −S n, G k,ℓ,ξ (with 0 being the largest). Let W ( k, ℓ, ξ, n ) denote the reciprocalof λ k,ℓ,ξ,n , which is set to ∞ if λ k,ℓ,ξ,n = 0. Then the associated Poincar´e-inequality reads asVar( f, ν k,ℓ,ξ ) ≤ W ( k, ℓ, ξ, n ) D n ( f, ν k,ℓ,ξ ) , (2.5)where Var( f, ν k,ℓ,ξ ) is the variance of f with respect to ν k,ℓ,ξ and the canon-ical Dirichlet form D n ( f, ν k,ℓ,ξ ) is given by D n ( f, ν k,ℓ,ξ ) := 12 X x,x +1 ∈ Λ ℓ E ν k,ℓ,ξ [ b R,nx ( η )( ∇ x,x +1 f ( η )) ] . When W ( k, ℓ, ξ, n ) < ∞ , the process is ergodic and ν k,ℓ,ξ is the unique in-variant measure.Denote the “outside variables” by η cℓ = { η ( x ) : x / ∈ Λ ℓ } . We will assumethe following condition on W ( k, ℓ, ξ, n ).(G) Suppose there is a constant C = C ( ρ ) such that, for n ≥
1, we have E ν ρ (cid:20) W (cid:18) X x ∈ Λ ℓ η ( x ) , ℓ, η cℓ , n (cid:19) (cid:21) ≤ Cℓ . We remark a sufficient condition to verify (G) would be the uniform boundsup k,ξ,n ℓ − W ( k, ℓ, ξ, n ) < ∞ , which holds for some types but not all of thespecific models discussed. Equivalence of ensembles.
We will also assume an “equivalence of ensem-bles” estimate between the canonical and grand-canonical measures. Define,for ℓ ≥ η ∈ Ω, the empirical average η ( ℓ ) = 12 ℓ + 1 X y ∈ Λ ℓ η ( y ) . P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN (EE) For local L ( ν ρ ) functions f , supported on { η ( x ) : x ∈ Λ ℓ } , suchthat ϕ f ( ρ ) = ϕ ′ f ( ρ ) = 0, and ℓ ≥ ℓ , there exist constants α > C = C ( ρ, ℓ , α ) where (cid:13)(cid:13)(cid:13)(cid:13) E ν ρ [ f | η ( ℓ ) , η cℓ ] − ϕ ′′ f ( ρ )2 (cid:20) ( η ( ℓ ) − ρ ) − σ ℓ ( ρ )2 ℓ + 1 (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν ρ ) ≤ C k f k L ( ν ρ ) ℓ α / . On the other hand, when only ϕ f ( ρ ) = 0 is known, k E ν ρ [ f | η ( ℓ ) , η cℓ ] − ϕ ′ f ( ρ )( η ( ℓ ) − ρ ) k L ( ν ρ ) ≤ C k f k L ( ν ρ ) ℓ / α / . We remark, a weaker version, where the L ( ν ρ ) norm, instead of the L ( ν ρ )norm of the difference, is say less than the same right-hand side expressionswith k f k L ( ν ρ ) in place of k f k L ( ν ρ ) would be sufficient for our purposes ifthere is a uniform bound on the inverse gap: sup k,ξ,n ℓ − W ( k, ℓ, ξ, n ) < ∞ .Usually, such estimates follow from a local central limit theorem. InProposition 5.1, we show, when ν ρ is a nondegenerate product measure,that (EE) holds with α = 1. In Proposition 5.2, with respect to a Marko-vian measure, we prove (EE) holds with α = 1 − ε for any fixed 0 < ε < Initial conditions.
We will start from initial measures { µ n } which havebounded relative entropy H ( µ n ; ν ρ ) with respect to ν ρ .(BE) Suppose { µ n } satisfiessup n H ( µ n ; ν ρ ) = sup n E ν ρ (cid:20) dµ n dν ρ log dµ n dν ρ (cid:21) < ∞ . In addition, we presume a diffusive initial limit starting from { µ n } .(CLT) Under initial measures { µ n } , we suppose Y n converges weakly toa spatial Gaussian process ¯ Y with covariance C ( G, H ) = lim n ↑∞ C nµ n ( G, H )for compactly supported functions
G, H .Of course, if µ n ≡ ν ρ , (BE) and (CLT) trivially hold with C ( G, H ) = C ν ρ ( G, H ). When ν ρ is a product measure, a possible way to get nontriv-ial examples of measures { µ n } satisfying (BE) and (CLT) is the following.For simplicity, we consider the case on which ν ρ is a Bernoulli product mea-sure on { , } Z . Let { κ nx : x ∈ Z } be a given bounded sequence and define µ n as the nonhomogeneous Bernoulli product measure satisfying µ n ( η ( x ) = 1) = ρ + κ nx √ n . A simple computation shows that H ( µ n ; ν ρ ) ≤ C ( k κ k ℓ ∞ ) n X x ∈ Z ( κ nx ) . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Therefore, taking κ nx = κ ( x/n ), where κ : R → R is bounded and in L ( R ), wesee that sup n H ( µ n ; ν ρ ) < ∞ , and (BE) is satisfied. On the other hand, sincethe measure µ n is product, a simple computation shows that, under { µ n } ,the process Y n converges in distribution to ¯ Y + κ , where ¯ Y is a white noisewith variance ρ (1 − ρ ). In [48], the Cole–Hopf solution of KPZ is consideredstarting from such initial conditions.One may relate probabilities of events A under µ n with those under ν ρ by an application of the entropy inequality: P µ n ( A ) ≤ log 2 + H ( µ n ; ν ρ )log(1 + P ν ρ ( A ) − ) . (2.6)For instance, let r ∈ L ( ν ρ ) be a local function. By the spatial mixing as-sumption (IM2), under ν ρ , we have the convergence in probability,lim n →∞ Z T n + 1 X x ∈ Λ n τ x r ( η ns ) ds = E ν ρ [ r ( η )] . (2.7)Then, by the entropy relation, also under { µ n } , the same limit also holds inprobability.Of course, given that we begin from nearly the invariant measure ν ρ , (2.7)is a trivial case of “hydrodynamics.” Formally, starting from more generalmeasures, the hydrodynamic equation for the limiting empirical density ρ = ρ ( x, t ) would read ∂ t ρ ( x, t ) + a ∇ ϕ b ( ρ ( x, t )) = 12 ∆ ϕ c ( ρ ( x, t )) . (2.8)In a sense, the main results of the paper are on the different fluctuationsfrom the law of large numbers (2.7) which arise for different regimes of thestrength asymmetry parameters a and γ .2.2. Results.
Denote by S ( R ) the standard Schwarz space of rapidly de-creasing functions equipped with the usual metric, and let S ′ ( R ) be its dual,namely the set of tempered distributions in R , endowed with the strongtopology. Denote the density fluctuation field acting on functions H ∈ S ( R )as Y nt ( H ) = 1 √ n X x ∈ Z H (cid:18) xn (cid:19) ( η nt ( x ) − ρ ) . Denote by D ([0 , T ] , S ′ ( R )) and C ([0 , T ] , S ′ ( R )) the spaces of right continuousfunctions with left limits and continuous functions respectively from [0 , T ]to S ′ ( R ).We now state a result from the literature which has been proved for someprocesses (cf. [23], Chapter 11 in [34] for zero-range processes with bounded P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN rate, [20, 50] for simple exclusion processes, and Section II.2.10 of [57] forexclusion systems with speed-change), sometimes from more general initialconditions, when the asymmetry is of order O ( n − ). Proposition 2.1.
For γ = 1 , starting from { µ n } , the sequence {Y nt ; n ≥ } converges in the uniform topology on D ([0 , T ] , S ′ ( R )) to the process Y t which solves the Ornstein–Uhlenbeck equation ∂ t Y t = 12 ϕ ′ c ( ρ )∆ Y t + a ϕ ′ b ( ρ ) ∇Y t + r ϕ b ( ρ ) ∇ ˙ W t , (2.9) where ˙ W t is a space–time white noise with unit variance, and Y = ¯ Y , thefield given in (CLT) . The Ornstein–Uhlenbeck equation (2.9) has a drift term coming from theweak asymmetry of the jump rates. The drift, as is well known, can beunderstood in terms of a characteristic velocity υ = ( a/ ϕ ′ b ( ρ ) from con-sidering the linearization of the hydrodynamic equation (2.8) (cf. ChapterII.2 of [57]). However, it can be removed from the limit field by observingthe density fluctuation field in the frame of an observer moving along theprocess characteristics. Define Y n,γt ( H ) = 1 √ n X x ∈ Z H (cid:18) xn − n (cid:26) aϕ ′ b n ( ρ ) tn n γ (cid:27)(cid:19) ( η nt ( x ) − ρ ) . If γ = 1, Proposition 2.1 is equivalent to the statement that Y n,γt convergesin the uniform topology on D ([0 , T ] , S ′ ( R )) to Y t , the unique solution of thedrift-removed Ornstein–Uhlenbeck equation ∂ t Y t = ϕ ′ c ( ρ )∆ Y t + q ϕ b ( ρ ) ∇ ˙ W t . (2.10)This equation of course corresponds to (2.9) with a = 0, is well posed andhas a unique solution (cf. [61]).Now we increase the strength of the asymmetry in the jump rates by de-creasing the value of γ . We show for 1 / < γ <
1, starting from the measures { µ n } , that there is no effect in the convergence result of the fluctuation field. Theorem 2.2 (Crossover fluctuations).
For / < γ < , starting frominitial measures { µ n } , the sequence {Y n,γt ; n ≥ } converges in the uniformtopology on D ([0 , T ] , S ′ ( R )) to the process Y t which is the solution of theOrnstein–Uhlenbeck equation (2.10) with initial condition Y = ¯ Y given in (CLT) . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS However, for γ = 1 /
2, which is a threshold, a much different qualitativelimit behavior is obtained as the strength of the weak asymmetry in thejump rates is big enough to influence the limit field. As mentioned in theIntroduction, the limit field Y t should satisfy, in some sense, a stochasticBurgers equation, written in our framework as ∂ t Y t = ϕ ′ c ( ρ )2 ∆ Y t + a ϕ ′′ b ( ρ ) ∇Y t + r ϕ b ( ρ ) ∇ ˙ W t , (2.11)although it is ill-posed.We now detail in what sense we mean to “solve” (2.11) in terms of a mar-tingale formulation. Let ι : R → [0 , ∞ ) be the function ι ( z ) = (1 / [ − , ( z ).Also, for 0 < ε ≤
1, define ι ε ( z ) = ε − ι ( ε − z ) and let G ε : R → [0 , ∞ ) be asmooth compactly supported function in S ( R ) which approximates ι ε : Thatis, k G ε k L ( R ) ≤ k ι ε k L ( R ) = ε − andlim ε ↓ ε − / k G ε − ι ε k L ( R ) = 0 . Such choices can be readily found by convoluting ι ε with smooth kernels.Also, for x ∈ R , define the shift τ x so that τ x G ε ( z ) = G ε ( x + z ).Consider now an S ′ ( R )-valued process {Y t ; t ∈ [0 , T ] } and for 0 ≤ s ≤ t ≤ T let A εs,t ( H ) = Z ts Z R ∇ H ( x )[ Y u ( τ − x G ε )] dx du. We say the process Y · satisfies the probability energy condition if for each H ∈ S ( R ), {A εs,t ( H ) } is Cauchy in probability as ε ↓ { G ε } . This limit defines the process {A s,t ; 0 ≤ s ≤ t ≤ T } given by A s,t ( H ) := lim ε ↓ A εs,t ( H ) , which is S ′ ( R ) valued (cf. pages 364–365; Theorem 6.15 of [61]).We will say that {Y t ; t ∈ [0 , T ] } is a probability energy solution of (2.11) ifthe following conditions hold:(i) Initially, Y is a spatial Gaussian process with covariance C ( G, H )for
G, H ∈ S ( R ).(ii) The process {Y t ; t ∈ [0 , T ] } satisfies the probability energy condition(2.12). P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN (iii) Then, the S ′ ( R ) valued process {M t : t ∈ [0 , T ] } where M t ( H ) := Y t ( H ) − Y ( H ) − ϕ ′ c ( ρ )2 Z t Y s (∆ H ) ds − aϕ ′′ b ( ρ )2 A ,t ( H )(2.13)is a continuous martingale with quadratic variation hM t ( H ) i = ϕ b ( ρ ) t k∇ H k L ( R ) . In particular, condition (iii) specifies by L´evy’s theorem that M t ( H ) is aBrownian motion with variance ( ϕ b ( ρ ) / t k∇ H k L ( R ) .We also define a stronger notion of solution to (2.11) which may be verifiedin some cases. We say that Y t satisfies the L energy condition if in (2.12),instead of in the probability sense, we assert {A εs,t ( H ) } is Cauchy in L withrespect to the underlying probability measure, and A s,t ( H ) is its L limit.Then we say Y t is an L energy solution of (2.11) if (i) holds as before, (ii)the L energy condition holds and (iii) holds with respect to the L limit A s,t ( H ). Theorem 2.3 (KPZ fluctuations).
For γ = 1 / , starting from initialmeasures { µ n } , the sequence of processes {Y n,γt : n ≥ n } is tight in the uni-form topology on D ([0 , T ] , S ′ ( R )) . Moreover, any limit point of Y n,γt is aprobability energy solution with respect to (2.11) with initial field ¯ Y givenin (CLT) .If the initial measure is µ n ≡ ν ρ , any limit point of Y n,γt is an L energysolution of (2.11) with initial field ¯ Y given in (CLT) . Remark 2.4.
We now make the following comments:1. Formally, equation (2.13) corresponds to the stochastic Burgers equa-tion (2.11) where the nonlinear term is represented by A ,t . We remark, asin [3], by taking a fast subsequence in ε , one may write A ,t as a function of {Y u : u ≤ t } , and form an equation in which Y t satisfies (2.11) a.s. on a typeof negative order Hermite Hilbert space.2. We also remark, as alluded to in the Introduction, if there were aunique probability or L energy solution, that is uniqueness of process in theassociated “martingale formulation,” since with respect to simple exclusionthe fluctuation field limit is known in terms of the “Cole–Hopf” solution ofthe KPZ equation [14], not only could one conclude a unique fluctuation fieldlimit in Theorem 2.3 in the framework of the particle systems considered,but also identify it in terms of the “Cole–Hopf” apparatus. What is requiredto show uniqueness of Y t is to determine uniquely its finite dimensionaldistributions (cf. Section 4.4 of [21]), which the nonlinearity of A ,t makesdifficult. TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS
3. We also note that the statement of Theorem 2.3 is nontrivial when a = 0 and b is such that ϕ ′′ b ( ρ ) = 0 . (2.14)Otherwise, when ϕ ′′ b ( ρ ) = 0, the limit field Y t satisfies the Ornstein–Uhlenbeckequation (2.10). Examples, fitting in our framework, where the second deriva-tive vanishes include types of zero-range, that is independent particle sys-tems where ϕ b ( ρ ) = 2 ρ which are in the EW class.2.3. Model : Zero-range processes. The one-dimensional weakly asym-metric zero-range process η nt , on the state space Ω := N Z , consists of a col-lection of random walks which interact in that the jump rate of a particle atvertex x only depends on the number of particles at x . More precisely, thegenerator is in form (2.1) where b R,nx ( η ) = g ( η ( x )) and b L,nx ( η ) = g ( η ( x + 1))do not depend on n and are fixed with respect to a function g : N → R + such that g (0) = 0, g ( k ) > k ≥ g is Lipschitz,(LIP) sup k ≥ | g ( k + 1) − g ( k ) | < ∞ .Under this specification, a Markov process η nt can be constructed (on asubset of Ω) [2]. Hence, (R1) holds and we identify the fixed function c n ≡ c as c ( η ) = g ( η (0)) . The zero-range process possesses a family of invariant measures which arefairly explicit product measures. For α ≥
0, define Z ( α ) := X k ≥ α k g ( k )! , where g ( k )! = g (1) · · · g ( k ) for k ≥ g (0)! = 1. Let α ∗ be the radius ofconvergence of this power series and notice that Z increases on [0 , α ∗ ). Fix0 ≤ α < α ∗ and let ¯ ν α be the product measure on N Z whose marginal at thesite x is given by¯ ν α { η : η ( x ) = k } = Z ( α ) α k g ( k )! , when k ≥ , Z ( α ) , when k = 0 . We now reparameterize these measures in terms of the “density.” Let ρ ( α ) := E ¯ ν α [ η (0)] = α Z ′ ( α ) / Z ( α ). By computing the derivative, we obtain that ρ ( α )is strictly increasing on [0 , α ∗ ). Then let α ( · ) denote its inverse. Define ν ρ ( · ) := ¯ ν α ( ρ ) ( · ) , P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN so that { ν ρ : 0 ≤ ρ < ρ ∗ } is a family of invariant measures parameterized bythe density. Here, ρ ∗ = lim α ↑ α ∗ ρ ( α ), which may be finite or infinite depend-ing on whether lim α → α ∗ Z ( α ) converges or diverges.Note, since ν ρ is a product measure, that ν λ ( z ) ρ = ν z for 0 ≤ z < ρ ∗ , andcondition (D) holds. One can readily check that (R2) holds: g ( η x +1 ,x ( x )) dν x +1 ,xρ dν ρ = g ( η ( x ) + 1) g ( η ( x ))! g ( η ( x + 1))! g ( η ( x ) + 1)! g ( η ( x + 1) − g ( η ( x + 1)) . Also, by the construction in [55], which extends the construction in [2] toan L ( ν ρ ) process, we have that L n is a Markov L ( ν ρ ) generator whose corecan be taken as the space of all local L ( ν ρ ) functions. Indeed, in [55], a coreof bounded Lipschitz functions is identified; however, since any local L ( ν ρ )function is a limit of bounded Lipschitz functions, and the formula (2.1) iswell defined and L ( ν ρ )-bounded for a local L ( ν ρ ) function, by dominatedconvergence the core can be extended. It follows that the measures { ν ρ : 0 ≤ ρ < ρ ∗ } are invariant for the zero-range process. Also, (IM) holds as ν ρ is a product measure whose marginal has some exponential moments. Inaddition, one can check that (EE) holds by Proposition 5.1.We now address the spectral gap properties of the system. Since the modelinteractions are range 0, the gap does not depend on the outside variables ξ . However, the gap depends on g , as it should since g controls the rate ofjumps. We identify three types of rates for which a spectral gap bound hasbeen proved. Let β = k/ (2 ℓ + 1) d . • If g is not too different from the independent case, for which the gap isof order O ( ℓ − ) uniform in k , one expects similar behavior as for a singleparticle. This has been proved for d ≥ x and ε > g ( x + x ) − g ( x ) ≥ ε for all x ≥ • If g is sublinear, that is C − x γ ≤ g ( x + 1) − g ( x ) ≤ Cx γ for 0 < γ < C >
0, then it has been shown that the spectral gap depends on thenumber of particles k , namely the gap for d ≥ O ((1 + β ) − γ ℓ − ) [45]. • If g ( x ) = 1( x ≥ d ≥ O ((1 + β ) − ℓ − ) [43]. In d = 1, this is true because of the connection between thezero-range and simple exclusion processes for which the gap estimate iswell known [47]: The number of spaces between consecutive particles insimple exclusion correspond to the number of particles in the zero-rangeprocess.In all these cases, (G) follows readily by straightforward moment calcula-tions. TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Model : Kinetically constrained exclusion systems. We consider atype of exclusion process, which may be thought of as a microscopic modelfor porous medium behavior, developed in [29] and references therein, inone dimension on Ω = { , } Z where particles more likely hop to unoccupiednearest-neighbor sites when at least m − ≥ m = 2, the rates are in the form b R,nx ( η ; θ ) = η ( x )(1 − η ( x + 1)) (cid:20) η ( x −
1) + η ( x + 2) + θ n (cid:21) ,b L,nx ( η ; θ ) = η ( x + 1)(1 − η ( x )) (cid:20) η ( x −
1) + η ( x + 2) + θ n (cid:21) , with respect to a parameter θ >
0. If θ would vanish, particles can jumpfrom site x to x + 1 exactly when there is at least 1 particle in the vicinityof the bond ( x, x + 1). However, with θ >
0, the jump from x to x + 1 mayalso occur irrespective of the neighboring particle structure with a small rate θ/ (2 n ).When m ≥
2, the rates generalize to b R,nx ( η ; θ ) = η ( x )(1 − η ( x + 1)) A n ( η ; θ ) ,b L,nx ( η ; θ ) = η ( x + 1)(1 − η ( x )) A n ( η ; θ ) , where A n ( η ; θ ) equals − Y j = − ( m − η ( x + j ) + Y j = − ( m − j =0 , η ( x + j ) + · · · + m − Y j = − j =0 , η ( x + j ) + m Y j =2 η ( x + j ) + θ n . The role of θ > θ = 0, there would bean infinite number of invariant measures, such as Dirac measures supportedon configurations which cannot evolve under the dynamics. The hydrody-namic limit for this model corresponds to the porous medium equation, ∂ t ρ t ( t, u ) = ∆ ρ m ( t, u ).Now, one may calculate that b R,nx ( η ; θ ) − b L,nx ( η ; θ ) = c nx ( η ) − c nx +1 ( η ) where,for m ≥ c n ( η ; θ ) = Y j = − ( m − η ( j ) + · · · + m − Y j =0 η ( j ) − Y j = − ( m − j =0 η ( j ) − · · · − m − Y j = − j =0 η ( j ) + θ n η (0) . In the case m = 2, the last formula reduces to c n ( η ; θ ) = η ( − η (0)+ η (0) η (1) − η ( − η (1) + θ n η (0). P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
Of course, uniformly in η , as n ↑ ∞ , the terms involving θ vanish, b R,nx ( η ; θ ) → b Rx ( η ) := b R, x ( η ; 0) , b L,nx ( η ; θ ) → b Lx ( η ) := b L, x ( η ; 0) and c n ( η ; θ ) → c := c ( η ; 0) . Consider now the Bernoulli product measure on Ω: ν ρ = Y x ∈ Z µ ρ where µ ρ (1) = 1 − µ ρ (0) = ρ for ρ ∈ [0 , L n is aMarkov L ( ν ρ ) generator. One may also inspect that condition (R2) holdswith respect to ν ρ . Hence, ν ρ is invariant for ρ ∈ [0 , ν ρ supports two-state configurations. In addition, as ν ρ is a productmeasure, ν λ ( z ) ρ = ν z and (D) holds. Also, by Proposition 5.1, (EE) is satisfied.We now discuss the spectral gap behavior of the process. Proposition 2.5.
For kinetically constrained exclusion processes evolv-ing on Λ ℓ , when m ≥ , there exists a constant C , uniform over ξ and n ,such that W ( k, ℓ, ξ, n ) ≤ Cℓ (cid:18) ℓk (cid:19) m k ≥ . When m = 2 and k ≤ ℓ/
3, the above spectral gap estimate is alreadygiven in Proposition 6.2 of [29]. However, a straightforward modificationof the proof of Proposition 6.2 in [29] yields the more general estimate inProposition 2.5. Indeed, the difference when m ≥ Cj m − ways to arrange m − j . Now, a similaroptimization on j as given in the proof of Proposition 6.2 of [29] leads tothe desired generalized spectral gap estimate. Lemma 2.6.
For the kinetically constrained exclusion model, the spectralgap condition (G) is satisfied.
Proof.
With respect to a constant C , which may change line to line, E ν ρ (cid:20)(cid:18) W (cid:18) X x ∈ Λ ℓ η ( x ) , ℓ, ξ, n (cid:19)(cid:19) (cid:21) ≤ Cℓ E ν ρ (cid:20) (cid:18) ℓ + 1 ≤ η ( ℓ ) (cid:19) ( η ℓ ) − m (cid:21) ≤ Cℓ (cid:26) ε − m + E ν ρ (cid:20) (cid:18) ℓ + 1 ≤ η ( ℓ ) < ε (cid:19) ( η ( ℓ ) ) − m (cid:21)(cid:27) ≤ Cℓ { ε − m + ℓ m P ν ρ ( η ( ℓ ) < ε ) } TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS for a fixed ε < ρ . Then, as ν ρ is a Bernoulli product measure with density ρ , by a large deviations estimate say, E ν ρ [ W ( P x ∈ Λ ℓ η ( x ) , ℓ, ξ, n ) ] ≤ Cℓ forall ℓ ≥ (cid:3) Model 3: Gradient exclusion with speed change.
In this version ofexclusion on Ω = { , } Z , rates are chosen which correspond to a Hamiltonianwith nearest-neighbor interactions, Q β ( η ) = − β X x ∈ Z ( η ( x ) − / η ( x + 1) − / β ∈ R , which will be reversible with respect to a stationary Markovianmeasure ν / . That is, specify ν / by its finite-dimensional distributions ν / ( η ( x ) = e ( x ) : x ∈ Λ ℓ | η ( y ) = ξ ( y ) for y / ∈ Λ ℓ ) = e − Q β,ℓ ( e,ξ ) Z , where Q β,ℓ ( e, ξ ) = − β X x,x +1 ∈ Λ ℓ ( e ( x ) − / e ( x + 1) − / − β ( ξ ( − ℓ − − / e ( − ℓ ) − / − β ( e ( ℓ ) − / ξ ( ℓ + 1) − / ,e, ξ ∈ Ω and Z = Z ( ℓ, ξ ) is the normalization. It is not difficult to see that ν / is Markovian with transition matrix P = 1 e β/ + e − β/ (cid:20) e β/ e − β/ e − β/ e β/ (cid:21) , and marginal distribution h / , / i so that E ν / [ η (0)] = 1 / b R,nx = b Rx and b L,nx = b Lx which do not depend on n as b Rx ( η ) = η ( x )(1 − η ( x + 1)) × [ α η ( x − η ( x + 2) + α (1 − η ( x − η ( x + 2)+ α η ( x − − η ( x + 2)) + α (1 − η ( x − − η ( x + 2))] ,b Lx ( η ) = η ( x + 1)(1 − η ( x )) × [ α η ( x − η ( x + 2) + α (1 − η ( x − η ( x + 2)+ α η ( x − − η ( x + 2)) + α (1 − η ( x − − η ( x + 2))] , where α , α = e β α , α >
0. The condition (R1) also follows if we also as-sume that α − α − α + α = 0 so that, as can be checked, c ( η ) takes the P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN form c ( η ) = α η (0) + ( α − α ) η ( − η (0) + ( α − α ) η (0) η (1)+ ( α − α ) η ( − η (1) + ( α − α ) η ( − η (0) η (1) . Again, by [40], L n is a Markov L ( ν ρ ) generator for the process. We notewhen β = 0 and α i = 1 for i = 1 , , ,
4, the model is the simple exclusionprocess and ν / is the Bernoulli product measure with density 1 / λ . Define ν λ / , again specified by its finite-dimensional distributions, through the relation dν λ / dν / ( η ( x ) = e ( x ) : x ∈ Λ ℓ | η ( y ) = ξ ( y ) for y / ∈ Λ ℓ ) = e λ P x ∈ Λ ℓ ( e ( x ) − / Z ′ , where e, ξ ∈ Ω and Z ′ = Z ′ ( ℓ, ξ ) is another normalization. These measuresare also Markovian with transition matrix P λ = (cid:20) − u u v − v (cid:21) , (2.15)where u = r ( λ, β ) + sinh( λ/ λ/
2) + r ( λ, β ) , v = r ( λ, β ) − sinh( λ/ λ/
2) + r ( λ, β )and r ( λ, β ) = q sinh ( λ/
2) + e − β . The stationary distribution equals π λ =( v + u ) − h v , u i , which is the marginal distribution of ν λ / . These deriva-tions are performed in [12] and [62].The measures { ν λ / : λ ∈ R } are uniformly mixing: Indeed, the eigenvaluesof P λ are 1 and 1 − u − v , and the spectral gap u + v is uniformly boundedaway from 0 for λ ∈ R .One can calculate E ν λ / [ η (0)] = u / ( u + v ) strictly increases in λ . Toparameterize in terms of “density,” recall ν λ ( z )1 / = ν z where λ = λ ( z ) is chosenso that E ν z [ η (0)] = z . Here, as z ↓ ρ ∗ , λ ( z ) ↓ −∞ and, as z ↑ ρ ∗ , λ ( z ) ↑ ∞ ; also λ (1 /
2) = 0. Hence, since also ν z is exponentially mixing, both(IM) and (D) hold.From the defining relation for λ ( z ), u / ( u + v ) = z , one can differentiateat z = 1 / λ ′ (1 / e − β/ /
4] = 1.Also, we note the additive functional variance σ ( z ) [cf. (IM2)] satisfiesthe formula σ ( z ) = E π λ ( z ) [ u ] − E π λ ( z ) [( P λ ( z ) u ) ] where ( I − P λ ( z ) ) u = f and f = h− z, − z i represents the values of the function f ( η ) = η (0) − z ; see Sec-tion 6.5 of [60]. In fact, we find σ (1 /
2) = e − β/ / λ ′ (1 / σ (1 /
2) = 1.
TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS The spectral gap for a more general process, including this one, has beenbounded as follows [41]: Uniformly over k and ξ (it does not depend on n ),we have W ( k, ℓ, ξ, n ) ≤ Cℓ . Hence, (G) holds.Also, in Proposition 5.2, we show that (EE) holds.
3. Proofs-outline.
The strategies of the proofs for Theorems 2.2 and 2.3are similar. We consider the stochastic differential of Y n,γt and represent it interms of corrector and martingale terms. Tightness is shown for each term inthe decomposition of Y n,γt . Under the assumption that the initial measure isthe invariant state ν ρ , limit points are identified using a Boltzmann–Gibbsprinciple, and shown to satisfy (2.10) when 1 / < γ ≤ γ = 1 /
2. When the initial measures { µ n } satisfy(BE), the entropy inequality then allows to characterize the limit points asdesired.In the following Sections 3.1–3.3, associated martingales, Boltzmann–Gibbs principles and tightness are discussed. In Section 3.4, limit pointsare identified and Theorems 2.2 and 2.3 are proved.To reduce some of the notation, we will drop the superscript “ n ” in therate functions and write b R,nx = b Rx , b L,nx = b Lx , b nx = b x , b n = b , c nx = c x and c n = c until Section 3.4.3.1. Associated martingales.
For H ∈ S ( R ), x ∈ Z and n ≥
1, define∆ nx H = n (cid:26) H (cid:18) x + 1 n (cid:19) + H (cid:18) x − n (cid:19) − H (cid:18) xn (cid:19)(cid:27) , ∇ nx H = n (cid:26) H (cid:18) x + 1 n (cid:19) − H (cid:18) xn (cid:19)(cid:27) . Define also, for γ, s ≥
0, the functions H γ,s ( · ) = H (cid:18) · − n (cid:22) aϕ ′ b ( ρ ) sn n γ (cid:23)(cid:19) and(3.1) e H γ,s ( · ) = H (cid:18) · − n (cid:26) aϕ ′ b ( ρ ) sn n γ (cid:27)(cid:19) . We note, in H γ,s , the process characteristic shift is along n − Z , which helpsmake tidy some proofs [in applying a Boltzmann–Gibbs principle (Theo-rem 3.2) in proofs of Propositions 3.3 and 3.5], instead of along R as in e H γ,s . P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
Let F ( s, η ns ; H, n ) = Y n,γs ( H ), and F ( η ; H, n ) = n − / P x ∈ Z H ( xn )( η ( x ) − ρ ). Although F ( η ; H, n ) is an L ( ν ρ ) function, in general, it is not a localfunction. However, by approximation with local functions and noting bycondition (R1) that | b ( η ) | ≤ C P | x |≤ R η ( x ), one may conclude F ( η ; H, n )and also F ( η ; H, n ) belong to the domain of L n . In particular, L n F ( s, η ns ; H, n ) = 12 √ n X x ∈ Z c x ( η ns )∆ nx e H γ,s + a n γ − / X x ∈ Z b x ( η ns ) ∇ nx e H γ,s . Also, ∂∂ s F ( s, η ns ; H, n ) = (cid:26) − aϕ ′ b ( ρ ) n n γ (cid:27) n / X x ∈ Z ∇ e H γ,s (cid:18) xn (cid:19) ( η ns ( x ) − ρ ) . Then M n,γt ( H ) := F ( t, η nt ; H, n ) − F (0 , η n ; H, n ) − Z t ∂∂ s F ( s, η ns ; H, n ) + L n F ( s, η ns ; H, n ) ds is a martingale. We may decompose M n,γt ( H ) = Y n,γt ( H ) − Y n,γ ( H ) − I n,γt ( H ) − B n,γt ( H ) − K n,γt ( H ) , (3.2)where I n,γt ( H ) = 12 Z t √ n X x ∈ Z ( c x ( η ns ) − ϕ c ( ρ ))∆ nx H γ,s ds, B n,γt ( H ) = a n γ − / Z t X x ∈ Z ( b x ( η ns ) − ϕ b ( ρ ) − ϕ ′ b ( ρ )( η ns ( x ) − ρ )) ∇ nx H γ,s ds, K n,γt ( H )= Z t (cid:20) √ n X x ∈ Z κ n, x ( H, s )( c x ( η ns ) − ϕ c ( ρ ))+ a n γ − / X x ∈ Z κ n, x ( H, s )( b x ( η ns ) − ϕ b ( ρ ) − ϕ ′ b ( ρ )( η ns ( x ) − ρ )) (cid:21) ds. Here, we introduced the centering constants ϕ c ( ρ ) and ϕ b ( ρ ) in I n,γt and B n,γt as ∆ nx H γ,s and ∇ nx H γ,s both sum to zero. Also, κ n, x ( H, s ) = ∆ nx ( e H γ,s − H γ,s ) = O ( n − ) · ∆ nx H ′ γ,s + O ( n − ) · H (4) γ,s ( x ′ /n ) ,κ n, x ( H, s ) = ∇ nx ( e H γ,s − H γ,s )= O ( n − ) · ∆ H γ,s ( x/n ) + O ( n − ) · H ′′′ γ,s ( x ′′ /n ) , TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS where | x ′ − x | , | x ′′ − x | ≤ hM n,γt i , we compute L n F ( s, η ns ; H, n ) − F ( s, η ns ; H, n ) L n F ( s, η ns ; H, n )= 12 n X x ∈ Z b x ( η ns )( ∇ nx e H γ,s ) + a n γ X x ∈ Z ( c x ( η ns ) − c x +1 ( η ns ))( ∇ nx e H γ,s ) so that ( M n,γt ( H )) − hM n,γt ( H ) i is a martingale with hM n,γt ( H ) i = Z t n X x ∈ Z ( ∇ nx e H γ,s ) b x ( η ns ) ds + Z t a n γ X x ∈ Z ( c x ( η ns ) − c x +1 ( η ns ))( ∇ nx e H γ,s ) ds. When starting from the invariant measure ν ρ , noting the bounds in (R1),we have E ν ρ [( M n,γt ( H ) − M n,γs ( H )) ] ≤ (cid:26)Z ts (cid:18) n X x ∈ Z ( ∇ nx e H γ,s ) (cid:19) ds (cid:27) (3.3) × (cid:20) E ν ρ [ b ( η )] + a n γ E ν ρ [ | c ( η ) − c ( η ) | ] (cid:21) ≤ C ( a ) k b k L ( ν ρ ) Z ts (cid:18) n X x ∈ Z ( ∇ nx e H γ,s ) (cid:19) ds. To express an exponential martingale, we now observe for 0 ≤ λ ≤ λ ( H, n )small that exp { λF ( η ; H, n ) } is in the domain of L n . Indeed, if H is a localfunction, as ν ρ is assumed in (IM) to have small parameter exponentialmoments, then exp { λF ( η ; H, n ) } ∈ L ( ν ρ ) for all small λ . Again, an approx-imation argument when H ∈ S ( R ) is not local shows also exp { λF ( η ; H, n ) } belongs to the domain of L n . We calculateexp {− λF ( u, η nu ; H, n ) } (cid:18) ∂∂ u + L n (cid:19) exp { λF ( u, η nu ; H, n ) } = n X x ∈ Z [ b Rx ( η ) p n (exp { λn − / ( ∇ nx e H γ,u ) } − b Lx ( η ) q n (exp {− λn − / ( ∇ nx e H γ,u ) } − − n / (cid:26) aλϕ ′ b ( ρ ) n n γ (cid:27) X x ∈ Z ∇ e H γ,u ( x/n )( η nu ( x ) − ρ ) , P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN which, given the assumptions on b in (R1) and on moments of ν ρ in (IM),belongs to L ( ν ρ ).Hence, by the proof of Lemma IV.3.2 of [21], Z s,t = exp (cid:26) λF ( t, η nt ) − λF ( s, η ns ) − Z ts e − λF ( u,η nu ) (cid:18) ∂∂ u + L n (cid:19) e λF ( u,η nu ) du (cid:27) is a martingale. We may expand Z s,t in terms of λ as Z s,t = exp (cid:26) λ ( M n,γt ( H ) − M n,γs ( H )) − λ hM n,γt ( H ) − M n,γs ( H ) i + λ Z ts R du + λ Z ts R du + λ Z ts R du (cid:27) , where R ( u ) = n n / X x ∈ Z ( b Rx ( η ) − b Lx ( η ))( ∇ nx e H γ,u ) + an n / / γ ) X x ∈ Z b x ( η )( ∇ nx e H γ,u ) , R ( u ) = n n X x ∈ Z b x ( η )( ∇ nx e H γ,u ) + an n / γ ) X x ∈ Z ( b Rx ( η ) − b Lx ( η ))( ∇ nx e H γ,u ) . By the gradient condition and the bound on b in assumption (R), one maycompute for i = 1 , kR i ( u ) k L ( ν ρ ) ≤ C ( a ) n / k b ( η ) k L ( ν ρ ) (cid:18) n X x |∇ nx e H γ,u | i (cid:19) . (3.4)Since E ν ρ [ Z s,t ] = 1, by expanding in powers of λ , using Schwarz inequality,the bound on the quadratic variation (3.3), bounds on R i (3.4) and invari-ance of ν ρ , we obtain a bound for the fourth moment of M n,γt ( H ) − M n,γs ( H ): E ν ρ [( M n,γt ( H ) − M n,γs ( H )) ](3.5) ≤ C ( a, H ) k b k L ( ν ρ )( | t − s | + n − / | t − s | ) . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Generalized Boltzmann–Gibbs principles.
To treat the stochastic dif-ferential of Y n,γt , we replace the spatial terms of form P x ∈ Z h ( x ) τ x f ( η ),where h is a function on Z and f is a local function, in terms of the fluctua-tion field itself to close the evolution equations. Such replacements fall underthe term “Boltzmann–Gibbs principles” coined by Brox–Rost in [18] whichhave general validity. For instance, the following result forms the backbone ofthe argument for Proposition 2.1, when starting from the invariant measure ν ρ , with respect to the papers cited just before the proposition statement. Proposition 3.1.
Let f be a local L ( ν ρ ) function. For t ≥ and h ∈ ℓ ( Z ) , we have lim n →∞ E ν ρ (cid:20)(cid:18)Z t √ n X x ∈ Z ( τ x f ( η ns ) − ϕ f ( ρ ) − ϕ ′ f ( ρ )( η ns ( x ) − ρ )) h ( x ) ds (cid:19) (cid:21) = 0 . We now state a main result of this paper which provides a sharper esti-mate, perhaps of independent interest, when starting from ν ρ . To simplifyexpressions, we will use the notation( η ns ) ( ℓ ) ( x ) := 12 ℓ + 1 X y ∈ Λ ℓ η ns ( x + y ) . Theorem 3.2 ( L generalized Boltzmann–Gibbs principle). Let f be alocal L ( ν ρ ) function supported on sites Λ ℓ such that ϕ f ( ρ ) = ϕ ′ f ( ρ ) = 0 .There exists a constant C = C ( ρ, ℓ ) such that, for t ≥ , ℓ ≥ ℓ and h ∈ ℓ ( Z ) ∩ ℓ ( Z ) , E ν ρ (cid:20)(cid:18)Z t X x ∈ Z (cid:18) τ x f ( η ns ) − ϕ ′′ f ( ρ )2 (cid:26) (( η ns ) ( ℓ ) ( x ) − ρ ) − σ ℓ ( ρ )2 ℓ + 1 (cid:27)(cid:19) h ( x ) ds (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) (cid:18) tℓn (cid:18) n X x ∈ Z h ( x ) (cid:19) + t n ℓ α (cid:18) n X x ∈ Z | h ( x ) | (cid:19) (cid:19) . On the other hand, when only ϕ f ( ρ ) = 0 is known, E ν ρ (cid:20)(cid:18)Z t X x ∈ Z ( τ x f ( η ns ) − ϕ ′ f ( ρ ) { ( η ns ) ( ℓ ) ( x ) − ρ } ) h ( x ) ds (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) (cid:18) tℓ n (cid:18) n X x ∈ Z h ( x ) (cid:19) + t n ℓ α (cid:18) n X x ∈ Z | h ( x ) | (cid:19) (cid:19) . Here, α > is the power in assumption (EE) . P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
The proof of Theorem 3.2 is given in Section 4. We note, if the uniformspectral gap holds, sup k,ξ,n ℓ − W ( k, ℓ, ξ, n ) < ∞ , then the argument showsone can replace in the right-hand sides above k f k L ( ν ρ ) with k f k L ( ν ρ ) .3.3. Tightness.
We prove tightness of the fluctuation fields, first startingfrom the invariant measure ν ρ , using the L generalized Boltzmann–Gibbsprinciple. Then by the relative entropy bound (2.6), we deduce tightnesswhen beginning from initial measures { µ n } . Proposition 3.3.
The sequences {Y n,γt : t ∈ [0 , T ] } n ≥ , {M n,γt : t ∈ [0 , T ] } n ≥ , {I n,γt : t ∈ [0 , T ] } n ≥ , {B n,γt : t ∈ [0 , T ] } n ≥ , {K n,γt : t ∈ [0 , T ] } and {hM n,γt i : t ∈ [0 , T ] } n ≥ , when starting from the invariant measure ν ρ , aretight in the uniform topology on D ([0 , T ] , S ′ ( R )) . Proof.
By Mitoma’s criterion [42], to prove tightness of the sequenceswith respect to the uniform topology on D ([0 , T ] , S ′ ( R )), it is enough to showtightness of {Y n,γt ( H ); t ∈ [0 , T ] } n ≥ , {M n,γt ( H ) : t ∈ [0 , T ] } n ≥ , {I n,γt ( H ) : t ∈ [0 , T ] } n ≥ , {B n,γt ( H ) : t ∈ [0 , T ] } n ≥ , {K n,γt ( H ) : t ∈ [0 , T ] } and {hM n,γt ( H ) i : t ∈ [0 , T ] } n ≥ , with respect to the uniform topology for all H ∈ S ( R ). Notethat all initial values vanish, except Y n,γ ( H ).Tightness of Y n,γt ( H ), in view of the decomposition Y n,γt ( H ) = Y n,γ ( H ) + I n,γt ( H ) + B n,γt ( H ) + K n,γt ( H ) + M n,γt ( H ), will follow from tightness of eachterm. The tightness of Y n,γ ( H ), given that we begin under ν ρ , follows fromassumption (IM).For the martingale term, we use Doob’s inequality and stationarity toobtain P ν ρ (cid:16) sup | t − s |≤ δ ≤ s,t ≤ T |M n,γt ( H ) − M n,γs ( H ) | > ε (cid:17) ≤ ε − E ν ρ h sup | t − s |≤ δ ≤ s,t ≤ T |M n,γt ( H ) − M n,γs ( H ) | i ≤ Cε − δ − E ν ρ [( M n,γδ ( H )) ] . Now, by the fourth moment estimate (3.5), we have δ − E ν ρ [( M nδ ( H )) ] ≤ C k b k L ( ν ρ ) ( δ + n − / ) , which vanishes as n ↑ ∞ and then δ ↓
0. This is enough to conclude that {M n,γt ( H ) : t ∈ [0 , T ] } n ≥ is tight in the uniform topology.We now prove tightness for B n,γt ( H ) through the Kolmogorov–Centsov cri-terion. The argument for I n,γt ( H ) is similar. Also, the proofs for hM n,γt ( H ) i TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS and K n,γt ( H ), given their forms, are simpler and can be done using invari-ance of ν ρ by squaring all terms. We focus on the case γ = 1 /
2, given thatthe estimates are analogous and simpler when 1 / < γ ≤
1. Let V b ( η ) = b ( η ) − ϕ b ( ρ ) − ϕ ′ b ( ρ )( η (0) − ρ ) . By assumption (R1), V b has range R . Also, by its form, ϕ V b ( ρ ) = ϕ ′ V b ( ρ ) = 0and also ϕ ′′ V b ( ρ ) = ϕ ′′ b ( ρ ).Then B n,γt ( H ) = a Z t X x ∈ Z ( ∇ nx H γ,s ) τ x V b ( η s ) ds. By invoking Theorem 3.2 and translation-invariance of ν ρ which allows toreplace ∇ nx H γ,s with ∇ nx H (which does not depend on time s ), for ℓ ≥ ℓ = R , with respect to a constant C = C ( a, ρ, R ), we have E ν ρ (cid:20)(cid:18) B n,γt ( H ) − a Z t X x ∈ Z ( ∇ nx H γ,s ) ϕ ′′ b ( ρ ) (cid:26) (( η ns ) ( ℓ ) ( x ) − ρ ) − σ ℓ ( ρ )2 ℓ + 1 (cid:27) ds (cid:19) (cid:21) (3.6) ≤ C k b k L ( ν ρ ) (cid:26) tℓn + t n ℓ α (cid:27)(cid:20)(cid:18) n X x ∈ Z ( ∇ nx H ) (cid:19) + (cid:18) n X x ∈ Z |∇ nx H | (cid:19) (cid:21) . On the other hand, given sup ℓ ≥ R E ν ρ [( √ ℓ ( η ℓ − ρ )) ] < ∞ by assumption(IM) and | ϕ ′′ b ( ρ ) | ≤ C k b k L ( ν ρ ) by assumption (D), and the Schwarz inequal-ity ( P x h ( x ) r ( x )) ≤ ( P x | h ( x ) | ) P x | h ( x ) | r ( x ), we have for ℓ > R that E ν ρ (cid:20)(cid:18)Z t X x ∈ Z ( ∇ nx H γ,s ) ϕ ′′ b ( ρ )2 (cid:26) (( η ns ) ( ℓ ) ( x ) − ρ ) − σ ℓ ( ρ )2 ℓ + 1 (cid:27) ds (cid:19) (cid:21) ≤ C ( ρ ) k b k L ( ν ρ ) t n ℓ (cid:18) n X x ∈ Z |∇ nx H | (cid:19) . Hence, for ℓ > R , we have E ν ρ [( B n,γt ( H )) ] ≤ C ( a, ρ, R, H ) k b k L ( ν ρ ) [ tℓ/n + t n /ℓ ], noting the domination n /ℓ α ≤ n /ℓ . Then, if ℓ is taken as ℓ = t / n > R , we conclude E ν ρ [( B n,γt ( H )) ] ≤ C ( a, ρ, R, H ) k b k L ( ν ρ ) t / .However, when t / n ≤ R , we have by the same Schwarz bound that E ν ρ [( B n,γt ( H )) ] ≤ C ( ρ, a ) k b k L ( ν ρ ) t n (cid:18) n X x |∇ nx H | (cid:19) ≤ C ( ρ, a, H, R ) k b k L ( ν ρ ) t / . P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
This shows tightness of B n,γt ( H ).Combining these estimates, we conclude the proof of the proposition. (cid:3) We now update to when the process begins from the measures { µ n } . Proposition 3.4.
The fluctuation field sequences {Y n,γt : t ∈ [0 , T ] } n ≥ , {M n,γt : t ∈ [0 , T ] } n ≥ , {I n,γt : t ∈ [0 , T ] } n ≥ , {B n,γt : t ∈ [0 , T ] } n ≥ , {K n,γt : t ∈ [0 , T ] } n ≥ and {hM n,γt i : t ∈ [0 , T ] } n ≥ are tight in the uniform topology on D ([0 , T ] , S ′ ( R )) when starting from { µ n } satisfying assumption (BE) . Proof.
As before, all initial values vanish except Y n,γ which, however, istight by (CLT). Next, by Proposition 3.3, we have lim δ ↓ lim n ↑∞ P ν ρ ( O nδ,ε ) =0 where O nδ,ε = n sup | t − s |≤ δs,t ∈ [0 ,T ] k X nt − X ns k > ε o , and X nt may be equal to Y n,γt , M n,γt , I n,γt , B n,γt , K n,γt or hM n,γt i . Then wehave by the entropy inequality (2.6) that also lim δ ↓ lim n ↑∞ P µ n ( O δ,ε ) = 0which allows to conclude. (cid:3) Identification of limit points: Proofs of Theorems 2.2 and 2.3.
Withtightness (Proposition 3.4) in hand, we now identify the limit points of {Y n,γt : t ∈ [0 , T ] } n ≥ and its parts in decomposition (3.2). Let Q n be thedistribution of( Y n,γt , M n,γt , I n,γt , B n,γt , K n,γt , hM n,γt i : t ∈ [0 , T ]) , and let n ′ be a subsequence where Q n ′ converges to a limit point Q . Let also Y t , M t , I t , B t , K t and D t be the respective limits in distribution of the com-ponents. Since tightness is shown in the uniform topology on D ([0 , T ] , S ′ ( R )),we have that Y t , M t , I t , B t , K t and D t have a.s. continuous paths.Let now G ε : R → [0 , ∞ ) be a smooth compactly supported function for0 < ε ≤ ι ε ( z ) = ε − [ − , ( zε − ) as in the definition ofenergy solution before Theorem 2.3. That is, k G ε k L ( R ) ≤ k ι ε k L ( R ) = ε − and lim ε ↓ ε − / k G ε − ι ε k L ( R ) = 0. Define A n,γ,εs,t ( H ) := Z ts n X x ∈ Z ( ∇ nx H )[ τ x Y n,γu ( G ε )] du. Since for fixed 0 < ε ≤ π · R ts du R dx ( ∇ H ( x )) { π u ( τ − x G ε ) } iscontinuous in the uniform topology on D ([0 , T ]; S ′ ( R )), we have subsequen-tially in distribution thatlim n ′ ↑∞ A n ′ ,γ,εs,t ( H ) = Z ts du Z dx ( ∇ H ( x )) {Y u ( τ − x G ε ) } =: A εs,t ( H ) . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Proposition 3.5.
Suppose the initial distribution is the invariant mea-sure ν ρ and t ∈ [0 , T ] .When γ = 1 / , there is a constant C = C ( a, ρ, R ) such that lim n ↑∞ E ν ρ (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) B n,γt ( H ) − aϕ ′′ b ( ρ )4 A n,γ,ε ,t ( H ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) ≤ Ct ( ε + ε − k G ε − ι ε k L ( R ) ) k b k L ( ν ρ ) [ k∇ H k L ( R ) + k∇ H k L ( R ) ] . Then, in L ( P ν ρ ) , A ε ,t ( H ) is a Cauchy ε -sequence. Hence, aϕ ′′ b ( ρ )4 A ,t ( H ) := lim ε ↓ aϕ ′′ b ( ρ )4 A ε ,t ( H ) = B t ( H ) . In particular, we conclude A s,t ( H ) d = A ,t − s ( H ) does not depend on the spe-cific smoothing family { G ε } . Moreover, when / < γ ≤ , we have B t ( H ) =0 . In addition, when / ≤ γ ≤ , lim n ↑∞ E ν ρ (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) I n,γt ( H ) − ϕ ′ c ( ρ )2 Z t Y n,γs (∆ H ) ds (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) = 0 , lim n ↑∞ E ν ρ (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) hM n,γt ( H ) i − ϕ b ( ρ )2 t k∇ H k L ( R ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) = 0 , lim n ↑∞ E ν ρ [ |K n,γt ( H ) | ] = 0 . Then, in L ( P ν ρ ) , K t ( H ) = 0 and I t ( H ) = ϕ ′ c ( ρ )2 Z t Y s (∆ H ) ds and D t ( H ) = ϕ b ( ρ )2 t k∇ H k L ( R ) . Proof.
We prove the limit display for B t ( H ) when γ = 1 / E ν ρ [ |B t ( H ) − ( aϕ ′′ b ( ρ ) / A ε ,t ( H ) | ] ≤ C ( a, ρ, R, H ) × t [ ε + ε − k G ε − ι ε k L ( R ) ]. Therefore, A ε ,t ( H ), as a sequence in ε , is Cauchy in L ( P ν ρ ). The arguments for I t ( H ), D t ( H ), and K t ( H ), noting their forms,are similar; for D t ( H ) and K t ( H ), one might also use spatial mixing assumedin (IM). To simplify notation, we will call n = n ′ .Note, for ℓ = εn , that X x ∈ Z ( ∇ nx H γ,s )(( η ns ) ( ℓ ) ( x ) − ρ ) = X x ∈ Z ( ∇ nx H γ,s ) (cid:18) nε + 1 X | z |≤ nε ( η ns ( z + x ) − ρ ) (cid:19) P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN = 1 + O ( n − ) n X x ∈ Z ( ∇ nx H )[ τ x Y n,γs ( ι ε )] . Here, the shift by n − ⌊ aϕ ′ b ( ρ ) sn / (2 n γ ) ⌋ in ∇ nx H γ,s [cf. (3.1)] was transferredto τ x Y n,γs ( ι ε ).Then, with ℓ = εn , by Theorem 3.2, as in the bound (3.6), we havelim n ↑∞ E ν ρ (cid:20)(cid:18) B n,γt ( H ) − aϕ ′′ b n ( ρ )4 Z t n X x ∈ Z ( ∇ nx H ) τ x Y n,γs ( ι ε ) ds (cid:19) (cid:21) = lim n ↑∞ E ν ρ (cid:20)(cid:18) B n,γt ( H ) − aϕ ′′ b n ( ρ )4 Z t n X x ∈ Z ( ∇ nx H ) τ x (cid:26) Y n,γs ( ι ε ) − σ ℓ ( ρ )2 ε (cid:27) ds (cid:19) (cid:21) ≤ lim n ↑∞ C ( a, ρ, R ) k b n k L ( ν ρ ) t (cid:18) ε + 1 ε α n α (cid:19) × (cid:20)(cid:18) n X x ∈ Z ( ∇ nx H ) (cid:19) + (cid:18) n X x ∈ Z |∇ nx H | (cid:19) (cid:21) . Here, as the sum of ∇ nx H γ,s on x vanishes, we introduced the centeringconstant (2 ε ) − σ ℓ ( ρ ) in the second line.Now, Y n,γs ( ι ε ) − Y n,γx ( G ε ) = [ Y n,γs ( ι ε ) − Y n,γs ( G ε )][ Y n,γs ( ι ε ) + Y n,γs ( G ε )]and by (IM2) C ν ρ ( ι ε − G ε , ι ε − G ε ) / C ν ρ ( ι ε + G ε , ι ε + G ε ) / ≤ C ( ρ ) ε − / k G ε − ι ε k L ( R ) . Hence, by Schwarz inequality,lim n ↑∞ E ν ρ (cid:20)(cid:18)Z t n X x ∈ Z ( ∇ nx H ) τ x Y n,γs ( ι ε ) ds − A n,γ,ε ,t ( H ) (cid:19) (cid:21) ≤ C ( ρ ) ε − k G ε − ι ε k L ( R ) t (cid:18) n X x ∈ Z |∇ nx H | (cid:19) . Finally, combining these estimates with the inequality ( a + b ) ≤ a + 2 b ,and by assumption (D) that lim n ↑∞ ϕ ′′ b n ( ρ ) = ϕ ′′ b ( ρ ), we complete the proof. (cid:3) Proposition 3.6.
Suppose the initial measures { µ n } satisfy assumption (BE) , and t ∈ [0 , T ] . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS When γ = 1 / , we have A ε ,t ( H ) is a Cauchy ε -sequence in probabilitywith respect to a limit measure Q , and hence aϕ ′′ b ( ρ )4 A ,t ( H ) := lim ε ↓ aϕ ′′ b ( ρ )4 A ε ,t ( H ) = B t ( H ) . On the other hand, when / < γ ≤ , we have B t ( H ) ≡ .When / ≤ γ ≤ , we have K nt ( H ) ≡ , I t ( H ) = ϕ ′ c ( ρ )2 Z t Y s (∆ H ) ds and D t ( H ) = ϕ b ( ρ )2 t k∇ H k L ( R ) . Proof.
By assumption (BE), and lower semicontinuity of entropy, thelimit measure Q also has bounded entropy with respect to P ν ρ , H ( Q ; P ν ρ ) < ∞ . When γ = 1 /
2, by the L ( P ν ρ ) statements in Proposition 3.5 and theentropy inequality (2.6), we have for δ > ε ↓ Q ( |B t ( H ) − ( aϕ ′′ b ( ρ ) / A εt ( H ) | > δ ) = 0, and so A εt ( H ) is Cauchy in probability with respect to Q . Therefore, lim ε ↓ ( aϕ ′′ b ( ρ ) / A εt ( H ) = B t ( H ).The other claims follow similarly. (cid:3) Proof of Theorems 2.2 and 2.3.
Let H ∈ S ( R ), t ∈ [0 , T ], and sup-pose the initial measures are { µ n } . When γ = 1 /
2, by the decomposition(3.2), Proposition 3.6, and tightness of the constituent processes Y n,γt , M n,γt , I n,γt , B n,γt , K n,γt and hM n,γt i in the uniform topology, any limit point of( Y n,γt , M n,γt , I n,γt , B n,γt , K n,γt , hM n,γt i : t ∈ [0 , T ])satisfies M t ( H ) = Y t ( H ) − Y ( H ) − ϕ ′ c ( ρ )2 Z t Y s (∆ H ) ds − ( aϕ ′′ b ( ρ ) / A t ( H ) . However, when 1 / < γ ≤ M t ( H ) = Y t ( H ) − Y ( H ) − ϕ ′ c ( ρ )2 Z t Y s (∆ H ) ds. (3.7)Also, in both cases, Y ( H ) = ¯ Y ( H ) by assumption (CLT).We also claim in both cases that M t ( H ) is a continuous martingale witha quadratic variation hM t ( H ) i = ϕ b ( ρ )2 t k∇ H k L ( R ) . Indeed, by Proposition 3.6, any limit point of the quadratic variation se-quence equals D t ( H ) = ( ϕ b ( ρ ) / t k∇ H k L ( R ) . Next, M t ( H ) as the limit of P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN martingales with respect to the uniform topology is a continuous martin-gale. Also, by the triangle inequality, Doob’s inequality and the quadraticvariation bound (3.3),sup n E ν ρ h sup ≤ s ≤ t |M n,γs ( H ) − M n,γs − ( H ) | i ≤ n E ν ρ h sup u ∈ [0 ,t ] |M n,γu ( H ) | i / ≤ n E ν ρ [ h M n,γt ( H ) i ] / ≤ C ( a, T ) k b k L ( ν ρ ) k∇ H k L ( R ) . Then, by Corollary VI.6.30 of [31], ( M n,γt ( H ) , hM n,γt ( H ) i ) converges on asubsequence in distribution to ( M t ( H ) , hM t ( H ) i ). Since, also hM n,γt ( H ) i converges on a subsequence in distribution to D t ( H ) = ( ϕ b ( ρ ) / t k∇ H k L ( R ) ,we have hM t ( H ) i = ( ϕ b ( ρ ) / t k∇ H k L ( R ) .By Proposition 3.6, when γ = 1 / Y t is a “probability energy solution”corresponding to the stochastic Burgers equation (2.11). But, if initially µ n ≡ ν ρ , by Proposition 3.5, Y t is an “ L energy solution.” This completesthe proof of Theorem 2.3.However, when 1 / < γ ≤
1, by the form of M t ( H ) in (3.7), we conclude Y t ( H ) solves the Ornstein–Uhlenbeck equation (2.10). By uniqueness, allsubsequences converge to the same limit, and we obtain Theorem 2.2. (cid:3)
4. Proof of the generalized Boltzmann–Gibbs principle.
We start by re-calling the notion of H ,n and H − ,n spaces [35]. For n ≥ n , recall S n =( L n + L ∗ n ) / H ,n seminorm k · k ,n on L ( ν ρ )functions by k f k ,n := E ν ρ [ f ( − S n ) f ] = n D ν ρ ( f ) . The Hilbert space H ,n is then the completion of functions with finite H ,n norm modulo norm-zero functions. In particular, local bounded functionsare dense in H ,n .Correspondingly, one can define the dual seminorm k · k − ,n with respectto the L ( ν ρ ) inner-product by k f k − ,n := sup (cid:26) E ν ρ [ f φ ] k φ k ,n : φ = 0 local, bounded (cid:27) , and the Hilbert space H − ,n which is the completion over those functionswith finite k · k − ,n norm modulo norm-zero functions.We now state a helping lemma for the results in this section. Define therestricted Dirichlet form on local, bounded functions with respect to the TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS grand canonical measure ν ρ as D ν ρ ,ℓ ( φ ) = X x,x +1 ∈ Λ ℓ E ν ρ [ b R,nx ( η )( ∇ x,x +1 φ ( η )) ] . Recall the collection η cr := { η ( x ) : x / ∈ Λ r } . Proposition 4.1.
Let r : Ω → R be an L ( ν ρ ) function and ℓ ≥ . Sup-pose that E ν ρ [ r | η ( ℓ ) , η cℓ ] = 0 a.s. Then, for local, bounded functions φ , wehave | E ν ρ [ r ( η ) φ ( η )] | ≤ E ν ρ (cid:20) W (cid:18) X x ∈ Λ ℓ η ( x ) , ℓ , η cℓ , n (cid:19) (cid:21) / k r k L ( ν ρ ) D / ν ρ ,ℓ ( φ ) . Proof.
Recall, from Section 2.1, for k ≥ ℓ ≥ ξ ∈ Ω, the space G k,ℓ ,ξ = (cid:26) η : X x ∈ Λ ℓ η ( x ) = k, η ( y ) = ξ ( y ) for y / ∈ Λ ℓ (cid:27) and generator S n, G := S n, G k,ℓ ,ξ which governs the evolution of the sym-metrized process on G k,ℓ ,ξ . Suppose W ( k, ℓ , ξ, n ) < ∞ so that the measure ν k,ℓ ,ξ is the unique invariant measure for the process.Given E ν ρ [ r | P | x |≤ ℓ η ( x ) = k, η ( y ) = ξ ( y ) for y / ∈ Λ ℓ ] = E ν k,ℓ ,ξ [ r ] = 0, wehave r restricted to G k,ℓ ,ξ is orthogonal to constant functions and thereforebelongs to the range of − S n, G , that is the equation r = − S n, G u can be solvedfor some function u : G k,ℓ ,ξ → R .Now, with k = P x ∈ Λ ℓ η ( x ) and ξ = η cℓ , W ( k, ℓ , η cℓ , n ) < ∞ a.s. by as-sumption (G). Hence, | E ν ρ [ rφ ] | = | E ν ρ [ E ν ρ [ rφ | η ( ℓ ) , η cℓ ]] | = | E ν ρ [ E ν ρ [( − S n, G u ) φ | η ( ℓ ) , η cℓ ]] |≤ E ν ρ [ E ν ρ [ u ( − S n, G u ) | η ( ℓ ) , η cℓ ] / E ν ρ [ φ ( − S n, G φ ) | η ( ℓ ) , η cℓ ] / ] . The last line follows as − S n, G is a nonnegative symmetric operator and,therefore, has a square root.Further, since W ( k, ℓ , ξ, n ) is the reciprocal of the spectral gap for − S n, G ,we have E ν ρ [ ru | η ( ℓ ) , η cℓ ] ≤ W ( k, ℓ , η cℓ , n ) E ν ρ [ r | η ( ℓ ) , η cℓ ] . Therefore, we conclude | E ν ρ [ rφ ] | ≤ E ν ρ (cid:20) W (cid:18) X x ∈ Λ ℓ η ( x ) , ℓ , η cℓ , n (cid:19) E ν ρ [ r | η ( ℓ ) , η cℓ ] (cid:21) / D / ν ρ ,ℓ ( φ ) . The desired bound now follows from Schwarz inequality. (cid:3) P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
The following bound on the variance of additive functionals is the mainway we control the fluctuations of several quantities in the sequel. A proofof Proposition 4.2 can be found in Appendix 1.6 of [34].To simplify notation, for the rest of the section, we will drop the super-script “ n ” and write η n = η . Proposition 4.2.
Let r : Ω → R be a mean-zero L ( ν ρ ) function, ϕ r ( ρ ) = 0 .Then E ν ρ (cid:20)(cid:18)Z t r ( η s ) ds (cid:19) (cid:21) ≤ t k r k − ,n . The proof of Theorem 3.2, given at the end of the section, is made througha succession of steps, labeled “one-block,” “renormalization step,” “two-blocks” and “equivalence of ensembles” estimates.
Lemma 4.3 (One-block estimate).
Let f : Ω → R be a local L ( ν ρ ) func-tion supported on sites in Λ ℓ such that ϕ f ( ρ ) = 0 . Then there exists a con-stant C = C ( ρ ) such that for ℓ ≥ ℓ , t ≥ and h ∈ ℓ ( Z ) ∩ ℓ ( Z ) : E ν ρ (cid:20)(cid:18)Z t X x ∈ Z h ( x ) τ x { f ( η s ) − E ν ρ [ f ( η s ) | η ( ℓ ) s , ( η s ) cℓ ] } ds (cid:19) (cid:21) ≤ Ct ℓ n k f k L ( ν ρ ) X x ∈ Z h ( x ) . Proof.
By Proposition 4.2, we need only to estimate the H − ,n normof the integrand [which is in L ( ν ρ ) since h ∈ ℓ ( Z )]. Bound the H − ,n normmultiplied by n , using Proposition 4.1, as follows:sup φ D − / ν ρ ( φ ) E ν ρ (cid:20)X x ∈ Z h ( x ) τ x { f − E ν ρ [ f | η ( ℓ ) , η cℓ ] } φ (cid:21) = sup φ X x ∈ Z D − / ν ρ ( φ ) E ν ρ [ h ( x ) τ x ( f − E ν ρ [ f | η ( ℓ ) , η cℓ ]) φ ](4.1) ≤ sup φ D − / ν ρ ( φ ) × X x ∈ Z | h ( x ) | E ν ρ (cid:20) W (cid:18) X x ∈ Λ ℓ η ( x ) , ℓ, η cℓ , n (cid:19) (cid:21) / k f k L ( ν ρ ) D / ν ρ ,ℓ ( τ − x φ ) . Observe now, by translation-invariance of ν ρ , that X x ∈ Z D ν ρ ,ℓ ( τ − x φ ) ≤ (2 ℓ + 1) D ν ρ ( φ ) . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Then, noting the spectral gap assumption (G), and using the relation 2 ab =inf κ> [ a κ + κ − b ], we bound (4.1) bysup φ D − / ν ρ ( φ ) inf κ> (cid:26) κCℓ k f k L ( ν ρ ) X x ∈ Z h ( x ) + κ − CℓD ν ρ ( φ ) (cid:27) ≤ (cid:18) Cℓ k f k L ( ν ρ ) X x ∈ Z h ( x ) (cid:19) / , where C = C ( ρ ) is a constant. This completes the proof. (cid:3) Now we double the size of the box in the conditional expectation.
Lemma 4.4 (Renormalization step).
Let f : Ω → R be a local L ( ν ρ ) function supported on sites in Λ ℓ such that ϕ f ( ρ ) = ϕ ′ f ( ρ ) = 0 . There existsa constant C = C ( ρ, ℓ ) such that for ℓ ≥ ℓ , t ≥ and h ∈ ℓ ( Z ) ∩ ℓ ( Z ) : E ν ρ (cid:20)(cid:18)Z t X x ∈ Z τ x { E ν ρ [ f ( η s ) | η ( ℓ ) s , ( η s ) cℓ ] − E ν ρ [ f ( η s ) | η (2 ℓ ) s , ( η s ) c ℓ ] } h ( x ) ds (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) t ℓn X x ∈ Z h ( x ) . On the other hand, when only ϕ f ( ρ ) = 0 is known, E ν ρ (cid:20)(cid:18)Z t X x ∈ Z τ x { E ν ρ [ f ( η s ) | η ( ℓ ) s , ( η s ) cℓ ] − E ν ρ [ f ( η s ) | η (2 ℓ ) s , ( η s ) c ℓ ] } h ( x ) ds (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) t ℓ n X x ∈ Z h ( x ) . Proof.
We prove the first statement as the second is similar. Since E ν ρ [ E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ] | η (2 ℓ ) , η c ℓ ] = E ν ρ [ f ( η ) | η (2 ℓ ) , η c ℓ ] , we follow now the same steps as in the proof of Lemma 4.3 to the last line.To finish the proof, we now give an order O ( k f k L ( ν ρ ) ℓ − ) bound on thevariance k E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ] − E ν ρ [ f ( η ) | η (2 ℓ ) , η c ℓ ] k L ( ν ρ ) . P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
Adding and subtracting terms, and the inequality ( a + b + c ) ≤ a + 3 b +3 c , the variance is bounded by ≤ (cid:13)(cid:13)(cid:13)(cid:13) E ν ρ (cid:20) f ( η ) − ϕ ′′ f ( ρ )2 (cid:26) ( η ( ℓ ) − ρ ) − σ ℓ ( ρ )2 ℓ + 1 (cid:27)(cid:12)(cid:12)(cid:12) η ( ℓ ) , η cℓ (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν ρ ) + 3 (cid:13)(cid:13)(cid:13)(cid:13) E ν ρ (cid:20) f ( η ) − ϕ ′′ f ( ρ )2 (cid:26) ( η (2 ℓ ) − ρ ) − σ ℓ ( ρ )2(2 ℓ + 1) (cid:27)(cid:12)(cid:12)(cid:12) η (2 ℓ ) , η c ℓ (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν ρ ) + 3 (cid:13)(cid:13)(cid:13)(cid:13) ϕ ′′ f ( ρ )2 (cid:26) E ν ρ (cid:20) ( η ( ℓ ) − ρ ) − σ ℓ ( ρ )2 ℓ + 1 (cid:12)(cid:12)(cid:12) η ( ℓ ) , η cℓ (cid:21) + E ν ρ (cid:20) ( η (2 ℓ ) − ρ ) − σ ℓ ( ρ )2(2 ℓ + 1) (cid:12)(cid:12)(cid:12) η (2 ℓ ) , η c ℓ (cid:21)(cid:27)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν ρ ) . The last term, by the fourth moment bound of ( η ( k ) − ρ ) in (IM2) with k = ℓ and k = 2 ℓ and that | ϕ ′′ f ( ρ ) | ≤ C ( ρ ) k f k L ( ν ρ ) in (D), is of order O ( k f k L ( ν ρ ) ℓ − ).But the first two terms are of order O ( k f k L ( ν ρ ) ℓ − α ) by applying theequivalence of ensembles assumption (EE). (cid:3) Lemma 4.5 (Two-blocks estimate).
Let f : Ω → R be a local L ( ν ρ ) func-tion supported on sites in Λ ℓ such that ϕ f ( ρ ) = ϕ ′ f ( ρ ) = 0 . Then, there existsa constant C = C ( ρ, ℓ ) such that for ℓ ≥ ℓ , t ≥ and h ∈ ℓ ( Z ) ∩ ℓ ( Z ) : E ν ρ (cid:20)(cid:18)Z t X x ∈ Z τ x { E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ] − E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ] } h ( x ) ds (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) t ℓn X x ∈ Z h ( x ) . On the other hand, when only ϕ f ( ρ ) = 0 is known, E ν ρ (cid:20)(cid:18)Z t X x ∈ Z τ x { E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ] − E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ] } h ( x ) ds (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) t ℓ n X x ∈ Z h ( x ) . Proof.
We prove the first display as the second is analogous. Again,we invoke Proposition 4.2 and bound the square of the H − ,n norm. To thisend, write ℓ = 2 m +1 ℓ + r where 0 ≤ r ≤ m +1 ℓ −
1. Then E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ] − E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ]= E ν ρ [ f ( η ) | η (2 m +1 ℓ ) , η c m +1 ℓ ] − E ν ρ [ f ( η ) | η ( ℓ ) , η cℓ ] TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS + m X i =0 { E ν ρ [ f ( η ) | η (2 i ℓ ) , η c i ℓ ] − E ν ρ [ f ( η ) | η (2 i +1 ℓ ) , η c i +1 ℓ ] } . Now, by Minkowski’s inequality, with respect to the H − ,n norm, overthe m + 2 terms, and Lemma 4.4, we obtain that the left-hand side of thedisplay in the lemma statement is bounded by ((cid:18) Ct m +1 ℓ n (cid:19) / + m X i =0 (cid:18) Ct i ℓ n (cid:19) / ) k f k L ( ν ρ ) X x ∈ Z h ( x ) ≤ C k f k L ( ν ρ ) tℓn X x ∈ Z h ( x )to finish the proof. (cid:3) Lemma 4.6 (Equivalence of ensembles estimate).
Let f : Ω → R be alocal L ( ν ρ ) function supported on sites in Λ ℓ such that ϕ f ( ρ ) = ϕ ′ f ( ρ ) = 0 .Then, there exists a constant C = C ( ρ, ℓ ) such that for ℓ ≥ ℓ , t ≥ and h ∈ ℓ ( Z ) : E ν ρ (cid:20)(cid:18)Z t X x ∈ Z τ x (cid:26) E ν ρ [ f ( η s ) | η ( ℓ ) s , ( η s ) cℓ ] − ϕ ′′ f ( ρ )2 (cid:18) ( η ( ℓ ) s − ρ ) − σ ℓ ( ρ )2 ℓ + 1 (cid:19)(cid:27) h ( x ) ds (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) t n ℓ α (cid:18) n X x ∈ Z | h ( x ) | (cid:19) . On the other hand, when only ϕ f ( ρ ) = 0 is known, E ν ρ (cid:20)(cid:18)Z t X x ∈ Z τ x { E ν ρ [ f ( η s ) | η ( ℓ ) s , ( η s ) cℓ ] − ϕ ′ f ( ρ )( η ( ℓ ) s − ρ ) } h ( x ) ds (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) t n ℓ α (cid:18) n X x ∈ Z | h ( x ) | (cid:19) . Here, α > is the power mentioned in assumption (EE) . Proof.
By squaring and using invariance of ν ρ , the left-hand side ofthe display is bounded by2 t E ν ρ (cid:20)(cid:18)X x ∈ Z | h ( x ) || r ( x ) | (cid:19) (cid:21) , P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN where r ( x ) is the τ x -shifted expression in curly braces in the display ofLemma 4.6. Now, by Schwarz inequality, (cid:18)X x ∈ Z | h ( x ) | r ( x ) (cid:19) ≤ (cid:18)X x ∈ Z | h ( x ) | (cid:19) X x ∈ Z | h ( x ) | r ( x ) . Since ν ρ is translation-invariant, the desired bound is now obtained by notingthe form of r ( x ) and the equivalence of ensembles assumption (EE). (cid:3) Proof of Theorem 3.2.
By combining Lemma 4.3 with ℓ = ℓ , andLemmas 4.5 and 4.6, we straightforwardly obtain the result. (cid:3)
5. Equivalence of ensembles.
We prove, as a consequence of Proposi-tion 5.1, that condition (EE) holds for a large class of systems with productinvariant measures. In this case, ν k,ℓ,ξ does not depend on ξ , which simplifiesthe conditional expectation in the statement of (EE).Next, we show in Proposition 5.2 that (EE) also holds for the Markovchain measure ν / defined in Section 2.5. Some parts of the proofs of thesestatements are similar to those in [56].Define Λ + m = { x : 1 ≤ x ≤ m } . Proposition 5.1.
Let ν ρ be a product measure on Ω such that (IM) holds, and < ν ρ ( η (0) = j ) < for j = 0 , . Let also f be a local L ( ν ρ ) function, supported on sites Λ + ℓ , such that ϕ f ( ρ ) = ϕ ′ f ( ρ ) = 0 . Then thereexists a constant C = C ( ρ, ℓ ) , such that for n ≥ ℓ we have (cid:13)(cid:13)(cid:13)(cid:13) E ν ρ [ f ( η ) | y ] − (cid:26) y − σ ( ρ ) n (cid:27) ϕ ′′ f ( ρ )2 (cid:13)(cid:13)(cid:13)(cid:13) L ( ν ρ ) ≤ C k f k L ( ν ρ ) n / . On the other hand, when only ϕ f ( ρ ) = 0 is known, k E ν ρ [ f ( η ) | y ] − yϕ ′ f ( ρ ) k L ( ν ρ ) ≤ C k f k L ( ν ρ ) n . Here, y := n P x ∈ Λ + n η ( x ) − ρ . Proof.
We prove the first display as the second statement, followingthe same scheme, has a simpler argument. At the expense of the constant,we need only to consider all large n > ℓ . To simplify notation, we will call ℓ = ℓ . The proof follows in several steps. Step
1. Recall the tilted measures { ν z : ρ ∗ < z < ρ ∗ } given after assump-tion (D1) which are well defined as ν ρ is a product measure. Let σ ( z ) = E ν z [( η (0) − z ) ]. Note also the canonical expectation E ν z [ f | y ] does not de-pend on z , and that we are free to choose it as desired. TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Develop E ν ρ [ f ( η ) | y ] = E ν y + ρ (cid:20) f ( η ) (cid:12)(cid:12)(cid:12) n X x ∈ Λ + n η ( x ) − ρ = y (cid:21) = E ν y + ρ [ f ( η )1((1 /n ) P x ∈ Λ + n η ( x ) − ρ = y )] ν y + ρ ((1 /n ) P x ∈ Λ + n η ( x ) − ρ = y ) . Define θ m ( z ) = √ mν y + ρ ( P x ∈ Λ + m η ( x ) − ρ − y = z ), and write the last ex-pression as E ν y + ρ (cid:20) f ( η ) √ nθ n − ℓ ( − P x ∈ Λ + ℓ ( η ( x ) − y − ρ )) √ n − ℓθ n (0) (cid:21) . The goal will be now to expand θ n − ℓ ( z ) to recover the main terms approxi-mating E ν ρ [ f | y ] when | y | is small. We will treat the case when | y | is boundedaway from 0 afterward. Step
2. To expand θ m ( z ), let ψ y ( t ) = E ν y + ρ [ e it ( η ( x ) − ρ − y ) ] be the charac-teristic function. Then one can write θ m ( x ) = √ m π Z π − π e − itx ψ my ( t ) dt = 12 π Z π √ m − π √ m e − itx/ √ m ψ my ( t/ √ m ) dt. By Taylor expansion,2 πθ m ( x ) = Z π √ m − π √ m ψ my ( t/ √ m ) dt − Z π √ m − π √ m ixt √ m ψ my ( t/ √ m ) dt − Z π √ m − π √ m x t m ψ my ( t/ √ m ) dt (5.1) + O (cid:18) | x | m / (cid:19) Z π √ m − π √ m | t | | ψ my ( t/ √ m ) | dt. Step
3. Let δ > ρ − δ, ρ + δ ) ⊂ ( ρ ∗ , ρ ∗ ) and sufficiently smallin the following estimates. Let also 0 < ε ≤ π .First, sup | y |≤ δ,ε ≤| t |≤ π | ψ my ( t ) | < C m where C <
1: Write | ψ y ( t ) | ≤ | ν y + ρ ( η (0) = 0) + e it ν y + ρ ( η (0) = 1) | + X k ≥ ν y + ρ ( η (0) = k ) ≤ ( A − ν y + ρ ( η (0) = 0) ν y + ρ ( η (0) = 1)[1 − cos( t )]) / + 1 − A, where A = ν y + ρ ( η (0) = 0) + ν y + ρ ( η (0) = 1). By the proposition assumptionsand continuity of ν y + ρ ( η (0) = k ) in y , 0 < ν y + ρ ( η (0) = j ) < j = 0 , P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN uniformly for | y | ≤ δ . Hence, uniformly over ε ≤ | t | ≤ π , | y | ≤ δ , the right-hand side of the display above is strictly bounded by a constant C < ≤ | t/ √ m | < ε and | y | ≤ δ , ψ my ( t/ √ m ) = [1 − ( t σ ( y + ρ ) / (2 m )) + O ( C ( δ ) | t | m − / )] m so that | ψ my ( t/ √ m ) | ≤ e − C ( y,ε ) t . Here, by continuity in y and σ ( ρ ) > ε is small, inf | y |≤ δ C ( y, ε ) >
0. Similarly, we note sup | y |≤ δ σ ( y + ρ ) < ∞ , and inf | y |≤ δ σ ( y + ρ ) > m ↑∞ θ m (0) = (2 πσ ( y + ρ )) − / . Step
4. We now observe, for | y | ≤ δ and m ≥
1, as a consequence of the es-timates in step 3, the integral in the last term in (5.1) is uniformly bounded:Split the integral over the ranges | t/ √ m | < ε and | t/ √ m | ≥ ε and boundeach part separately.Also, similarly, we split the second integral in (5.1), when | y | ≤ δ , overranges | t/ √ m | ≥ ε and | t/ √ m | < ε . On the first range, the restricted integralexponentially decays, and on the range | t/ √ m | < ε , the restricted integralis almost the integral of an odd function since here ψ my ( t/ √ m ) = (cid:18) − t σ ( y + ρ )2 m (cid:19) m [1 + O ( C ( δ ) | t | m − / )] . Therefore, we conclude that the second integral in (5.1) is of order O ( m − / ). Step
5. Then, for | y | ≤ δ , we have E ν ρ [ f ( η ) | y ] = κ E ν y + ρ [ f ( η )] + κ √ n − ℓ E ν y + ρ (cid:20) f ( η ) (cid:18) X x ∈ Λ + ℓ η ( x ) − ρ − y (cid:19)(cid:21) + κ n − ℓ E ν y + ρ (cid:20) f ( η ) (cid:18) X x ∈ Λ + ℓ η ( x ) − ρ − y (cid:19) (cid:21) + ε f ( n ) , where | ε f ( n ) | ≤ C ( ρ, ℓ, δ ) k f k L ( ν ρ ) n − / and κ i = κ i ( n ) for i = 0 , , κ ( n ) = √ n √ n − ℓ θ n − ℓ (0) θ n (0) = 1 + O ( n − / ) ,κ ( n ) = √ nθ n (0) √ n − ℓ π Z π √ n − ℓ − π √ n − ℓ itψ n − ℓy (cid:18) t √ n − ℓ (cid:19) dt = O ( n − / ) ,κ ( n ) = −√ n θ n (0) √ n − ℓ π Z π √ n − ℓ − π √ n − ℓ t ψ n − ℓy (cid:18) t √ n − ℓ (cid:19) dt = − σ ( y + ρ ) + O ( n − / ) . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Step
6. We now develop expansions of E ν y + ρ [ h ] for a local L ( ν ρ ) function h supported on coordinates in Λ + ℓ . The “tilting” given in the Introduction,(2.3) reduces to E ν y + ρ [ h ] = E ν ρ (cid:20) h ( η ) e λ ( y + ρ ) P x ∈ Λ+ ℓ ( η ( x ) − ρ ) M ℓ ( λ ( y + ρ )) (cid:21) , where λ ( y + ρ ) is the “tilt” chosen to change the density to y + ρ and M ( λ ) = E ν ρ [ e λ ( η ( x ) − ρ ) ]. Note that z − ρ = M ′ ( λ ( z )) /M ( λ ( z )) and λ ′ ( z ) = (cid:20) M ′′ ( λ ( z )) M ( λ ( z )) − (cid:18) M ′ ( λ ( z )) M ( λ ( z )) (cid:19) (cid:21) − = 1 σ ( z ) . Consider the first and second derivatives of E ν y + ρ [ h ] given exactly in (2.4)as ν y + ρ is a product measure. The third derivative takes the form d dy E ν y + ρ [ h ( η )] = λ ′′′ ( y + ρ ) E ν y + ρ (cid:20) ¯ h ( η ) (cid:18) X x ∈ Λ + ℓ η ( x ) − y − ρ (cid:19)(cid:21) + 3 λ ′ ( y + ρ ) λ ′′ ( y + ρ ) E ν y + ρ (cid:20) ¯ h ( η ) (cid:18) X x ∈ Λ + ℓ η ( x ) − y − ρ (cid:19) (cid:21) + ( λ ′ ( y + ρ )) E ν y + ρ (cid:20) ¯ h ( η ) (cid:18) X x ∈ Λ + ℓ η ( x ) − y − ρ (cid:19) (cid:21) − λ ′ ( y + ρ )) E ν y + ρ (cid:20) ¯ h ( η ) (cid:18) X x ∈ Λ + ℓ η ( x ) − y − ρ (cid:19)(cid:21) × E ν y + ρ (cid:20)(cid:18) X x ∈ Λ + ℓ η ( x ) − y − ρ (cid:19) (cid:21) , where ¯ h ( η ) = h ( η ) − E ν y + ρ [ h ].Then, for | y | ≤ δ , when ϕ h ( ρ ) = ϕ ′ h ( ρ ) = 0, we may expand around y = 0: E ν y + ρ [ h ( η )] = ( λ ′ ( ρ )) y E ν ρ (cid:20) h ( η ) (cid:18) X x ∈ Λ + ℓ η ( x ) − ρ (cid:19) (cid:21) + | y | r ( ρ, δ, h ) . When only ϕ h ( ρ ) = 0 is known, E ν y + ρ [ h ( η )] = λ ′ ( ρ ) yE ν ρ (cid:20) h ( η ) (cid:18) X x ∈ Λ + ℓ η ( x ) − ρ (cid:19)(cid:21) + | y | r ( ρ, δ, h ) . P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
When possibly ϕ h ( ρ ) = 0, E ν y + ρ [ h ( η )] = E ν ρ [ h ( η )] + | y | r ( ρ, δ, h ) . Here, as the first and second derivatives in (2.4) and the third deriva-tive above are bounded for | y | ≤ δ , we may conclude that the remainders | r ( ρ, δ, h ) | ≤ C ( ρ, δ ) k h k L ( ν ρ ) .We now relate the terms E ν ρ [ h ( η )( P x ∈ Λ + ℓ ( η ( x ) − ρ )) k ] to derivatives ϕ ( k ) h ( ρ ):From (2.4), for k = 1 ,
2, when ϕ ( k − h ( ρ ) = ϕ h ( ρ ) = 0, we have ϕ ( k ) h ( ρ ) = ( λ ′ ( ρ )) k E ν ρ (cid:20) h ( η ) (cid:18) X x ∈ Λ + ℓ ( η ( x ) − ρ ) (cid:19) k (cid:21) . (5.2) Step
7. Consider the expansion of E ν ρ [ f | y ] in step 5 when | y | ≤ δ . With h equal to variously f , f ( η )( P x ∈ Λ + ℓ ( η ( x ) − ρ )), and f ( η )( P x ∈ Λ + ℓ ( η ( x ) − ρ )) ,we may write E ν ρ [ f | y ] = κ λ ′ ( ρ )) y E ν ρ (cid:20) f ( η ) (cid:18)X Λ + ℓ ( η ( x ) − ρ ) (cid:19) (cid:21) + κ | y | r ( f )+ κ λ ′ ( ρ ) y √ n − ℓ E ν ρ (cid:20) f ( η ) (cid:18)X Λ + ℓ ( η ( x ) − ρ ) (cid:19) (cid:21) + κ √ n − ℓ | y | r ( f )+ κ n − ℓ E ν ρ (cid:20) f ( η ) (cid:18)X Λ + ℓ ( η ( x ) − ρ ) (cid:19) (cid:21) + κ n − ℓ | y | r ( f ) + ε f ( n ) , where | r ( f ) | ≤ C ( ρ, ℓ, δ ) k f k L ( ν ρ ) .Hence, noting the assumptions on ϕ f ( ρ ), (5.2), and E ν ρ [ y p ] = O ( n − p ) sothat each y factor is O ( n − / ), we can group the dominant terms so that E ν ρ (cid:20) | y | ≤ δ ) × (cid:18) E ν ρ [ f ( η ) | y ] − (cid:26) κ y λ ′ ( ρ ) κ y √ n + 1( λ ′ ( ρ )) κ n (cid:27) ϕ ′′ f ( ρ ) (cid:19) (cid:21) ≤ C ( ρ, δ ) k f k L ( ν ρ ) n − . Noting κ ( n ) = 1 + O ( n − / ) , κ ( n ) = O ( n − / ), formula λ ′ ( ρ ) = σ − ( ρ ) instep 6, | ϕ ′′ ( ρ ) | ≤ C k f k L ( ν ρ ) and, by Taylor expansion of σ ( y + ρ ) around y = 0, κ ( n ) = − − σ − ( ρ ) + O ( n − / ), we have further E ν ρ (cid:20) | y | ≤ δ ) (cid:18) E ν ρ [ f ( η ) | y ] − (cid:26) y − σ ( ρ ) n (cid:27) ϕ ′′ f ( ρ )2 (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) n − . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS Step
8. On the other hand, by say large deviations estimates, we bound E ν ρ (cid:20) | y | > δ ) (cid:18) E ν ρ [ f ( η ) | y ] − (cid:26) y − σ ( ρ ) n (cid:27) ϕ ′′ f ( ρ )2 (cid:19) (cid:21) ≤ C k f k L ( ν ρ ) O ( n − )to complete the proof. (cid:3) We now prove the equivalence ensembles estimate (EE) with respect toa Markovian measure. Recall the Gibbs measures ν / and ν z = ν λ ( z )1 / , andtransition matrix P defined in Section 2.5. To see how the next propositioncan be used to satisfy assumption (EE), we note (1) the estimate is uni-form in the “outside variables” η cℓ , and (2) since the transition matrix P ispositive, the L ∞ norm of any local function supported on sites Λ ℓ can bebounded k f k L ∞ ≤ C ( ℓ , β ) k f k L p ( ν / ) for p >
0. Recall also the definitionsof ϕ f ( ρ ) and its derivatives in (2.4). Proposition 5.2.
Let f be a local function, supported on sites indexedby Λ ℓ , such that ϕ f (1 /
2) = ϕ ′ f (1 /
2) = 0 . Then, for each < ε < , there isa constant C = C ( ℓ , ε ) such that for a, b ∈ { , } and n ≥ ℓ , (cid:13)(cid:13)(cid:13)(cid:13) E ν / [ f | y, η ( − n −
1) = a, η ( n + 1) = b ] − ϕ ′′ f (1 / (cid:20) y − σ n (1 / n + 1 (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν / ) ≤ C k f k L ∞ n / − ε . On the other hand, when only ϕ f (1 /
2) = 0 is known, (cid:13)(cid:13)(cid:13)(cid:13) E ν / [ f | y, η ( − n −
1) = a, η ( n + 1) = b ] − yϕ ′ f (cid:18) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν / ) ≤ C k f k L ∞ n − ε . Here, y = (2 n + 1) − P x ∈ Λ n ( η ( x ) − ) . Proof.
The argument has the same structure as for Proposition 5.1. Wewill concentrate on the first display for all large n ; the second statement hasa similar argument. Since ν / corresponds to an ergodic finite-state Markovchain with uniform invariant measure, it is exponentially mixing and allowsstandard approximations, which are used in many steps. Step
1. Let χ > n ′ = n − n χ . Write E ν / [ f | y, η ( − n −
1) = a, η ( n + 1) = b ]= E ν y +1 / [ f ( η ) | y, η ( − n −
1) = a, η ( n + 1) = b ] P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN = E ν y +1 / (cid:20) f ( η ) √ n + 1 θ χn,y,a,b ( − P x ∈ Λ nχ ( η ( x ) − y − / √ n ′ θ n,y,a,b (0) (cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) , where θ χn,y,a,b ( z ) = √ n ′ ν y +1 / (cid:18) X n χ < | x |≤ n η ( x ) − y − / z (cid:12)(cid:12)(cid:12) η ( n χ ) , η ( − n χ ) , η ( − n −
1) = a,η ( n + 1) = b (cid:19) ,θ n,y,a,b ( z ) = √ n + 1 ν y +1 / (cid:18) X x ∈ Λ n η ( x ) − y − / z (cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:19) . Step
2. Let the characteristic function ψ n,y,χ,a,b ( t ) for | t | ≤ π be definedby E ν y +1 / [ e it P nχ< | x |≤ n ( η ( x ) − y − / | η ( n χ ) , η ( − n χ ) , η ( − n −
1) = a, η ( n + 1) = b ] . Let δ > ρ − δ, ρ + δ ) ⊂ (0 ,
1) and sufficiently small in thefollowing estimates. Suppose | y | ≤ δ . Let also r > | t | < r , we claim ψ n,y,χ,a,b (cid:18) t √ n ′ (cid:19) = (cid:18) − t σ ( y + 1 / n ′ ) (cid:19) n ′ [1 + O ( n ′− / )] , (5.3)where σ ( z ) = lim n ↑∞ (2 n + 1) − E ν z [( P x ∈ Λ n η ( x ) − z ) ] is the limiting vari-ance of the additive functional n − / P nx =1 η ( x ) − z with respect to measure ν z (cf. formula in Section 2.5).Therefore, for | t | < r and C = C ( δ, r ) >
0, we have | ψ n,y,χ,a,b ( t/ √ n ′ ) | < exp {− Ct } .Next, we claim, for r ≤ | t | ≤ π that | ψ n,y,χ,a,b ( t ) | < A n ′ , (5.4)where A = A ( δ, r ) < n ↑∞ θ n,y,a,b (0) = 1 p πσ ( y + 1 / . TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS We now give an argument for the above claims which may be skipped onfirst reading. Recall u and v near (2.15) with λ = λ ( y + 1 / e P ( s ) = (cid:20) (1 − u ) e s ( − / − y ) u e s (1 / − y ) v e s ( − / − y ) (1 − v ) e s (1 / − y ) (cid:21) . By the Markov property, one writes ψ n,y,χ,a,b ( t ) = e P ( it ) n ′ ( η ( − n χ ) , a ) e P ( it ) n ′ ( η ( n χ ) , b ) e P (0) n ′ ( η ( − n χ ) , a ) e P (0) n ′ ( η ( n χ ) , b ) . (5.5)We may diagonalize e P ( it/ √ m ) m = Q ( it/ √ m ) D m ( it/ √ m ) Q − ( it/ √ m ),for large m , where D ( t ) is a diagonal matrix with eigenvalues w ( t ) and w ( t ) and Q ( t ) is the matrix of the corresponding eigenvectors. Of course,when t = 0, 1 = w (0) > w (0) = 1 − u − v with corresponding eigenvec-tors h , i and h u , − v i . For large m , w ( it/ √ m ) is the eigenvalue withmaximum absolute value and is expressed as e − ity/ √ m − u ) e − it/ (2 √ m ) + (1 − v ) e it/ (2 √ m ) + { ((1 − u ) e − it/ (2 √ m ) − (1 − v ) e it/ (2 √ m ) ) + 4 u v } / ] . It is not difficult to check that w ′ (0) = − ity/ √ m + it/ (2 √ m )[( u − v ) / ( u + v )]= ( it/ √ m ) E π ( y +1 / [ η (0) − / − y ] = 0 , where π ( y + 1 /
2) = ( u + v ) − h v , u i is the marginal of ν y +1 / (cf. Sec-tion 2.5). One now expands, as all quantities are smooth, w ( it/m ) = 1 − ( t / m ) w ′′ (0) + O ( m − / ) where the error is uniform for | y | ≤ δ and | t | ≤ π .Similarly, Q ( it/ √ m ) = Q (0) + O ( m − / ). One can identify w ′′ (0) as the vari-ance σ ( y + 1 /
2) since we know E ν y +1 / [ e ( it/ √ m ) P mx =1 η ( x ) − y − / ]= π ( y + 1 / P ( it/ √ m ) m = (1 − t w ′′ (0) / (2 m )) m (1 + O ( m − / ))must converge to e − t σ ( y +1 / / . Here, π ( y + 1 /
2) is thought of as a rowvector, and is the column vector with entries equal to 1.Putting these estimates together, we may conclude (5.3). To verify (5.4),from equation (5.5), we need only show the moduli | w ( it ) | , | w ( it ) | < r ≤ | t | ≤ π and | y | ≤ δ . One way is the following. Suppose y = 0and note that the moduli are less than 1 at | t | = π . For r ≤ | t | ≤ π , from the P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN determinant of e P ( it ), w ( it ) w ( it ) = 1 − u − v . In particular, if w ( it ) say isof the form e iθ with | θ | ≤ π , then w ( it ) = e − iθ (1 − u − v ). From the trace,we obtain equation e iθ + e − iθ (1 − u − v ) = (1 − u ) e − it/ + (1 − v ) e it/ whichis absurd: The real part is cos( θ )(2 − u − v ) = cos( t/ − u − v ) whichyields θ = t/
2. But the imaginary part is sin( θ )( u + v ) = sin( t/ u − v )which is a contradiction as r/ < | θ | = | t | / ≤ π/ v = 0 for y = 0.Hence, by continuity, for | y | ≤ δ small, we conclude the claim. Step
3. Now, write θ χn,y,a,b ( x ) = √ n ′ π Z π − π e − itx ψ n,y,χ,a,b ( t ) dt = 12 π Z π √ n ′ − π √ n ′ e − itx/ √ n ′ ψ n,y,χ,a,b ( t/ √ n ′ ) dt. The last expression is rewritten as12 π Z π √ n ′ − π √ n ′ ψ n,y,χ,a,b ( t/ √ n ′ ) dt − ix π √ n ′ Z π √ n ′ − π √ n ′ tψ n,y,χ,a,b ( t/ √ n ′ ) dt − x πn ′ Z π √ n ′ − π √ n ′ t ψ n,y,χ,a,b ( t/ √ n ′ ) dt + r ( x ) n − / in terms of error r ( x ) which, by the estimates in step 2, is of order O ( | x | ).The second integral in the last display is also estimated of order O ( n ′− / )by the same argument as given in step 3 of the proof of Proposition 5.1. Step
4. Then, for | y | ≤ δ , we have E ν / [ f | y, η ( − n −
1) = a, η ( n + 1) = b ]= κ E ν y +1 / [ f ( η ) | η ( − n −
1) = a, η ( n + 1) = b ]+ κ √ n ′ E ν y +1 / (cid:20) f ( η ) (cid:18) X | x |≤ n χ η ( x ) − − y (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a,η ( n + 1) = b (cid:21) + κ n ′ E ν y +1 / (cid:20) f ( η ) (cid:18) X | x |≤ n χ η ( x ) − − y (cid:19) (cid:12)(cid:12)(cid:12) η ( − n −
1) = a,η ( n + 1) = b (cid:21) + ε f ( n ) , TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS where | ε f ( n ) | ≤ C k f k L ∞ ( ν ρ ) n − / χ and κ i = κ i ( n ) for i = 0 , , Step
5. Recall the tilted measures and the formula for the tilt λ ( z ) inSection 2.5. Recall also the definitions of ϕ ( i ) f ( ρ ) (2.4). For | y | ≤ δ and i =0 , ,
2, using the uniform exponentially mixing property of the measures { ν y +1 / : | y | ≤ δ } and ϕ f (1 /
2) = ϕ ′ f (1 /
2) = 0, we claim E ν y +1 / (cid:20) f ( η ) (cid:18) X | x |≤ n χ η ( x ) − / − y (cid:19) i (cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) = λ ′ (1 / − i y − i (2 − i )! E ν / (cid:20) f ( η ) (cid:18) X | x |≤ n χ η ( x ) − / (cid:19) (cid:21) (5.6) + | y | − i r ( f, n ) + r ( f, n ) . Here, the error r ( f, n ) stands for the error made first in Taylor approxima-tion around y = 0 with respect to the conditioned measure: Using that ν y +1 / is exponentially mixing, one can bound the first, second and third derivativesbelow (5.9), (5.10) and (5.11), uniformly in a, b and | y | ≤ δ after a calcula-tion so that | r ( f, n ) | ≤ C ( δ ) n χ k f k L ∞ . The error r ( f, n ) represents othererrors made by exponential approximations and | r ( h, n ) | ≤ C k f k L ∞ n − / .The reader, on first reading, may like to skip now to step 6.Indeed, in more detail, when i = 1, E ν y +1 / (cid:20) f ( η ) (cid:18) X | x |≤ n χ η ( x ) − / − y (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) = E ν / (cid:20) f ( η ) (cid:18) X | x |≤ n χ η ( x ) − / (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) (5.7) + By + y r ( f, n ) , where, referring to the first derivative expression (5.9), B equals λ ′ (cid:18) (cid:19) E ν / (cid:20) f ( η ) (cid:18) X | x |≤ n χ η ( x ) − (cid:19)(cid:18) X | x |≤ n e η ( x ) (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) (5.8) − n χ E ν / [ f | η ( − n −
1) = a, η ( n + 1) = b ]and e η ( x ) = η ( x ) − E ν / [ η ( x ) | η ( − n −
1) = a, η ( n + 1) = b ]. The error r ( f, n )is less than the bound on the second derivative (5.10) with h = f ( η ) × ( P | x |≤ n χ η ( x ) − /
2) plus 2 n χ times the bound on the first derivative (5.9) P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN with h = f . We now bound the second derivative; estimating the first deriva-tive is similar. See notation ¯ h and ¯ η ( x ) above (5.9).For | y | ≤ δ , from the formula for the tilt λ ( z ) in Section 2.5, the deriva-tives λ ( k ) ( y + 1 /
2) for k = 1 , , E ν y +1 / [¯ h ( η )( P | x |≤ n ¯ η ( x )) | η ( − n −
1) = a, η ( n + 1) = b ] in (5.10) is handledas follows. By splitting the sum P | x |≤ n ¯ η ( x ) over indices | x | ≤ n χ , n χ < | x | ≤ n − n χ and | x | > n − n χ , and using the uniform exponentially mixingproperty of ν y +1 / , for | y | ≤ δ , and that all variables | η ( x ) | ≤
1, one boundsthis term as O ( k f k L ∞ n χ ).Consider now the other term E ν y +1 / [¯ h ( η )( P | x |≤ n ¯ η ( x )) | η ( − n −
1) = a , η ( n + 1) = b ] in (5.10). Split the sum over | x | ≤ n into sums over | x | ≤ n (1+ u ) χ and | x | > n (1+ u ) χ , and square to yield three terms. Bounding the cross termis the most involved, the other two being straightforward. The cross term is2 E ν y +1 / (cid:20) ¯ h ( η ) (cid:18) X | x |≤ n (1+ u ) χ ¯ η ( x ) (cid:19)(cid:18) X n (1+ u ) χ < | x |≤ n ¯ η ( x ) (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) . By splitting the sum over n (1+ u ) χ < | x | ≤ n into sums on n (1+ u ) χ < | x | < n (1+2 u ) χ , n (1+2 u ) χ ≤ | x | ≤ n − n uχ and | x | > n − n uχ , and using theexponentially mixing property of ν y +1 / , one can bound the cross term O ( k f k L ∞ n (3+3 u ) χ ) which for u < / ϕ ′ f (1 /
2) and ϕ ′′ f (1 /
2) [cf. (2.4)], usingthe exponentially mixing property. It is straightforward that the differencebetween the first conditional expectation on the right-hand side of (5.7) and λ ′ (1 / − ϕ ′ f (1 /
2) = 0 is exponentially close. Also, as ϕ ′ f (1 /
2) = 0, the firstconditional expectation in the expression B in (5.8) is exponentially close to( λ ′ (1 / − ϕ ′′ f (1 / B is exponentiallysmall.The cases i = 0 , h supported on sites in Λ n χ , and notation¯ h ( η ) = h ( η ) − E ν y +1 / [ h | η ( − n −
1) = a, η ( n + 1) = b ] and¯ η ( x ) = η ( x ) − E ν y +1 / [ η ( x ) | η ( − n −
1) = a, η ( n + 1) = b ] , the first derivative is ddy E ν y +1 / [ h ( η ) | η ( − n −
1) = a, η ( n + 1) = b ]= λ ′ (cid:18) y + 12 (cid:19) (5.9) TOCHASTIC BURGERS FROM MICROSCOPIC INTERACTIONS × E ν y +1 / (cid:20) ¯ h ( η ) (cid:18) X | x |≤ n ¯ η ( x ) (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) . The second derivative is d dy E ν y +1 / [ h ( η ) | η ( − n −
1) = a, η ( n + 1) = b ]= λ ′′ (cid:18) y + 12 (cid:19) E ν y +1 / (cid:20) ¯ h ( η ) (cid:18) X | x |≤ n ¯ η ( x ) (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) (5.10) + (cid:18) λ ′ (cid:18) y + 12 (cid:19)(cid:19) × E ν y +1 / (cid:20) ¯ h ( η ) (cid:18) X | x |≤ n ¯ η ( x ) (cid:19) (cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) . The third derivative is d dy E ν y +1 / [ h ( η ) | η ( − n −
1) = a, η ( n + 1) = b ]= λ ′′′ (cid:18) y + 12 (cid:19) E ν y +1 / (cid:20) ¯ h ( η ) (cid:18) X | x |≤ n ¯ η ( x ) (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) + 3 λ ′ (cid:18) y + 12 (cid:19) λ ′′ (cid:18) y + 12 (cid:19) × E ν y +1 / (cid:20) ¯ h ( η ) (cid:18) X | x |≤ n ¯ η ( x ) (cid:19) (cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) + (cid:18) λ ′ (cid:18) y + 12 (cid:19)(cid:19) (5.11) × E ν y +1 / (cid:20) ¯ h ( η ) (cid:18) X | x |≤ n ¯ η ( x ) (cid:19) (cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) − (cid:18) λ ′ (cid:18) y + 12 (cid:19)(cid:19) × E ν y +1 / (cid:20) ¯ h ( η ) (cid:18) X | x |≤ n ¯ η ( x ) (cid:19)(cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) × E ν y +1 / (cid:20)(cid:18) X | x |≤ n ¯ η ( x ) (cid:19) (cid:12)(cid:12)(cid:12) η ( − n −
1) = a, η ( n + 1) = b (cid:21) . P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN
Step
6. By the exponentially mixing property of ν / and the assumption ϕ f (1 /
2) = ϕ ′ f (1 /
2) = 0 [cf. (2.4)], we have λ ′ (1 / E ν / (cid:20) f ( η ) (cid:18) X | x |≤ n χ η ( x ) − / (cid:19) (cid:21) = ϕ ′′ f (1 /
2) + O ( n − / ) . Also, note the relation λ ′ (1 / σ (1 /
2) = 1 (cf. Section 2.5), and by ex-ponential mixing that | σ (1 / − σ n (1 / | = O ( n − ). Recall the asymptoticbehaviors of κ , κ and κ (cf. step 5 of proof of Proposition 5.1). In ad-dition, a factor n χ y is of order O ( n − (1 / − χ ) ) in L ( ν / ). Hence, with theparameter χ chosen small enough, dominant terms may be gathered, as donein the proof of Proposition 5.1, to obtain for all large n that (cid:13)(cid:13)(cid:13)(cid:13) | y | ≤ δ ) (cid:18) E ν / [ f | y, η ( − n −
1) = a, η ( n + 1) = b ] − ϕ ′′ f (1 / (cid:20) y − σ n (1 / n + 1 (cid:21)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν / ) ≤ C k f k L ∞ n − / ε . On the other hand, large deviation estimates yield (cid:13)(cid:13)(cid:13)(cid:13) | y | > δ ) (cid:18) E ν / [ f | y, η ( − n −
1) = a, η ( n + 1) = b ] − ϕ ′′ f (1 / (cid:20) y − σ n (1 / n + 1 (cid:21)(cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν / ) ≤ C k f k L ∞ n − / . (cid:3) Acknowledgments.
P. Gon¸calves and M. Jara are grateful to Capes(Brazil) and FCT (Portugal) for the research project “Non-Equilibrium Sta-tistical Mechanics of Stochastic Lattice Systems” no. FCT 291/11. Also,P. Gon¸calves thanks the Research Centre of Mathematics, University ofMinho, and “Feder” for support of the “Programa Operacional Factores deCompetitividade—COMPETE.” Thanks also to the Editors and referees fortheir constructive comments. REFERENCES [1]
Amir, G. , Corwin, I. and
Quastel, J. (2011). Probability distribution of the freeenergy of the continuum directed random polymer in 1 + 1 dimensions.
Comm.Pure Appl. Math. [2] Andjel, E. D. (1982). Invariant measures for the zero range processes.
Ann. Probab. Assing, S. (2011). A rigorous equation for the Cole–Hopf solution of the conservativeKPZ dynamics. Available at arXiv:1109.2886.[4]
Assing, S. (2002). A pregenerator for Burgers equation forced by conservative noise.
Comm. Math. Phys.
Assing, S. (2007). A limit theorem for quadratic fluctuations in symmetric simpleexclusion.
Stochastic Process. Appl.
Baik, J. , Deift, P. and
Johansson, K. (1999). On the distribution of the lengthof the longest increasing subsequence of random permutations.
J. Amer. Math.Soc. Baik, J. and
Rains, E. M. (2000). Limiting distributions for a polynuclear growthmodel with external sources.
J. Stat. Phys.
Bal´azs, M. , Quastel, J. and
Sepp¨al¨ainen, T. (2011). Fluctuation exponent of theKPZ/stochastic Burgers equation.
J. Amer. Math. Soc. Bal´azs, M. , Rassoul-Agha, F. and
Sepp¨al¨ainen, T. (2006). The random averageprocess and random walk in a space–time random environment in one dimension.
Comm. Math. Phys.
Bal´azs, M. and
Sepp¨al¨ainen, T. (2009). Fluctuation bounds for the asymmet-ric simple exclusion process.
ALEA Lat. Am. J. Probab. Math. Stat. Bal´azs, M. and
Sepp¨al¨ainen, T. (2010). Order of current variance and diffusivityin the asymmetric simple exclusion process.
Ann. of Math. (2)
Baxter, R. J. (1982).
Exactly Solved Models in Statistical Mechanics . AcademicPress, London. MR0690578[13]
Bertini, L. and
Cancrini, N. (1995). The stochastic heat equation: Feynman–Kacformula and intermittence.
J. Stat. Phys. Bertini, L. and
Giacomin, G. (1997). Stochastic Burgers and KPZ equations fromparticle systems.
Comm. Math. Phys.
Borodin, A. and
Corwin, I. (2012). Macdonald processes. Preprint. Available atarXiv:1111.4408.[16]
Borodin, A. , Corwin, I. and
Sasamoto, T.
Duality to determinants for q-TASEPand ASEP. Preprint. Available at arXiv:1207.5035.[17]
Borodin, A. , Ferrari, P. L. , Pr¨ahofer, M. and
Sasamoto, T. (2007). Fluctua-tion properties of the TASEP with periodic initial configuration.
J. Stat. Phys.
Brox, T. and
Rost, H. (1984). Equilibrium fluctuations of stochastic particle sys-tems: The role of conserved quantities.
Ann. Probab. Corwin, I. (2012). The Kardar–Parisi–Zhang equation and universality class.
Ran-dom Matrices Theory Appl. Dittrich, P. and
G¨artner, J. (1991). A central limit theorem for the weakly asym-metric simple exclusion process.
Math. Nachr.
Ethier, S. N. and
Kurtz, T. G. (1986).
Markov Processes: Characterization andConvergence . Wiley, New York. MR0838085[22]
Ferrari, P. A. and
Fontes, L. R. G. (1994). Current fluctuations for the asym-metric simple exclusion process.
Ann. Probab. P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN[23]
Ferrari, P. A. , Presutti, E. and
Vares, M. E. (1988). Nonequilibrium fluctu-ations for a zero range process.
Ann. Inst. Henri Poincar´e Probab. Stat. Ferrari, P. L. and
Spohn, H. (2006). Scaling limit for the space–time covariance ofthe stationary totally asymmetric simple exclusion process.
Comm. Math. Phys.
G¨artner, J. (1988). Convergence towards Burgers’ equation and propagation ofchaos for weakly asymmetric exclusion processes.
Stochastic Process. Appl. Gonc¸alves, P. (2008). Central limit theorem for a tagged particle in asymmetricsimple exclusion.
Stochastic Process. Appl.
Gonc¸alves, P. and
Jara, M. (2010). Universality of KPZ equation. Available atarXiv:1003.4478.[28]
Gonc¸alves, P. and
Jara, M. (2012). Crossover to the KPZ equation.
Ann. HenriPoincar´e Gonc¸alves, P. , Landim, C. and
Toninelli, C. (2009). Hydrodynamic limit for aparticle system with degenerate rates.
Ann. Inst. Henri Poincar´e Probab. Stat. Hairer, M. (2013). Solving the KPZ equation.
Ann. of Math. (2)
Jacod, J. and
Shiryaev, A. N. (2003).
Limit Theorems for Stochastic Processes ,2nd ed.
Grundlehren der Mathematischen Wissenschaften . Springer, Berlin.MR1943877[32]
Jara, M. D. and
Landim, C. (2006). Nonequilibrium central limit theorem fora tagged particle in symmetric simple exclusion.
Ann. Inst. Henri Poincar´eProbab. Stat. Kardar, M. , Parisi, G. and
Zhang, Y. C. (1986). Dynamic scaling of growinginterfaces.
Phys. Rev. Lett. Kipnis, C. and
Landim, C. (1999).
Scaling Limits of Interacting Particle Sys-tems . Grundlehren der Mathematischen Wissenschaften . Springer, Berlin.MR1707314[35]
Kipnis, C. and
Varadhan, S. R. S. (1986). Central limit theorem for additivefunctionals of reversible Markov processes and applications to simple exclusions.
Comm. Math. Phys.
Kolmogorov, A. N. (1962). A local limit theorem for Markov chains. In
Select.Transl. Math. Statist. and Probability, Vol. 2
Kumar, R. (2011). Current fluctuations for independent random walks in multipledimensions.
J. Theoret. Probab. Landim, C. , Sethuraman, S. and
Varadhan, S. (1996). Spectral gap for zero-rangedynamics.
Ann. Probab. Lee, E. (2010). Distribution of a particle’s position in the ASEP with the alternatinginitial condition.
J. Stat. Phys.
Liggett, T. M. (1985).
Interacting Particle Systems . Grundlehren der Mathematis-chen Wissenschaften . Springer, New York. MR0776231[41]
Lu, S. L. and
Yau, H.-T. (1993). Spectral gap and logarithmic Sobolev inequality forKawasaki and Glauber dynamics.
Comm. Math. Phys. [42] Mitoma, I. (1983). Tightness of probabilities on C ([0 , Y ′ ) and D ([0 , Y ′ ). Ann.Probab. Morris, B. (2006). Spectral gap for the zero range process with constant rate.
Ann.Probab. Mueller, C. (1991). On the support of solutions to the heat equation with noise.
Stochastics Stochastics Rep. Nagahata, Y. (2010). Spectral gap for zero-range processes with jump rate g ( x ) = x γ . Stochastic Process. Appl.
Pr¨ahofer, M. and
Spohn, H. (2002). Current fluctuations for the totally asymmet-ric simple exclusion process. In
In and Out of Equilibrium (Mambucaba, 2000) ( V. Sidoraviˇcius , ed.).
Progress in Probability Quastel, J. (1992). Diffusion of color in the simple exclusion process.
Comm. PureAppl. Math. Quastel, J. and
Remenik, D. (2011). Local Brownian property of the narrowwedge solution of the KPZ equation.
Electron. Commun. Probab. Quastel, J. and
Valko, B. (2007). t / superdiffusivity of finite-range asymmetricexclusion processes on Z . Comm. Math. Phys.
Ravishankar, K. (1992). Fluctuations from the hydrodynamical limit for the sym-metric simple exclusion in Z d . Stochastic Process. Appl. Rost, H. and
Vares, M. E. (1985). Hydrodynamics of a one-dimensional near-est neighbor model. In
Particle Systems, Random Media and Large Deviations(Brunswick, Maine, 1984) . Contemp. Math. Sasamoto, T. and
Spohn, H. (2010). One-dimensional KPZ equation: An exactsolution and its universality.
Phys. Rev. Lett.
Sasamoto, T. and
Spohn, H. (2010). The crossover regime for the weakly asym-metric simple exclusion process.
J. Stat. Phys.
Sepp¨al¨ainen, T. (2005). Second-order fluctuations and current across characteristicfor a one-dimensional growth model of independent random walks.
Ann. Probab. Sethuraman, S. (2001). On extremal measures for conservative particle systems.
Ann. Inst. Henri Poincar´e Probab. Stat. Sethuraman, S. and
Xu, L. (1996). A central limit theorem for reversible exclusionand zero-range particle systems.
Ann. Probab. Spohn, H. (1991).
Large Scale Dynamics of Interacting Particles . Springer, Berlin.[58]
Tracy, C. A. and
Widom, H. (2011). Formulas and asymptotics for the asymmetricsimple exclusion process.
Math. Phys. Anal. Geom. van Beijeren, H. , Kutner, R. and
Spohn, H. (1985). Excess noise for drivendiffusive systems.
Phys. Rev. Lett. Varadhan, S. R. S. (2001).
Probability Theory . Courant Lecture Notes in Math-ematics . New York Univ. Courant Institute of Mathematical Sciences, NewYork. MR1852999[61] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In ´Ecole D’´et´e de Probabilit´es de Saint-Flour, XIV—1984 . Lecture Notes in Math. P. GONC¸ ALVES, M. JARA AND S. SETHURAMAN[62]
Yin, M. (2013). A Markov chain approach to renormalization group transformations.
Phys. A
P. Gonc¸alvesDepartamento de MatemiticaUC-RIORua Marquzs de Sco Vicenteno. 225, 22453-900Givea, Rio de JaneiroBrazilandCMATCentro de Matem´aticaUniversidade do MinhoCampus de Gualtar4710-057 BragaPortugalE-mail: [email protected]
M. JaraIMPAEstrada Dona Castorina, 110Horto, Rio de JaneiroBrasilE-mail: [email protected]