aa r X i v : . [ m a t h . P R ] F e b Non-Stationary KPZ equationfrom ASEP with slow bonds
Kevin Yang
Stanford University A BSTRACT . We make progress on the weak-universality conjecture for the KPZ equation by proving the height functions associatedto a class of non-integrable and non-stationary generalizations of ASEP converge to the solution of the appropriate KPZ stochasticPDE. The models considered herein are variations on ASEP with a mesoscopic family of slow bonds, thus our results partially extendthose of [ ] to the non-stationary regime and add to the almost empty set of non-integrable and non-stationary interacting particlesystems for which weak KPZ universality has been established. Technically, in this current paper we develop further the dynamicalstrategy from [ ] and [ ] through spatial localization and combination with the two-blocks scheme of [ ] and [ ] with hopestowards weak KPZ universality in more generality. C ONTENTS
1. Introduction 12. Microscopic Stochastic Heat Equation 63. Heat Kernel Estimates 74. Local Probabilistic Estimates 165. Pathwise Comparison 286. Proof of Theorem 1.4 35Appendix A. Stochastic Continuity Lemma 35Appendix B. Technical Lemma for Tightness 36Appendix C. Index for Notation 37References 371. I
NTRODUCTION
The problem or conjecture of weak-universality for the Kardar-Parisi-Zhang equation, which we henceforth abbreviate asthe KPZ equation, proposes that for a large class of dynamic fluctuating interface models that exhibit some local smoothing,uncorrelated noise, and lateral growth mechanisms, with respect to the appropriate weak-type scaling, large-scale statisticsare equal to those of the KPZ equation, which is the following stochastic PDE on R > × R derived within [ ] in the physicsliterature for exactly the above purpose through non-rigorous RG methods; below, ξ is a space-time Gaussian white noise: ∂ T h = ∂ X h − | ∂ X h | + ξ . (1.1)The focus of the current article is the aforementioned weak-universality conjecture within the context of discrete latticemodels, and interacting particle systems in particular. To provide a brief context, in the seminal article of Bertini-Giacominof [ ] , the authors considered ASEP, which is an abbreviation for the asymmetric simple exclusion process. The fluctuationsof the canonically associated height function are shown therein to exhibit KPZ equation statistics at large space-time scales,providing the first step towards the weak-universality conjecture. However, analysis of ASEP performed within [ ] cruciallydepends on intrinsic algebraic structure of ASEP, and therefore fails to generalize beyond the very small set of models whichare referred to as partially solvable or integrable . Other algebraically rich models are studied in [ ] , [ ] , [ ] , [ ] .Provided our attention is dedicated towards universality, we are primarily interested in those models which fall outsidethe aforementioned class of algebraically rich systems. One first partial step towards analyzing these non-algebraic modelswas provided in [ ] , within which the main results are identical to that of [ ] although for non-simple variations on ASEP.This step was almost completed in [ ] , for which the step distribution for the underlying random walk for the interactingparticle system was extended to infinite-range. The procedure implemented in [ ] incorporated an input from the theoryof hydrodynamic limits in addition to the algebraic procedure within [ ] ; the procedure implemented in [ ] included yetan additional input from the theory of fluctuations of hydrodynamic limits. In particular, for each additional step, anothergeneral strategy and analysis was introduced. efore we provide some more context and necessary details, the main result within the current article is, again, identicalto that of [ ] , though for a generalization of ASEP which includes a family of slow bonds supported on a mesoscopic block,a model which also falls outside the class of algebraically rich systems and is a member of a family of particle systems thathas recently been attracting significant attention. We shortly comment on this and on the innovations of this article.1.1. Background.
The weak-universality conjecture considered in this article is of current significant interest to the prob-ability, stochastic PDE, and mathematical physics communities, effectively because of the singular quadratic nonlinearity.More precisely, the solution to the linearization of (1.1) exhibits the local behavior like a Brownian motion on micro-scales,so the derivative does not exist in any classical sense, thus making the quadratic nonlinearity ill-defined. However, in [ ] ,Hairer provided a robust theory of solutions for the KPZ equation through the theory of regularity structures, and this waslater developed in [ ] in significant generality; moreover, the output of such a theory agrees with whatever solution of theKPZ equation is adopted from [ ] , therefore providing the physical justification for the output of Hairer’s theory. However,the approach is currently restricted to compact spaces, and many systems are of most interest on infinite lattices.Unfortunately, to the author’s knowledge, although this approach provides some direct interpretation of the problematicnonlinearity from (1.1), Hairer’s theory of regularity structures has not yet been applied to the weak-universality conjecturefor the KPZ equation in the context of interacting particle systems courtesy of a number of obstructions. Another approachto remedy the inapplicability to this problem for interacting particle systems was genuinely initiated in [ ] , within which anonlinear martingale problem formulation of the KPZ equation was established for density fluctuations, or "height functiongradients", for a wide class of interacting particle systems given by ASEP with environment-dependent speed-changes thatsatisfy a structural gradient condition. However, as indicated by the last constraint, this approach, known as energy solutiontheory and whose martingale problem analysis was completed by Gubinelli-Perkowski in [ ] , crucially depends upon themodel at hand admitting an explicit, reasonable, and rapidly decorrelating invariant measure; moreover, the model is alsonecessarily assumed to start at or very close to the invariant measures; see [ ] , for example. Comparing these constraintsto the previous examples of models for which the weak-universality conjecture was verified, these aforementioned systemsconsidered in each of [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , and [ ] were assumed to begin quite far from the relevant invariantmeasure. Moreover, within [ ] and [ ] , the problem of understanding the invariant measure itself is already challenging.We introduce yet another article in [ ] to conclude the current discussion; the model studied therein does not exhibit therich algebra like the partially integrable models whose analyses completely rely on, and effectively nothing about invariantmeasures is known explicitly. A valid high-level interpretation for the main result within this article is another step towardsweak KPZ universality for non-solvable and non-stationary systems which does not seriously depend on understanding anyinvariant measures; the single result to the author’s knowledge of such type is the result of the aforementioned article [ ] .For this paper, we will instead adopt for the KPZ equation the following Cole-Hopf solution developed in [ ] . • We define the solution of the KPZ equation (1.1) to be h T , X = − log Z T , X , and Z denotes the solution to the followingstochastic PDE which we refer to as the stochastic heat equation, or SHE for an abbreviation, on R > × R : ∂ T Z = ∂ X Z + Z · ξ . (1.2) • The SHE (1.2) admits a solution which is continuous in space-time with probability 1 as courtesy of the Ito calculus;moreover, the solution via this aforementioned Ito calculus is positive with probability 1 provided the initial datais non-zero and non-negative as proved by Mueller in [ ] . This justifies the Cole-Hopf transform defining h .Adopting the Cole-Hopf solution theory introduced above for the KPZ equation, similar to [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , and [ ] , verification of weak-universality for systems considered therein and this article amounts to implementing atype of exponential transform for the microscopic height function, or more precisely its fluctuations, and then establishingthe convergence for such microscopic Cole-Hopf transform to the solution of the SHE in (1.2). What then makes the particlesystem partially integrable is miraculously the dynamics of the microscopic Cole-Hopf transform match a microscopic copyof the SHE itself. For models which are not partially integrable, dynamics of the microscopic Cole-Hopf transform are equalto a microscopic copy of the SHE although with additional error terms, at least if holding a belief in weak KPZ universality,which are not obviously error terms generically and thus require tailor-made analysis; in particular, this yields approximate microscopic Cole-Hopf transforms in general when beyond integrability. We emphasize that approximate microscopic Cole-Hopf transforms are certainly not unique as we illustrate later for the models studied herein, and a main part of the problemof weak KPZ universality seems to be in finding and working with an appropriately accessible choice of transform.We conclude this introductory section by reemphasizing and reiterating that the main goal for this article is to establishthe weak-universality conjecture for variations on ASEP with a mesoscopic family of slow bonds; the analysis we apply doesnot seriously depend on the model exhibiting explicit invariant measures, and moreover, the models herein are genericallynot partially-integrable, therefore adding to the almost empty set of models for which this is done. Concerning our motivation, we first remark that for one individual slow bond, the stationary version of our resultwas established in [ ] with the aforementioned theory of energy solutions. Moreover, trivial absence of any slowbonds specializes the models discussed in this article to the original model ASEP within [ ] . We thus expand uponresults in [ ] to the non-stationary regime and also recover the result in [ ] . • Moreover, for some specialization for the strength of the assumed individual slow bond, the non-stationary versionof this previous result in [ ] was established in the article [ ] , though after complete removal of the "asymmetry"within this model. The large-scale statistics are therefore equal to those of the linear infinite-dimensional Ornstein-Uhlenbeck process for those models which is non-singular, well-posed, and emphatically not the KPZ equation.In wider perspective, exclusion processes with slow bonds are prevalent in statistical physics and probabilistic communities;see [ ] , [ ] , [ ] , [ ] , [ ] for just a couple of such papers, at least the first two of which keep an eye towards KPZ statistics.Because the content of these papers is not directly related to the content herein, we only briefly remark that a lesson learnedwithin the above articles is that the presence of any individual slow bond already significantly affects microscopic statisticsof the original unperturbed model. Weak KPZ universality is thus a homogenization statement with respect to appropriatespace-time scaling. In view of this last sentence, we conclude by citing [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] which studyother non-KPZ aspects of homogenization theory for these models, thus opening the door for studying KPZ-related aspects.Beyond [ ] , which again specializes to stationary models, this KPZ-relevant literature is almost if not totally empty.Therefore, in view of the above discussion, the problem considered in the current article addresses a gap in the literatureon the homogenization and fluctuation theory for variations on ASEP with a family of slow bonds and a gap in the literatureconcerning the weak-universality of the KPZ equation among non-integrable and non-stationary models discussed before.An interesting feature, from where our analysis begins, of the interacting particle systems considered within this paper isa two-fold approximate microscopic Cole-Hopf transform. In particular, these models, or more precisely the correspondingheight functions, admit two distinct approximate microscopic Cole-Hopf transforms; the first is essentially employed within [ ] and "more obvious and natural" but problematic for reasons to be detailed later, and the second is novel to this article.We expand on this, as well as the non-integrable nature of the particle systems herein, towards the end of this section.We finish with a short introduction of technical innovations in this paper. The first is spatial localization of the dynamicalanalysis introduced within [ ] to analyze spatially local fluctuations of systems with hopes towards weak KPZ universalityin general settings when the invariant measures may not admit any obvious symmetries, for example. A potentially robusttool we obtain as consequence is a spatially local version of the classical one-block estimate in [ ] . The second innovationwe develop are heat kernel estimates for a near-non-elliptic, non-divergence-form operator possibly of external interest.1.2. The Model.
Throughout the rest of this article, we will denote by N ∈ Z > the underlying scaling parameter. We firstintroduce the interacting particle systems of interest with the following prescription as a Markov process. • Provided any sub-lattice I ⊆ Z , let us define Ω I • = {± } I as the space of particle configurations on the sub-latticeat hand. Elements of Ω will be denoted by η ∈ Ω , and for any x ∈ Z , we denote by η x ∈ {± } the particle number;the interpretation is that η x = x ∈ Z , and otherwise η x = −
1. We lastly define Ω • = Ω Z . • We observe that any inclusion of sub-lattices I ⊆ I induces the canonical projection Ω I → Ω I in the "opposite"direction. In particular, when we specify any particle configuration, it does not depend on which Ω I -space we arecurrently working with. Notationally, we let Π I denote the pushforward map on probability measures induced bythe map Ω → Ω I for convenience, where I ⊆ Z is a generic sub-lattice. • To introduce precisely the infinitesimal generator of the model, we introduce the following preliminary data. – First, we consider the slow-bond parameter β ⋆ ∈ R > . We always require that β ⋆ ∈ R > satisfies 0 β ⋆ ,although for this discussion, we assume β ⋆ ∈ R > is arbitrarily and sufficiently small but still universal. – Provided the parameter β ⋆ ∈ R > , we additionally introduce the parameter ǫ ⋆ ∈ R > satisfying ǫ ⋆ > β ⋆ + ǫ with ǫ ∈ R > here any arbitrarily small although universal constant, and then we afterwards define the sub-lattice I ⋆ • = J N − ǫ ⋆ K ⊆ Z . Finally, we define I ⋆ ,2 • = { ⋆ j } N ǫ⋆ ,2 j = ⊆ I ⋆ to be any generic though fixed set, where ǫ ⋆ ,2 β ⋆ . We emphasize these constants 99 and are arbitrary to some extent. – Provided x , y ∈ Z , define L x , y as the symmetric Kawasaki spin-exchange dynamic for the bond { x , y } .The infinitesimal generator L N ,!! : L ( Ω ) → L ( Ω ) for the interacting particle system of interest here is definedby L N ,!! ϕ • = L N ,!!Sym ϕ + L N ,!! ∧ ϕ for any generic test function ϕ ∈ L ( Ω ) , where L N ,!!Sym ϕ : η • N X x I ⋆ ,2 L x , x + ϕ + N − β ⋆ X x ∈ I ⋆ ,2 L x , x + ϕ ; (1.3a) L N ,!! ∧ ϕ : η • N X x ∈ Z − η x + η x + L x , x + ϕ . (1.3b) oreover, provided any T ∈ R > , we denote with η NT the particle configuration obtained through time- T evolutionof this Markov process with initial configuration η N ∈ Ω . We let F • denote the canonical filtration. Remark . Although we will assume β ⋆ ∈ R > is sufficiently small, it is not the case that the model defined above dependscontinuously on β ⋆ ∈ R > uniformly in the scaling parameter N ∈ Z > at the microscopic or even generic mesoscopic scalenecessarily. Indeed, the challenges we overcome in this article are essentially to show that the model exhibits this uniformcontinuity-type property for β ⋆ ∈ R > sufficiently small though still independent of the scaling parameter N ∈ Z > .We additionally provide the following colloquial description of the interacting particle systems of interest in this paper. • Initially, every point in the one-dimensional lattice Z ⊆ R contains at most one particle. • The particles perform independent asymmetric simple random walks subject to the following exclusion principle– any attempted jump onto an already occupied point is suppressed. • The step distribution of the aforementioned asymmetric simple random walk is a symmetric simple random walkof a constant universal rate with additional asymmetry of rate N − , where N ∈ Z > is the scaling parameter fromwhich we see Kardar-Parisi-Zhang equation statistics at large scales N → ∞ . • At the present time, we have defined the classical ASEP model; the presence of the slow bonds slow the symmetriccomponent of the aforementioned random walk in the previous bullet point by the factor of N − β ⋆ , where β ⋆ ∈ R > is a universal underlying parameter. The asymmetric components of the random walk are completely unchanged.Having introduced the relevant particle system, we proceed to introduce the canonical height function of relevant interestcombined with a suitable microscopic version of the Cole-Hopf transform. Definition 1.2.
Provided T ∈ R > , we first define h NT ,0 as the net flux of particles crossing the origin where we adopt theconvention that leftward motion counts as positive flux. Moreover, we define, for x ∈ Z , h NT , x • = N − h NT ,0 + N − X y x η NT , y ; (1.4a) Z NT , x • = e − h NT , x + ν N T , (1.4b)where ν N • = N − . Moreover, we linearly interpolate Z NT , • to construct a function Z NT , • ∈ C ( R ) .1.3. Main Results.
We start the current presentation of the main results with the following classes of relevant initial data.
Definition 1.3.
We say any probability measure µ N on the state space Ω = Ω Z is a near-stationary probability measure or near-stationary initial data if provided any p ∈ R > along with any sufficiently large though universal κ ∈ R > and u ∈ R > satisfying 0 u < , we have the following estimates:sup x ∈ Z e − κ | x | N (cid:13)(cid:13)(cid:13) Z N x (cid:13)(cid:13)(cid:13) L p ω . κ , p
1; (1.5)sup x , y ∈ Z e − κ | x | + | y | N (cid:13)(cid:13)(cid:13) Z N x − Z N y (cid:13)(cid:13)(cid:13) L p ω . κ , p , u N − u | x − y | − u . (1.6)Additionally, we require the existence of a function Z ∞ ∈ C ( R ) such that lim N →∞ Z N N • = Z ∞ locally uniformly.We define the narrow-wedge initial data as the probability measure µ NW on Ω = Ω Z supported on η • = • > − • < .Before introducing the main result, we recall the following Skorokhod topological spaces of [ ] for T f ∈ R > ∪ { + ∞} : D T f • = D ([ T f ] , C ( R )) ; (1.7a) D ◦ T f • = D (( T f ] , C ( R )) . (1.7b) Theorem 1.4.
Assume β ⋆ ∈ R > is sufficiently small but universal; we will consider two regimes concerning initial data. • Assume first near-stationary initial data; this sequence of random fields Z N • , N • converges to the unique solution Z ∞ of SHE with initial data Z ∞ • in the Skorokhod space D . • Assume the narrow-wedge initial data and consider the rescaled field of ¯ Z N • = N Z N . This sequence of random fields ¯ Z N • , N • converges to the unique solution Z ∞ of SHE with initial data of the Dirac point mass δ supported at the originin the Skorokhod space D ◦ . We conclude this subsection with the following commentary. • We first emphasize Theorem 1.4 is not perturbative of [ ] , as will be shown shortly; the non-integrable feature ofthe particle system of interest is manifest in some quantity which does not vanish as β ⋆ → N ∈ Z > ,so that the model herein is "genuinely non-integrable". Similarly to [ ] , we specialize to the regime of near-stationary initial data, as the regime of narrow-wedge initialdata follows from the adaptation performed in detail in [ ] almost identically; the extension to the latter singularinitial data therefore effectively constitutes an exercise provided our analysis for near-stationary initial data. • We reemphasize the specialization of our model to a single slow bond supported at a fixed point is the KPZ-relatedcontent of [ ] with a larger set bond-strengths considered in that article but restricting to the stationary models. • Courtesy of Theorem 1.4, under appropriate "double-limit" procedure, statistics of the height function h N convergeto those of the so-called KPZ fixed point, as a consequence of the main result within [ ] . • Understanding the invariant measures will not play a substantial role in our analysis. Thus, we allow small thoughmicroscopically detectable perturbations of the slow bonds, and our analysis would be almost entirely unaffected. • In addition to the slow bond problems in [ ] , there is recent interest in hydrodynamics theory and correspondingfluctuations for generalizations of ASEP though with a slow-boundary dynamic; see [ ] , [ ] , [ ] , and [ ] . Webelieve that large-scale KPZ statistics are within reach with the methods developed herein for appropriate weakly-asymmetric versions of at least some of these models in the non-integrable and non-stationary regimes as well.1.4. Additional Commentary.
Rather than provide any complete outline to the proof of Theorem 1.4, we instead providea few illustrations to just become familiar with the problem of proving Theorem 1.4 in the context of the papers [ ] , [ ] , [ ] , [ ] , and [ ] . This begins with the next preliminary observation, which we will not actually depend upon or employand therefore is stated without any real proof beyond the brief commentary that it follows via elementary calculations.For simplicity, we first assume there is exactly one slow bond at 0 ∈ Z ; below, d B ≈ ξ as noise quantities;d h NT , x ≈ ∆ h NT , x d T − |∇ h T , x | d T + d B T , x + ν N d T + x = N g NT ,0 d T + x = N f NT ,0 d T ; (1.8)roughly speaking, the quantity g NT ,0 is a local fluctuation, and f NT ,0 vanishes in expectation with respect to a relevant invariantmeasure of the model at hand. In particular, the energy solution theory from [ ] suggests the latter two quantities withinthe RHS of (1.8) vanish in the large- N limit, and thus the renormalized height function h NT , x − ν N T converges to the solutionof the KPZ equation. Interpreting the gradient nonlinearity appropriately, this is what is done in [ ] , again through energysolution theory, and thus requires the model to exactly be initially distributed according to a relevant invariant measure.The procedure we adopt in the current article is to attack the problem of establishing Theorem 1.4 through a microscopicvariation of the Cole-Hopf transform introduced above, similar to the aforementioned articles of [ ] , [ ] , [ ] , and [ ] .The problem with this approach is that the latter two error-type terms within the RHS of (1.8) are manifest in the stochasticdynamics of the aforementioned microscopic Cole-Hopf transform.The error term g N therein, in particular, is currently inaccessible unless the system is initially distributed according to anyprobability measure quite close to some appropriate invariant measure, and generally for this article we will avoid assumingsuch a constraint. To avoid this obstruction, we actually match the stochastic differential for h N with the action of a slightlyperturbed almost-non-elliptic operator; before we proceed and write down this matching precisely, we emphasize that thiscurrent model of ASEP with a slow bond actually exhibits two distinct approximate microscopic Cole-Hopf transforms, firstin (1.8) as suggested by [ ] and another in the following equation, which is actually one main novelty of this article:d h NT , x ≈ a N , x ∆ h NT , x d T − |∇ h NT , x | d T + d B T , x + ν N d T + x = N f NT ,0 d T ; (1.9)above, we have introduced the inhomogeneity a N , • • = •6∈ I ⋆ ,2 + N − β ⋆ •∈ I ⋆ ,2 . Comparing the two equations of (1.8) and (1.9),we have swapped the problematic error term within the former equation for the slightly non-elliptic approximation for thecontinuum Laplacian; ultimately, we exchange a problematic probabilistic problem into some manageable analytic problemof heat kernels, which is actually accessible for β ⋆ ∈ R > sufficiently small although still universal, with some considerableamount of analysis at least. Moreover, studying the remaining error quantity of f N additionally requires analysis in its ownright, in particular because the lack of spatial averaging of this error obstructs the hydrodynamic approach of [ ] . Maybethis is an appropriate moment to emphasize that f N does not vanish in the limit β ⋆ → N ∈ Z > , so the modelis not perturbative from the integrable system which would intuitively be the ASEP within [ ] ; something probabilisticallyinteresting is going on for Theorem 1.4 to hold which our analysis provides some insight on. At a high-level, the statistics of f N are ultimately accessed through its mesoscopic dynamic fluctuations and global density statistics of the particle system.Technically, we employ a two-step approach which consists of localization of the dynamical one-block scheme from [ ] and [ ] with a mesoscopic probabilistic estimate we deduce from a stochastic-analytic estimate for the height function h N and the classical two-blocks estimate in [ ] . This, as above, is done at the level of the microscopic Cole-Hopf transform.A particular takeaway from this impressionistic discussion is that our proof for Theorem 1.4 transfers some probabilisticproblem, which is currently inaccessible outside the specialization within which the model is initially distributed accordingto an appropriate invariant measure, to an analytic problem and an accessible probabilistic problem. .5. Outline.
In Section 2, we compute the dynamics for the microscopic Cole-Hopf transform and approximately match itwith a microscopic stochastic heat equation. In Section 3, we establish heat kernel estimates for the relevant asymptoticallynon-parabolic operator introduced above. In Section 4, we develop the local dynamical analysis of [ ] and [ ] combinedwith considerations in spirit of the two-blocks scheme within [ ] and [ ] . In Section 5 and 6, we then establish Theorem1.4 by a pathwise argument and the analysis from [ ] and [ ] , respectively.1.6. Acknowledgements.
The author thanks Amir Dembo for advice, guidance, and useful discussion.The author and this work are funded by a fellowship from the Northern California chapter of the ARCS Foundation.2. M
ICROSCOPIC S TOCHASTIC H EAT E QUATION
The proof of Theorem 1.4 begins with some microscopic approximation of the SHE dynamic for the stochastic evolutionof the microscopic Cole-Hopf transform.
Proposition 2.1.
Provided any T ∈ R > and x ∈ Z , we have d Z NT , x = a N , x ∆ !! x Z NT , x d T + Z NT , x d ξ NT , x + c N x ∈ I ⋆ ,2 Q NT , x Z NT , x d T , (2.1) where we have introduced the following quantities. • We define the static field a N , • • = •6∈ I ⋆ ,2 + N − β ⋆ •∈ I ⋆ ,2 . We also have c N • = N − N − β ⋆ . • The differential martingale d ξ NT , x is defined via defining ξ NT , x as the compensation of the Poisson process correspondingto the Poisson clocks for particle-jumps for both directions at space-time coordinate ( T , x ) ∈ R > × Z . • We have defined the random field of Q N • , x • = q N • , x + N − e q N • , x , where q N • , x • = η N • , x η N • , x + and e q N • , x : Ω → R is a uniformlybounded functional.Proof. Provided x I ⋆ ,2 , the result of Proposition 2.1 follows via the calculation employed to establish (3.13) in [ ] . Thus,it suffices to assume x ∈ I ⋆ ,2 . We proceed with the following direct calculation. • An explicit calculation with the underlying Poisson clocks analogous to the proof for (3.13) in [ ] and Proposition2.2 in [ ] provides the following stochastic differential equation for Z N without the N -factor for convenience: N − d Z NT , x = Φ Sym T , x Z NT , x d T + Φ ∧ T , x Z NT , x d T + ν N Z NT , x d T + Z NT , x d ξ NT , x , (2.2)where Φ Sym T , x • = N − β ⋆ + η NT , x + − η NT , x (cid:16) e − N − − (cid:17) + N − β ⋆ − η NT , x + + η NT , x (cid:16) e N − − (cid:17) ; (2.3a) Φ ∧ T , x • = N − + η NT , x + − η NT , x (cid:16) e − N − − (cid:17) − N − − η NT , x + + η NT , x (cid:16) e N − − (cid:17) . (2.3b) • Meanwhile, we compute the corresponding action of the discrete differential operator a N , x ∆ !! on the microscopicCole-Hopf transform; we deduce and emphasize the discrete Laplacian is not parabolically rescaled:12 N − β ⋆ ∆ Z NT , x = N − β ⋆ (cid:16) e − N − η NT , x + − (cid:17) Z NT , x + N − β ⋆ (cid:16) e N − η NT , x − (cid:17) Z NT , x . (2.4)Application of Taylor expansion analogous to the proof for Proposition 2.2 in [ ] allows to suitably match the expressionsobtained in the two bullet points above. Though this expansion is a long calculation, it is elementary and follows identicallythe proof for Proposition 2.2 within [ ] ; we briefly remark that this calculation differs from the proof of Proposition 2.2within [ ] in the following minor technical fashion, which is presented below as a back-of-the-envelope calculation. • Considering N -dependence of the terms resulting from Taylor expansion of the exponential from (2.4), only thoseleading-order quantities of order ∼ N − β ⋆ − are relevant; any relevant remaining quantities are bounded uniformlyabove by N − up to universal constants even if they are random. • The previous bullet point above applies additionally to the quantity of Φ Sym T , x within (2.2). Taylor expansion appliedto both of (2.4) and (2.2) then establishes that N Φ Sym T , x Z NT , x and a N ,0 ∆ !! Z NT , x match up to the aforementioned errorsof order at most N − . • It remains to address the term Φ ∧ T , x Z NT , x and the drift term of ν N Z NT , x . We first observe that the presence of the slowbond is absent from both of these "infinitesimal" dynamics; in particular, identical to the proof for (3.13) from [ ] and the proof of Proposition 2.3 from [ ] , this pair of quantities Φ ∧ T , x Z NT , x and ν N Z NT , x mutually cancel each otherout up to the proposed error terms in c N Q N .This completes the proof. (cid:3) . H EAT K ERNEL E STIMATES
Provided the stochastic differential equation in Proposition 2.1, we require estimates for the heat kernel P N associatedto the following parabolic problem, and this provides the purpose for the current section: ∂ T P NS , T , x , y = a N , x ∆ !! x P NS , T , x , y ; (3.1a) P NS , S , x , y = x = y . (3.1b)Above, the understanding is S , T ∈ R > satisfy S T , and we further assume that x , y ∈ Z . Classical parabolic theorythen provides the following alternative PDEs for U N , which we freely apply without explicit reference, in which ∂ ∗ S • = − ∂ S : ∂ T P NS , T , x , y = ∆ !! y • a N , y P NS , T , x , y ˜ ; (3.2a) ∂ ∗ S P NS , T , x , y = a N , x ∆ !! x P NS , T , x , y ; (3.2b) ∂ ∗ S P NS , T , x , y = ∆ !! y • a N , y P NS , T , x , y ˜ . (3.2c)Within this current section, more precisely we are primarily interested in establishing Nash-type off-diagonal heat kernelestimates for this heat kernel P N along with suitable regularity estimates. For convenience, we provide a short outline. • We first establish some preliminaries; these amount to some discrete version of the Nash-Sobolev inequality alongwith a perturbative formula for P N in terms of some auxiliary heat kernel ¯ P N which admits estimates via a spectralanalysis, for example. Precisely, we now introduce this heat kernel through the corresponding parabolic problem: ∂ T ¯ P NS , T , x , y = ∆ !! x ¯ P NS , T , x , y ; (3.3a)¯ P NS , S , x , y = x = y . (3.3b) • The second step within this section consists of establishing sub-optimal although global on-diagonal-type estimatesfor P N through the strategy with Nash inequalities. Equipped with such an on-diagonal-type estimate, we establishthe corresponding off-diagonal estimate outside of a mesoscopic neighborhood of the inhomogeneities within I ⋆ ,2 via comparison with ¯ P N ; this is optimal but depends heavily on the sub-optimal Nash-type on-diagonal estimate. • We emphasize the comparison from the previous bullet point is moreover important in its own right; in particular,at least through this mechanism, we establish important regularity estimates for P N for the proof of Theorem 1.4.3.1. Preliminary Estimates.
As mentioned prior, the first preliminary inequality that we require is some discrete versionof the Nash-Sobolev inequality. The proof requires the heat kernel estimates for ¯ P N in Proposition A.1 of [ ] . Lemma 3.1.
Provided any function ϕ : Z → C , we have the following estimate with universal implied constant: X x ∈ Z | ϕ x | . –X x ∈ Z | ϕ x | ™ –X x ∈ Z |∇ ϕ x | ™ ; (3.4) above, we recall the discrete gradient ∇ k ϕ x • = ϕ x + k − ϕ x provided any k ∈ Z .Proof. We consider any function ϕ : Z → C and then define the heat flow ϕ T , x • = P y ∈ Z ¯ P N T , x , y ϕ y ; in particular, we observethat ϕ T , x solves the same PDE as ¯ P N T , x , y but with initial condition ϕ • = ϕ • . Differentiating then gives, for any T ∈ R > : X x ∈ Z ϕ T , x ϕ x = X x ∈ Z | ϕ x | + Z T X x ∈ Z ∆ !! x ϕ S , x · ϕ x d S (3.5) = X x ∈ Z | ϕ x | − Z T X x ∈ Z ∇ ! + ϕ S , x · ∇ ! + ϕ x d S ; (3.6)we have introduced the rescaled gradient operator ∇ ! • = N ∇ , and indeed the identity (3.6) follows through a summation-by-parts. To take advantage of (3.6), we introduce the following pair of observations. • Courtesy of Proposition A.1 within [ ] , we first establish the following estimate uniformly in space-time: | ϕ T , x | . N − ̺ − T X x ∈ Z | ϕ x | . (3.7) Denote by e T ∆ !! the operator corresponding to the semigroup action with kernel ¯ P N T , x , • , so in particular we have ϕ T , • = e T ∆ !! ϕ . We observe the commutation relation [ e T ∆ !! , ∇ + ] = P N from Appendix A in [ ] or the spectral calculus. As consequence, we have X x ∈ Z (cid:12)(cid:12) ∇ ! + ϕ S , x (cid:12)(cid:12) = X x ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X y ∈ Z ¯ P N S , x , y · ∇ ! + ϕ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.8) X y ∈ Z (cid:12)(cid:12) ∇ ! + ϕ y (cid:12)(cid:12) ; (3.9)indeed, the final estimate (3.9) is consequence of convexity properties of the heat flow operator e T ∆ !! .Combining the previous two observations with (3.6), we obtain the following estimate with a universal implied constant: X x ∈ Z | ϕ x | . N − ̺ − T –X x ∈ Z | ϕ x | ™ + Z T X x ∈ Z (cid:12)(cid:12) ∇ ! + ϕ S , x (cid:12)(cid:12) · (cid:12)(cid:12) ∇ ! + ϕ x (cid:12)(cid:12) d S (3.10) N − ̺ − T –X x ∈ Z | ϕ x | ™ + Z T –X x ∈ Z |∇ ! + ϕ S , x | ™ · –X x ∈ Z |∇ ! + ϕ x | ™ d S (3.11) N − ̺ − T –X x ∈ Z | ϕ x | ™ + N ̺ T X x ∈ Z |∇ + ϕ x | . (3.12)The desired estimate is now immediate consequence of optimizing the upper bound within (3.12) with respect to T ∈ R > ;this completes the proof. (cid:3) The second preliminary ingredient we that will require is actually some immediate consequence of the classical Duhamelprinciple, so we provide only a brief description of the proof.
Lemma 3.2.
Provided any S , T ∈ R > satisfying S T along with any x , y ∈ Z , we have P NS , T , x , y = ¯ P NS , T , x , y − Z TS X w ∈ I ⋆ ,2 P NR , T , x , w · ¯ a N , w ∆ !! ¯ P NS , R , w , y d R ; (3.13) above we have introduced the quantity ¯ a N , • = − a N , • which is supported on the sub-lattice I ⋆ ,2 ⊆ Z .Proof. Apply the fundamental theorem of calculus with respect to the time-coordinate R ∈ R > to the following quantity: Φ R • = X w ∈ Z P NR , T , x , w ¯ P NS , R , w , y . (3.14)This completes the proof after a short and elementary calculation. (cid:3) Nash Estimates I.
The primary estimate for the current subsection is the following sub-optimal on-diagonal estimatefor P N ; we recall that its proof consists of some suitable application of the Nash inequality of within Lemma 3.1 combinedwith the Chapman-Kolmogorov equation. Proposition 3.3.
Provided S , T ∈ R > satisfying S T , we establish the following estimate, within which the implied constantis universal: sup x , y ∈ Z P NS , T , x , y . N − + β ⋆ ̺ − S , T . (3.15a) Proof.
We assume first S =
0; indeed, the heat kernel P N is time-homogeneous, thus such a specialization suffices towardsthe proof for Proposition 3.3. Consequence of the existence and uniqueness properties for those equations defining the heatkernel P N , we observe the following Chapman-Kolmogorov equation for P N along with the estimate appearing afterwardscourtesy of the Cauchy-Schwarz inequality: P N T , x , y = X w ∈ Z P N T , T , x , w P N T , w , y (3.16) –X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , T , x , w (cid:12)(cid:12)(cid:12) a N , w ™ –X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , w , y (cid:12)(cid:12)(cid:12) a − N , w ™ . (3.17) hus, it remains to estimate each factor within the RHS of (3.17); beginning with the first such quantity, we first computeits time-derivative as follows, for which T ′ = T is interpreted as independent of T ∈ R > : ∂ T X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , T , x , w (cid:12)(cid:12)(cid:12) a N , w = X w ∈ Z P N T , T , x , w a N , w · h ∆ !! w (cid:16) P N T , T , x , w a N , w (cid:17)i (3.18) . − X w ∈ Z (cid:12)(cid:12)(cid:12) ∇ ! + (cid:16) P N T , T , x , w a N , w (cid:17)(cid:12)(cid:12)(cid:12) (3.19) . − N –X w ∈ Z P N T , T , x , w a N , w ™ − –X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , T , x , w (cid:12)(cid:12)(cid:12) a N , w ™ . (3.20)To proceed, we introduce the following pair of auxiliary observations. • Observe the Kolmogorov PDEs for P N implies the identities X w ∈ Z P N T , T , x , w = X w ∈ Z P N T , x , w =
1. (3.21)Indeed, both identities hold when the time-coordinates in the heat kernel P N within each summation coincide. Toestablish these identities globally in time, it therefore suffices to employ the following time-derivative calculationfor the first identity for example; the proof of the second identity follows from identical considerations: ∂ τ X w ∈ Z P N T , τ , x , w = X w ∈ Z P T , τ , x , w a N , w ∆ !! w =
0. (3.22)Recalling a N , w . Z , we deduce X w ∈ Z P N T , T , x , w a N , w .
1. (3.23) • Second, in similar fashion though upon realizing a N , w & N − β ⋆ again with a universal implied constant, we deduce X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , T , x , w (cid:12)(cid:12)(cid:12) a N , w & N − β ⋆ X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , T , x , w (cid:12)(cid:12)(cid:12) a N , w . (3.24)Combining the differential inequality (3.20) with (3.23) and (3.24), we deduce the differential estimate ∂ T X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , T , x , w (cid:12)(cid:12)(cid:12) a N , w . − N − β ⋆ –X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , T , x , w (cid:12)(cid:12)(cid:12) a N , w ™ , (3.25)from which integration and standard ODE principles provide the following estimate with a universal implied constant: X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , T , x , w (cid:12)(cid:12)(cid:12) a N , w . N − + β ⋆ ̺ − T . (3.26)It remains to provide a suitable estimate for the second factor within the RHS of (3.17). To this end, we again have ∂ T X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , w , y (cid:12)(cid:12)(cid:12) a − N , w . − X w ∈ Z (cid:12)(cid:12)(cid:12) ∇ ! + P N T , w , y (cid:12)(cid:12)(cid:12) (3.27) . − N –X w ∈ Z P N T , w , y ™ − –X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , w , y (cid:12)(cid:12)(cid:12) ™ . (3.28)As before, we require the following two auxiliary observations. • Observe the Kolmogorov PDEs for P N implies this following estimate: X w ∈ Z P N T , w , y . X w ∈ Z P N T , w , y a − N , w (3.29) = X w ∈ Z P N w , y a − N , w (3.30) . N β ⋆ . (3.31) • Second, in almost identical fashion, we have the following estimate with a universal implied constant as well: X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , w , y (cid:12)(cid:12)(cid:12) & N − β ⋆ X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , w , y (cid:12)(cid:12)(cid:12) a − N , w . (3.32) ombining the differential inequality in (3.28) with the estimates (3.31) and (3.32) within the previous two bullet points,we establish the following differential estimate: ∂ T X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , w , y (cid:12)(cid:12)(cid:12) a − N , w . − N − β ⋆ –X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , w , y (cid:12)(cid:12)(cid:12) a − N , w ™ , (3.33)from which, as with (3.26), we establish the following estimate with a universal implied constant: X w ∈ Z (cid:12)(cid:12)(cid:12) P N T , w , y (cid:12)(cid:12)(cid:12) a − N , w . N − + β ⋆ ̺ − T . (3.34)Upon combining (3.17), (3.26), and (3.34), this completes the proof. (cid:3) As an application of the previous Nash inequality in Proposition 3.3, we establish the following maximum principle forthe heat kernel P N . Lemma 3.4.
Provided S , T ∈ R > satisfying S T , with a universal implied constant, we have sup x , y ∈ Z P NS , T , x , y .
1. (3.35)
Moreover, we have P NS , T , x , y > for all space-time coordinates S , T ∈ R > satisfying S T and x , y ∈ Z .Proof. Courtesy of the initial condition P NS , S , x , y = x = y along with linearity of the PDE defining P N , it suffices to prove thefollowing variational identity: sup T > S sup x , y ∈ Z P NS , T , x , y = sup x , y ∈ Z P NS , S , x , y . (3.36)Observe that the supremum with respect to the time-coordinate within the LHS of (3.36) may be replaced by the supremumover times T ∈ R > satisfying T . N with a universal implied constant. Moreover, such supremum on the LHS of (3.36),if not attained at T = S , is attained on the interior of this interval 0 T . N for identical reason. In particular, if T ∈ R > denotes this time, then we deduce ∂ T P NS , T , x , y = x ∈ Z at which the maximum of P NS , T , • , y is attained.Thus, we have 0 = ∂ T P NS , T , x , y = a N , x ∆ !! x P NS , T , x , y . (3.37)Because x ∈ Z is a maximizer, this implies the identity P NS , T , x , y = P NS , T , x ± y . In particular, we deduce the neighboringpair of points x ± ∈ Z are also maximizers, from which we deduce that the relation (3.37) upon replacing x x ± P N is a multiple of the constant function, from which we deduce that P NS , • , • , y is constantupon applying the reverse heat flow. This is clearly a contradiction to the initial data; this completes the proof. (cid:3) Perturbative Estimates.
One primary goal for this current subsection is the following estimate through a perturbativeanalysis, in which we think of P N as a perturbation of ¯ P N provided we look outside some suitable mesoscopic neighborhoodof the slow bond defects in I ⋆ ⊆ Z .However, for organizational purposes, we first introduce the following glossary of parameters we will employ through-out the remainder of this section, all of which are universal. • First, we recall ǫ ⋆ ∈ R > defined by | I ⋆ | ∼ N − ǫ ⋆ satisfies the constraint ǫ ⋆ > β ⋆ + ǫ provided an arbitrarily smallbut universal constant ǫ ∈ R > ; again, 99 just denotes some large but universal number. • Second, we define β ∂ ,1 • = − ǫ ∂ ,1 , where ǫ ∂ ,1 ∈ R > is defined via ǫ ∂ ,1 • = β ⋆ + δ with δ ∈ R > another arbitrarilysmall but universal constant. We then define I ∂ ,1 • = J − N β ∂ ,1 , N β ∂ ,1 K ⊆ Z . • Third, we define a pair of mesoscopic time-scales τ N • = N − ǫ ∂ ,1 log − N and τ N , ⋆ • = N − + β ⋆ .Finally we introduce the following auxiliary time-differential operator which we employ throughout as well. Definition 3.5.
Provided any function ϕ S , T : R > → C , we define D τ ϕ S , T • = ϕ S , T + τ − ϕ S , T . Moreover, provided insteadany function ψ T : R > → C , define D τ ψ T • = ψ T + τ − ψ T . Proposition 3.6.
There exist universal constants ǫ , ǫ ∈ R > such that provided any S , T ∈ R > satisfying S T , we have e P NS , T • = sup x ∈ Z sup y I ∂ ,1 (cid:12)(cid:12)(cid:12) P NS , T , x , y − ¯ P NS , T , x , y (cid:12)(cid:12)(cid:12) . ǫ , ǫ N − − ǫ ̺ − + ǫ S , T . (3.38) Proof.
Identical to the proof for Proposition 3.3, we first assume S = P N and ¯ P N are both time-homogeneous. Moreover, we introduce the following quantities for notational convenience, for hich δ ∈ R > is an arbitrarily and sufficiently small but universal constant: Υ NS , T • = ̺ − δ S , T · sup x ∈ Z sup y I ∂ ,1 P NS , T , x , y ; (3.39a) Φ NS , T • = ̺ − δ S , T · e P NS , T . (3.39b)Courtesy of Lemma 3.2, we have P N T , x , y = ¯ P N T , x , y + Ψ + Ψ , where we have introduced the quantities Ψ • = − Z T τ N X w ∈ I ⋆ ,2 P NR , T , x , w · ¯ a N , w ∆ !! ¯ P N R , w , y d R ; (3.40a) Ψ • = − Z τ N X w ∈ I ⋆ ,2 P NR , T , x , w · ¯ a N , w ∆ !! ¯ P N R , w , y d R . (3.40b)We proceed to estimate each quantity within the RHS, beginning with Ψ . To this end, we employ the Chapman-Kolmogorovequation for P N , as employed within the proof behind Proposition 3.3; this provides the estimate | Ψ | . Ψ + Ψ , wherewe have introduced the additional quantities of Ψ • = Z T τ N X w ∈ I ⋆ ,2 X z I ∂ ,1 P N T + R , T , x , z P NR , T + R , z , w · (cid:12)(cid:12)(cid:12) ∆ !! ¯ P N R , w , y (cid:12)(cid:12)(cid:12) d R ; (3.41a) Ψ • = Z T τ N X w ∈ I ⋆ ,2 X z ∈ I ∂ ,1 P N T + R , T , x , z P NR , T + R , z , w · (cid:12)(cid:12)(cid:12) ∆ !! ¯ P N R , w , y (cid:12)(cid:12)(cid:12) d R . (3.41b)To estimate the quantity Ψ , we recall (3.31); along with Proposition A.1 in [ ] , this provides Ψ . Z T τ N Υ NR , T ̺ − + δ R , T X w ∈ I ⋆ ,2 X z I ∂ ,1 P NR , T + R , z , w · (cid:12)(cid:12)(cid:12) ∆ !! ¯ P N R , w , y (cid:12)(cid:12)(cid:12) d R (3.42) . N − + β ⋆ Z T τ N Υ NR , T ̺ − + δ R , T ̺ − R · | I ⋆ ,2 | d R (3.43) . N − + β ⋆ + ǫ ⋆ ,2 Z T τ N Υ NR , T · ̺ − + δ R , T ̺ − R d R . (3.44)To estimate the quantity Ψ , we again employ Proposition A.1 within [ ] along with Proposition 3.3 interpolated withthe maximum principle of Lemma 3.4; via Lemma C.2 in [ ] , this provides the following: Ψ . N − + β ⋆ + ǫ Z T τ N ̺ − + δ R , T ̺ − S , R · | I ⋆ ,2 || I ∂ ,1 | d R (3.45) . N − + ǫ ⋆ ,2 − ǫ ∂ ,1 + β ⋆ + δ Z T τ N ̺ − + δ R , T ̺ − S , R d R (3.46) . δ N − + ǫ ⋆ ,2 − ǫ ∂ ,1 + β ⋆ + δ τ − N ̺ − + δ S , T (3.47) . N − + ǫ ⋆ ,2 + β ⋆ + δ ̺ − + δ S , T . (3.48)We proceed to estimate Ψ . To this end, we employ the off-diagonal heat kernel estimates for ¯ P N from Proposition A.1 in [ ] ; because we have N τ N log N . | I ∂ ,1 | , this gives | Ψ | . e − log N . Combining this estimate with (3.44) and (3.48),along with Proposition A.1 in [ ] , we obtain Υ NS , T . δ N − + N − + ǫ ⋆ ,2 + β ⋆ + δ + N − + β ⋆ + ǫ ⋆ ,2 ̺ − δ S , T Z T τ N Υ NR , T · ̺ − + δ R , T ̺ − R d R . (3.49)From (3.49), with the singular Gronwall inequality we deduce Υ NS , T . δ N − . In particular, to establish the desired estimate,it suffices to estimate each of Ψ and Ψ , now equipped with this a priori estimate for Υ NS , T ; combining (3.44), (3.48), and he estimate | Ψ | . e − log N , along with Lemma C.2 in [ ] , once more yields Φ NS , T . δ N − + ǫ ⋆ ,2 + β ⋆ + δ + N − + β ⋆ + ǫ ⋆ ,2 ̺ − δ S , T Z T τ N Υ NR , T · ̺ − + δ R , T ̺ − R d R (3.50) . N − + ǫ ⋆ ,2 + β ⋆ + δ + N − + β ⋆ + ǫ ⋆ ,2 ̺ − δ S , T Z T τ N ̺ − + δ R , T ̺ − R d R (3.51) . N − + ǫ ⋆ ,2 + β ⋆ + δ + N − + β ⋆ + ǫ ⋆ ,2 τ − N (3.52) . N − + ǫ ⋆ ,2 + β ⋆ + δ + N − + β ⋆ + ǫ ⋆ ,2 + ǫ ∂ ,1 . (3.53)Recalling the parameters ǫ ⋆ ,2 , β ⋆ , ǫ ∂ ,1 ∈ R > and their relations, this completes the proof. (cid:3) Additionally to Proposition 3.6, we require auxiliary space-time regularity estimates for P N which we will later translateinto estimates for mesoscopic space-time behavior of the interacting particle system at hand. Technically, these space-timeregularity estimates, presented as follows, are an extension of the ideas behind proving Proposition 3.6. However, becausethe regularity estimates for P N required later in this article are quite delicate, we provide the argument below for completetransparency. Proposition 3.7.
For arbitrarily small but universal constants ǫ , ǫ ∈ R > , for S , T ∈ R > with S T and k ∈ Z , we have ∇ k e P NS , T • = sup x , y ∈ Z (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) . ǫ , ǫ N − + β ⋆ + ǫ ̺ − + ǫ S , T · | k | + N − − β ⋆ ̺ − + ǫ S , T ; (3.54) Proof.
Assume S =
0; we define following quantity in which δ ∈ R > is sufficiently small but universal: Υ NS , T • = ̺ − δ S , T sup x ∈ Z sup y I ∂ ,1 (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) . (3.55)Before we proceed, we make the following observation through the Chapman-Kolmogorov equation within which x , y ∈ Z are arbitrary points, and for which we employ Proposition 3.3 and the estimate (3.31) therein: (cid:12)(cid:12)(cid:12) ∇ k , x P N T , x , y (cid:12)(cid:12)(cid:12) X w ∈ I ∂ ,1 (cid:12)(cid:12)(cid:12) ∇ k , x P N T , T , x , w (cid:12)(cid:12)(cid:12) P N T , w , y + X w I ∂ ,1 (cid:12)(cid:12)(cid:12) ∇ k , x P N T , T , x , w (cid:12)(cid:12)(cid:12) P N T , w , y (3.56) . N − − ǫ ∂ ,1 + β ⋆ + δ ̺ − + δ T + N β ⋆ ̺ − + δ T · Υ N T , T . (3.57)Again, courtesy of Lemma 3.2 and Proposition A.1 in [ ] , provided S ∈ [ T ] , we have the following for y I ∂ ,1 ⊆ Z : (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) ∇ k , x ¯ P NS , T , x , y (cid:12)(cid:12)(cid:12) + Ψ + Ψ , (3.58) Ψ • = Z TS + τ N X w ∈ I ⋆ ,2 (cid:12)(cid:12)(cid:12) ∇ k , x P NR , T , x , w (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) ∆ !! w ¯ P NS , R , w , y (cid:12)(cid:12)(cid:12) d R ; (3.59) Ψ • = Z S + τ N S X w ∈ I ⋆ ,2 (cid:12)(cid:12)(cid:12) ∇ k , x P NR , T , x , w (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) ∆ !! w ¯ P NS , R , w , y (cid:12)(cid:12)(cid:12) d R . (3.60)To estimate Ψ , we proceed in identical fashion as for Proposition 3.6 to obtain the estimate Ψ Ψ + Ψ , where Ψ . N − + ǫ ⋆ ,2 + β ⋆ Z TS + τ N Υ NR , T · ̺ − + δ R , T ̺ − S , R d R ; (3.61a) Ψ . δ N − + ǫ ⋆ ,2 + β ⋆ + δ ̺ − + δ S , T ; (3.61b)indeed, the estimate (3.61b) is consequence of realizing the estimate for P N in Proposition 3.3 holds for ∇ k , x P N . Similarly,we have Ψ . e − log N , from which through (3.58), (3.61a), (3.61b), and Proposition A.1 in [ ] we deduce Υ NS , T . δ N − + δ · | k | + N − + ǫ ⋆ ,2 + β ⋆ + δ + N − ǫ ⋆ + β ⋆ ̺ − δ S , T Z TS + τ N Υ NR , T · ̺ − + δ R , T ̺ − S , R d R , (3.62)from which we deduce the following estimate Υ NS , T . δ N − + δ · | k | + N − + ǫ ⋆ ,2 + β ⋆ + δ . (3.63)We combine (3.57) and (3.63) to deduce the estimate (3.54); this completes the proof. (cid:3) roposition 3.8. There exist universal constants ǫ , ǫ ∈ R > such that provided S , T ∈ R > satisfying S T , we have D τ N , ⋆ P NS , T • = sup x , y ∈ Z (cid:12)(cid:12)(cid:12) D τ N , ⋆ P NS , T , x , y (cid:12)(cid:12)(cid:12) . ǫ , ǫ N − − β ⋆ ̺ − + ǫ S , T + N − + β ⋆ ̺ − + ǫ S , T τ N , ⋆ . (3.64) Remark . We briefly remark that the upon comparing P N ≈ ¯ P N , the optimal time-regularity of the latter heat kernel isnot reflected in Proposition 3.8; this reflects that P N ≈ ¯ P N only in a weaker sense. Proof.
We first introduce the following quantity for which δ ∈ R > is arbitrarily small but universal: Υ NS , T • = ̺ − δ S , T sup x ∈ Z sup y I ∂ ,1 (cid:12)(cid:12)(cid:12) D τ N , ⋆ P NS , T , x , y (cid:12)(cid:12)(cid:12) . (3.65)Like for Proposition 3.7, with the Chapman-Kolmogorov equation combined with Proposition 3.3 and (3.31) we have (cid:12)(cid:12)(cid:12) D τ N , ⋆ P NS , T , x , y (cid:12)(cid:12)(cid:12) . N − − ǫ ⋆ + β ⋆ + δ ̺ − + δ S , T + N β ⋆ ̺ − + δ S , T Υ N T + S , T . (3.66)We proceed almost in identical fashion as for the proof of Proposition 3.7. In particular, we first employ the decomposition D τ N , ⋆ P NS , T , x , y = D τ N , ⋆ ¯ P NS , T , x , y + Ψ + Ψ + Ψ , where courtesy of Lemma 3.2, Ψ • = − Z T + τ N , ⋆ T X w ∈ I ⋆ ,2 P NR , T + τ N , ⋆ , x , w · ¯ a N , w ∆ !! w ¯ P NS , R , w , y d R ; (3.67a) Ψ • = − Z TS + τ N X w ∈ I ⋆ ,2 D τ N , ⋆ P NR , T , x , w · ¯ a N , w ∆ !! w ¯ P NS , R , w , y d R ; (3.67b) Ψ • = − Z S + τ N S X w ∈ I ⋆ ,2 D τ N , ⋆ P NR , T , x , w · ¯ a N , w ∆ !! w ¯ P NS , R , w , y d R . (3.67c)To estimate Ψ , we employ Proposition 3.3 and Proposition A.1 in [ ] ; this gives | Ψ | . Z T + τ N , ⋆ T N − + β ⋆ ̺ − R , T + τ N , ⋆ · N δ ̺ − + δ S , R d R (3.68) . N − + β ⋆ + δ ̺ − + δ S , T Z T + τ N , ⋆ T ̺ − R , T + τ N , ⋆ d R (3.69) . N − + β ⋆ + δ ̺ − + δ S , T τ N , ⋆ . (3.70)To estimate Ψ , we proceed with the Chapman-Kolmogorov equation to obtain Ψ = Ψ + Ψ , where Ψ • = − Z TS + τ N X w ∈ I ⋆ ,2 X z I ∂ ,1 D τ N , ⋆ P N T + R , T , x , z P NR , T + R , z , w · ¯ a N , w ∆ !! w ¯ P NS , R , w , y d R ; (3.71a) Ψ • = − Z TS + τ N X w ∈ I ⋆ ,2 X z ∈ I ∂ ,1 D τ N , ⋆ P N T + R , T , x , z P NR , T + R , z , w · ¯ a N , w ∆ !! w ¯ P NS , R , w , y d R . (3.71b)Proceeding as in the proof of Proposition 3.7, we first establish the following estimates: | Ψ | . Z TS + τ N Υ N T + R , T · ̺ − + δ R , T X w ∈ I ⋆ ,2 X z I ∂ ,1 P NR , T + R , z , w (cid:12)(cid:12)(cid:12) ∆ !! w ¯ P NS , R , w , y (cid:12)(cid:12)(cid:12) d R (3.72) . N − + ǫ ⋆ ,2 + β ⋆ Z TS + τ N Υ N T + R , T · ̺ − + δ R , T ̺ − S , R d R ; (3.73) | Ψ | . Z TS + τ N | I ⋆ ,2 || I ∂ ,1 | N − + β ⋆ + δ ̺ − + δ R , T · N − ̺ − S , R d R (3.74) . N − − ǫ ∂ ,1 + ǫ ⋆ ,2 + β ⋆ + δ Z TS + τ N ̺ − + δ R , T ̺ − S , R d R (3.75) . δ N − + ǫ ⋆ ,2 + β ⋆ + δ ̺ − + δ S , T . (3.76)To estimate Ψ , we appeal to the off-diagonal heat kernel estimate from Proposition A.1 in [ ] to deduce | Ψ | . e − log N like in the proofs for Proposition 3.6 and Proposition 3.8. Combining this estimate with (3.73), (3.76), and the regularity stimates from Proposition A.1 in [ ] , we have Υ NS , T . δ N − + β ⋆ + δ τ N , ⋆ + N − + ǫ ⋆ ,2 + β ⋆ + δ + N − + ǫ ⋆ ,2 + β ⋆ ̺ − δ S , T Z TS + τ N Υ N T + R · ̺ − + δ R , T ̺ − S , R d R , (3.77)from which we deduce the following estimate via the singular Gronwall inequality: Υ NS , T . δ N − + β ⋆ + δ τ N , ⋆ + N − + ǫ ⋆ ,2 + β ⋆ + δ . (3.78)Combining (3.66) and (3.78) completes the proof. (cid:3) Nash Estimates II.
Provided those perturbative estimates within Proposition 3.6 and Proposition 3.6, by establishingsuitable off-diagonal estimates for ¯ P N of Proposition A.1 from [ ] and probabilistic hitting-time estimates for the randomwalk associated to the heat kernel of P N , we proceed to establish the following off-diagonal upgrade for the previous Nashestimate for P N from Proposition 3.3.First, however, we introduce the following convenient notation. Definition 3.10.
Provided any S , T ∈ R > satisfying S T along with any x , y ∈ Z and any κ ∈ R > , we define E N , κ S , T , x , y • = exp κ | x − y | N ̺ / S , T ∨ . (3.79)We now "lift" the estimates in Proposition 3.3 and Proposition 3.6 with respect to these exponential weights. Remark . We briefly comment here that the following analysis actually permits the aforementioned "lifting" to weightswhich are actually Gaussian rather than exponential because the employed martingale hitting-time estimates hold at sucha level. However, like with [ ] and [ ] , establishing off-diagonal estimates with respect to the exponential weights aboveare certainly sufficient for our purposes here. We mention, though, that establishing the optimal Gaussian estimates for P N may be of possible external or future interest for establishing parabolic Harnack inequalities for P N as a non-perturbativemechanism for establishing regularity estimates established in Proposition 3.14 below, for example. Proposition 3.12.
There exist a pair of universal constants ǫ , ǫ ∈ R > so that provided times S , T ∈ R > satisfying S Talong with any κ ∈ R > , we have sup x ∈ Z sup y I ∂ ,1 (cid:12)(cid:12)(cid:12) P NS , T , x , y E N , κ S , T , x , y − ¯ P NS , T , x , y E N , κ S , T , x , y (cid:12)(cid:12)(cid:12) . ǫ , ǫ , κ N − − ǫ ̺ − + ǫ S , T ∧
1; (3.80)sup x , y ∈ Z (cid:12)(cid:12)(cid:12) P NS , T , x , y E N , κ S , T , x , y − ¯ P NS , T , x , y E N , κ S , T , x , y (cid:12)(cid:12)(cid:12) . κ N − + β ⋆ ̺ − S , T ∧
1. (3.81)
As consequence of
Proposition A.1 in [ ] , we deduce sup x ∈ Z sup y I ∂ ,1 P NS , T , x , y E N , κ S , T , x , y . κ , ǫ N − ̺ − + ǫ S , T ∧
1; (3.82)sup x , y ∈ Z P NS , T , x , y E N , κ S , T , x , y . κ N − + β ⋆ ̺ − S , T ∧
1. (3.83)
Proof.
The estimate (3.82) follows from (3.80) and the ¯ P N -estimate of Proposition A.1 within [ ] ; in an identical fashion,the estimate (3.83) follows from (3.80) and Proposition A.1 in [ ] .First, we establish the following notation for any x , y ∈ Z and for any κ ∈ R > for convenience: Ξ N , κ S , T , x , y • = (cid:12)(cid:12)(cid:12) P NS , T , x , y E N , κ S , T , x , y − ¯ P NS , T , x , y E N , κ S , T , x , y (cid:12)(cid:12)(cid:12) ; (3.84a) P N , κ S , T , x , y • = P NS , T , x , y E N , κ S , T , x , y ; (3.84b)¯ P N , κ S , T , x , y • = ¯ P NS , T , x , y E N , κ S , T , x , y . (3.84c)We begin with the following observation valid for any δ ∈ R > , in which Q x , ℓ ⊆ Z is the neighborhood of radius ℓ ∈ Z > around x ∈ Z : Ξ N , κ S , T , x , y | Ξ N ,0 S , T , x , y | − δ X w Q x , | x − y | P N , κ S , T , x , y + X w Q x , | x − y | ¯ P N , κ S , T , x , y δ . (3.85) oreover, concerning each summation on the RHS of (3.85), we have the following elementary layer-cake-type estimate: X w Q x , | x − y | P N , κ S , T , x , y ∞ X j = E N , κ S , T , ( j + ) x , ( j + ) y X w Q x , j | x − y | P NS , T , x , y ; (3.86) X w Q x , | x − y | ¯ P N , κ S , T , x , y ∞ X j = E N , κ S , T , ( j + ) x , ( j + ) y X w Q x , j | x − y | ¯ P NS , T , x , y . (3.87)In view of these two estimates (3.86) and (3.87), it suffices to establish the following estimate for any κ ′ ∈ R > arbitrarilylarge but universal and for any ℓ ∈ Z > arbitrarily large but universal: X w Q x , ℓ P NS , T , x , y + X w Q x , ℓ ¯ P NS , T , x , y . κ ′ E N , − κ ′ S , T ,0, ℓ . (3.88)Indeed, provided the estimate (3.88), for κ ′ & κ κ ∈ R > , we have the followingbounds courtesy of an elementary calculation: X w Q x , | x − y | P N , κ S , T , x , y . κ ′ ∞ X j = E N , κ S , T , jx , j y E N , − κ ′ S , T , jx , j y (3.89) . κ , κ ′
1; (3.90) X w Q x , | x − y | ¯ P N , κ S , T , x , y . κ ′ ∞ X j = E N , κ S , T , jx , j y E N , − κ ′ S , T , jx , j y (3.91) . κ , κ ′
1. (3.92)Courtesy of Proposition 3.6 and Proposition 3.3, respectively, we havesup x ∈ Z sup y I ∂ ,1 (cid:12)(cid:12)(cid:12) Ξ N ,0 S , T , x , y (cid:12)(cid:12)(cid:12) − δ . N − − ǫ + δ + ǫ δ ̺ − + ǫ + δ − ǫ δ S , T ∧
1; (3.93a)sup x , y ∈ Z (cid:12)(cid:12)(cid:12) Ξ N ,0 S , T , x , y (cid:12)(cid:12)(cid:12) . N − + β ⋆ − δ + β ⋆ δ ̺ − + δ S , T ∧
1, (3.93b)from which, upon combination with (3.85), (3.86), (3.87), (3.90), and (3.92), we seesup x ∈ Z sup y I ∂ ,1 Ξ N , κ S , T , x , y . κ N − − ǫ + δ + ǫ δ ̺ − + ǫ + δ − ǫ δ S , T ∧
1; (3.94)sup x , y ∈ Z Ξ N , κ S , T , x , y . κ N − + β ⋆ − δ + β ⋆ δ ̺ − + δ S , T ∧
1; (3.95)choosing δ ∈ R > sufficiently small but universal, this completes the proof for both (3.80) and (3.81).For the first summation on the LHS of (3.88), we appeal to the following probabilistic argument. • Provided any x ∈ Z , define the stochastic process T B NT , x as the random walk with transition probabilities P ” B NT , x = y — • = P N T , x , y . (3.96)Moreover, define ¯ B NT , x • = B NT , x − x to be the associated martingale satisfying E ¯ B N • , x =
0. With this, observe X w Q x , ℓ P NS , T , x , y = P h(cid:12)(cid:12)(cid:12) ¯ B NT − S , x (cid:12)(cid:12)(cid:12) > ℓ i . (3.97)Thus, because ¯ B N • , x is a continuous-time mean-zero martingale with uniformly bounded increments, from standardmartingale estimates, such as the Azuma inequality, for any κ ′ ∈ R > we have X w Q x , ℓ P NS , T , x , y . κ ′ E N , − κ ′ S , T ,0, ℓ . (3.98)Combining (3.98) with the off-diagonal bounds in Proposition A.1 of [ ] , we obtain (3.88). This completes the proof. (cid:3) The final heat kernel estimate we require is an off-diagonal upgrade of the space-time regularity estimates from Proposi-tion 3.7 and Proposition 3.8, respectively. The estimates are certainly sub-optimal as we directly employ real interpolationas in the proof of (3.81) and (3.83) from Proposition 3.12, but they are beyond sufficient for our purposes.
Remark . We may probably proceed as within Section 4 of [ ] through a Chapman-Kolmogorov equation to obtainoptimal regularity estimates for P N , or even like in the proof of Proposition 3.12 with the Chapman-Kolmogorov equation. roposition 3.14. For arbitrarily small but universal constants ǫ , ǫ , ǫ , ǫ ∈ R > , provided any S , T ∈ R > satisfying S T ,provided k ∈ Z satisfying | k | . N ̺ / S , T with a universal implied constant, and provided any κ ∈ R > , we have sup x , y ∈ Z (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) E N , κ S , T , x , y . κ , ǫ , ǫ N − + β ⋆ + ǫ ̺ − + ǫ S , T · | k | + N − − β ⋆ + ǫ ̺ − + ǫ S , T ; (3.99)sup x , y ∈ Z (cid:12)(cid:12)(cid:12) D τ N , ⋆ P NS , T , x , y (cid:12)(cid:12)(cid:12) E N , κ S , T , x , y . κ , ǫ , ǫ N − − β ⋆ + ǫ ̺ − + ǫ S , T + N − + β ⋆ + ǫ ̺ − + ǫ S , T τ N , ⋆ . (3.100) Proof.
We begin by employing real interpolation like in the proof of Proposition 3.12. In particular, it suffices to prove thefollowing pair of estimates for any ℓ ∈ Z > and any κ ∈ R > ; we retain notation from the proof of Proposition 3.12: X w Q x , ℓ (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) . κ E N , − κ S , T ,0, ℓ ; (3.101) X W Q x , ℓ (cid:12)(cid:12)(cid:12) D τ N , ⋆ P NS , T , x , y (cid:12)(cid:12)(cid:12) . κ E N , − κ S , T ,0, ℓ . (3.102)To this end, we first observe X w Q x , ℓ (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) X w Q x , ℓ P NS , T , x , y + X w Q x , ℓ P NS , T , x + k , y , (3.103)from which we proceed with the probabilistic estimate (3.88) and the assumption | k | . N ̺ / S , T to deduce (3.101). As forthe estimate (3.102), we proceed in an identical fashion but instead with the bound | D τ N , ⋆ P NS , T , x , y | P NS , T + τ , x , y + P NS , T , x , y .This completes the proof. (cid:3)
4. L
OCAL P ROBABILISTIC E STIMATES
The current section returns to the dynamics of the microscopic Cole-Hopf transform Z N , because we now carry sufficientanalysis and substance concerning P N courtesy of the previous section to say something meaningful about this observable.Let us record the stochastic dynamics for Z N in Proposition 2.1 as the following stochastic integral equation: Z NT , x = X y ∈ Z P N T , x , y Z N y + Z T X y ∈ Z P NS , T , x , y · Z NS , y d ξ NS , y + E N , I T , x + E N , II T , x , (4.1)where we have introduced the proposed error quantities of E N , I T , x • = Z T X y ∈ I ⋆ ,2 P NS , T , x , y · c N q NS , y Z NS , y d S ; (4.2a) E N , II T , x • = Z T X y ∈ I ⋆ ,2 P NS , T , x , y · c N N − e q NS , y Z NS , y d S . (4.2b)The primary goal for this current section is to confirm these two quantities E N , • are genuinely error terms with respect toa suitable topology, which ultimately turns out to be the space-time uniform topology assuming a priori control on Z N . Wemake this precise in the following statement. First, however, we introduce the following notation which will be frequentlyemployed for the remainder of this article. Definition 4.1.
Provided any appropriate function ϕ : [
0, 1 ] × Z → C , let us define the one-parameter family of space-timenorms k ϕ k ∞ ; κ • = sup T ∈ [ ] sup x ∈ Z e − κ | x | N | ϕ T , x | . Proposition 4.2.
Provided κ ∈ R > sufficiently large, any ǫ ∈ R > arbitrarily small but universal, and β ∈ R > , consider thefollowing event defined via (cid:2) E κ , β , ǫ (cid:3) • = ”(cid:13)(cid:13) C N , I (cid:13)(cid:13) ∞ ; κ + (cid:13)(cid:13) C N , II (cid:13)(cid:13) ∞ ; κ & N − β + N − β (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ; κ — . (4.3) Provided β ⋆ ∈ R > is sufficiently small, there exist universal constants β u ,1 , β u ,2 ∈ R > such that provided κ ∈ R > arbitrarilylarge but universal and any ǫ ∈ R > arbitrarily small but universal:, we have P ” E κ , β u ,1 , ǫ — . κ , ǫ N − β u ,2 . (4.4)We provide the following outline for the rest of the section overviewing the high-level components behind the proof ofProposition 4.2, in which we introduce relevant quantities as well; in particular, the following outline may also be treatedas a central location for relevant notation and definitions in this section. First, the proof of the estimate for C N , II is straightforward provided the heat kernel estimates in Proposition 3.12.The following discussion thus focuses on C N , I and follows a one-block, two-blocks scheme though in the dynamicalspirit of [ ] and [ ] . • The one-block-component of the proof behind Proposition 4.2 above amounts to the replacement q N • , • A ℓ N ; X q N • , • ,where the spatial-averaging operator A ℓ N ; X , parameterized by ℓ N ∈ Z > , is defined by A ℓ N ; X q N • , • • = η N • , ± ⋆ ÝX ℓ N w = η N • , • + w ; (4.5)we have introduced the notation eP i ∈ I • = | I | − P w ∈ I , and we will take ℓ N • = N β ⋆ + ǫ with ǫ ∈ R > arbitrarily smallbut universal.This procedure is the usual goal for the classical one-block step. However, the term q N • , • is supported at a smallmesoscopic scale, thus obstructing the classical one-block step. We will circumvent such obstruction via exploiting dynamical fluctuations within the difference q N • , • − A ℓ N ; X q N • , • , spatially localizing the dynamical analysis within [ ] and [ ] . Technically, this replacement requires two preliminary ingredients in entropy production estimates andtime-regularity estimates for Z N . • The two-blocks component of the proof amounts to the replacement A ℓ N ; X q N • , • A m N ; X q N • , • given m N • = N − β ⋆ + ǫ ,where ǫ ∈ R > is another possibly different though still arbitrarily small and universal constant. This follows theclassical two-blocks estimate without any dynamical components. • Finally, we estimate A m N ; X q N • , • directly using regularity of the microscopic Cole-Hopf transform Z N ; this is a strategyfrom [ ] but we make it quantitative given the sub-optimal nature estimates in Proposition 3.14.We conclude this introduction with notation which will serve convenient. • Provided ℓ , ℓ ∈ Z > , we define the comparison operator DA ℓ ; ℓ q N • , • • = A ℓ ; X q N • , • − A ℓ ; X q N • , • . (4.6)In particular, if ℓ =
0, we have DA ℓ q N • , • = q N • , • − A ℓ ; X q N • , • . • We additionally introduce the following time-averaging operators A τ ; T parameterized by τ ∈ R and defined by A τ ; T f N • , • • = | τ | Z • + τ • f N • + R , • d R ; (4.7)above, we will specialize to f N • , • a local mesoscopic functional of the particle system. • Finally, we introduce the following list of quantities: E N , I ,1 T , x • = Z T X y ∈ I ⋆ ,2 P NS , T , x , y · N ” DA ℓ N q NS , y — Z NS , y d S ; (4.8a) E N , I ,2 T , x • = Z T X y ∈ I ⋆ ,2 P NS , T , x , y · N ” A τ N , ⋆ ; T DA ℓ N q NS , y — Z NS , y d S ; (4.8b) E N , I ,3 T , x • = Z T X y ∈ I ⋆ ,2 P NS , T , x , y · N ” DA ℓ N ; m N q NS , y — Z NS , y d S ; (4.8c) E N , I ,4 T , x • = Z T X y ∈ I ⋆ ,2 P NS , T , x , y · N ” A m N ; X q NS , y — Z NS , y d S . (4.8d)Moreover, define C N , I ,5 • = C N , I ,1 − C N , I ,2 .4.1. Entropy Production.
We begin the technical component of this section with an entropy production estimate classicalto the theory of hydrodynamic limits; see [ ] and [ ] , for example. This estimate is crucial to both one-block and two-blocks estimates, as it serves a tool of comparison to the local equilibrium.First, however, we introduce a few objects and their corresponding gadgets; we will adopt this notation throughout theremainder of this section. • Provided any possibly infinite sub-lattice I ⊆ Z , we define the grand-canonical ensemble µ gc I to be the probabilitymeasure on {± } I which is product, and whose one-dimensional marginals exhibit expectation equal to 0. • Provided any ̺ ∈ [ −
1, 1 ] and any possibly infinite sub-lattice I ⊆ Z , we define the associated canonical ensemble µ can ̺ , I as the probability measure on {± } I obtained by conditioning µ gc I on the hyperplane P x ∈ I η x = | I | ̺ . For a possibly infinite sub-lattice I ⊆ Z and probability measure µ I , N on {± } Z , we define the Dirichlet form via D I ‚ d µ I , N d µ gc I Œ • = − X x ∈ Z : x , x + ∈ I E µ gc I (cid:2) f I · L x , x + f I (cid:3) (4.9) = X x ∈ Z : x , x + ∈ I E ”(cid:12)(cid:12) L x , x + f I (cid:12)(cid:12) — , (4.10)where we have introduced the non-negative functional f I • = h d µ I , N d µ gc I i . Lemma 4.3.
We consider T ∈ R > , and we define the sub-lattice I N • = J − N , N K ⊆ Z ; provided any initial probability measure µ N on Ω , we have the following estimate with a universal implied constant and any ǫ ∈ R > : Z T D I N d µ I N S , N d µ gc I N ! d S . ǫ N − + β ⋆ + ǫ . (4.11) Above, we introduced the probability measure µ • , N • = e • [ L N ,!! ] ∗ µ N , and moreover µ I N • , N = Π I N µ • , N .Proof. Following the proof of Lemma 3.8 in [ ] , it suffices to assume that the initial relative entropy H Z of the measure µ N with respect to µ gc Z satisfies the a priori estimate H Z . ǫ N + ǫ with ǫ ∈ R > arbitrarily small although universal. Thus,Lemma 4.3 follows from standard procedure via the Kolmogorov forward equation as detailed in Section 9 of Appendix 1in [ ] , where the additional factor of N β ⋆ follows from the effect of the slow bonds. (cid:3) As a consequence of Lemma 4.3, we establish the following analog of Lemma 3.12 from [ ] . The primary discrepanciesbetween the following estimate and Lemma 3.12 within [ ] consist of the presence of the slow bonds inside I N ⊆ Z andthe lack of averaging in the spatial direction. Lemma 4.4.
Suppose the functional ϕ : Ω → R has support contained in I ϕ ⊆ I N ⊆ Z . We establish the following estimatewith a universal implied constant provided any κ ∈ R > , any T ∈ [
0, 1 ] , and any ǫ ∈ R > : E µ gc I ϕ ” ϕ · e f I ϕ T , N — . ǫ κ − N − + β ⋆ + ǫ ̺ − T | I ϕ | + κ − sup ̺ ∈ [ − ] log E µ can ̺ , I ϕ e κϕ . (4.12) Above, we have introduced the probability densities f I ϕ • , N • = d µ I ϕ • , N d µ gc I ϕ and e f I ϕ • , N • = • − R • f I ϕ S , N d S, where the initial condition µ N , asa probability measure on Ω , is generic.Proof. We appeal directly to the proof of Lemma 3.12 within [ ] upon employing Lemma 4.3 and observing that withoutaveraging e f I ϕ in the spatial direction, we do not improve on the estimate from Lemma 4.3, though we carry one less powerof | I ϕ | as compared to Lemma 3.12 in [ ] . (cid:3) One-Block Estimate.
To make the corresponding bullet point above precise, the primary goal for this first subsectionis the following estimate.
Proposition 4.5.
Provided κ ∈ R > sufficiently large and β ∈ R > , consider the following event defined via (cid:2) G κ , β (cid:3) • = ”(cid:13)(cid:13) C N , I ,1 (cid:13)(cid:13) ∞ ; κ & N − β (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ — . (4.13) Provided β ⋆ ∈ R > is sufficiently small, there exists a pair of universal constants β u ,1 , β u ,2 ∈ R > such that provided κ ∈ R > arbitrarily large but universal, we have P ” G κ , β u ,1 — . κ N − β u ,2 . (4.14)The first technical ingredient that we require is a time-regularity estimate for Z N ; the estimate is certainly sub-optimaland does not reflect the macroscopic regularity properties of the continuum SHE, although it is sufficient for our purposes.For example, see Proposition 6.2 in [ ] for an estimate which is close to optimal. Lemma 4.6.
Recall τ N , ⋆ = N − + β ⋆ ; consider the following event for any ǫ ∈ R > arbitrarily small but universal: ” G τ N , ⋆ , ǫ — • = ”(cid:13)(cid:13) T N (cid:13)(cid:13) ∞ ;0 & N + ǫ τ N , ⋆ — ; (4.15) above, we have introduced the quantity T N • = [ Z N ] − D τ N , ⋆ Z N . rovided any ǫ ∈ R > arbitrarily small but universal, we have P ” G τ N , ⋆ , ǫ — . ǫ e − log N . (4.16) Proof.
Observe that courtesy of Proposition 2.1 and its proof, or alternatively the definition of Z N , there are two componentsof the dynamic of Z N . The first, coming through the global drift with speed ν N ∈ R > , on time-scales τ N . N − contributesgrowth by a factor of at most ν N τ N . N τ N . The second component arises through the particle dynamic itself, where eachstep contributes growth by a factor of at most N − up to some universal constants. With respect to the time-scale τ N ∈ R > of interest, upon observing the parabolic scaling in space-time, the total contribution coming from this growth mechanismis bounded above by N − τ N , again up to universal factors, afterwards multiplied by the number of discrete jumps exhibitedby Z N at any particular point on the order-1 time-scales. As direct consequence of some classical large-deviations estimatesfor the Poisson distribution, with at most this proposed probability above, the number of these discrete jumps is at boundedabove by N + ǫ provided ǫ ∈ R > arbitrarily small but universal. This completes the proof. (cid:3) Provided Lemma 4.6 along with the previous time-regularity estimate for P N from Proposition 3.14, we may establishthe following dynamical replacement, which effectively reduces proving the estimate within Proposition 4.5 to an estimateupon replacing C N , I ,1 C N , I ,2 . Lemma 4.7.
Provided any κ ∈ R > arbitrarily large and any ǫ ∈ R > arbitrarily small with both universal, consider the event ” G τ N , ⋆ , κ , ǫ — • = ”(cid:13)(cid:13) C N , I ,5 (cid:13)(cid:13) ∞ ; κ & ǫ κ N , ǫ (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ — . (4.17) where κ N , ǫ • = N − β ⋆ + ǫ ⋆ ,2 + ǫ + N β ⋆ + ǫ ⋆ ,2 + ǫ τ N , ⋆ + N + β ⋆ + ǫ ⋆ ,2 + ǫ τ N , ⋆ . For any ǫ ∈ R > arbitrarily small but universal, we have P ” G τ N , ⋆ , κ , ǫ — . κ , ǫ e − log N . (4.18) Proof.
We assume κ = κ ∈ R > holds upon elementary adjustments like with theproof of Proposition 3.2 in [ ] . An elementary calculation gives us the following deterministic first step: (cid:13)(cid:13) C N , I ,5 (cid:13)(cid:13) ∞ ;0 . N τ N , ⋆ (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 + Ψ + Ψ , (4.19)where we introduced the following pair of quantities above: Ψ • = N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X y ∈ I ⋆ ,2 D τ N , ⋆ P NS , T , x , y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 ; (4.20a) Ψ • = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T X y ∈ I ⋆ ,2 P NS , T , x , y · N (cid:12)(cid:12)(cid:12) D τ N , ⋆ Z NS , y (cid:12)(cid:12)(cid:12) d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 . (4.20b)We proceed with another deterministic estimate consequence of Proposition 3.14 for Ψ as follows, in which ǫ ∈ R > is anarbitrarily small but universal constant: Ψ . N − β ⋆ + ǫ ⋆ ,2 + ǫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T ̺ − + ǫ S , T d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 + N β ⋆ + ǫ ⋆ ,2 + ǫ τ N , ⋆ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T ̺ − + ǫ S , T d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 (4.21) . ǫ N − β ⋆ + ǫ ⋆ ,2 + ǫ (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 + N β ⋆ + ǫ ⋆ ,2 + ǫ τ N , ⋆ (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 . (4.22)To estimate Ψ , we employ the heat kernel estimate for P N from Proposition 3.12 along with the high-probability estimateof Lemma 4.6, the latter of which holds with probability at least 1 − c ǫ e − log N with c ǫ ∈ R > a uniformly bounded constantand where ǫ ∈ R > is arbitrarily small but universal: Ψ . N + β ⋆ + ǫ ⋆ ,2 τ N , ⋆ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T ̺ − S , T d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 (4.23) . N + β ⋆ + ǫ ⋆ ,2 + ǫ τ N , ⋆ (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 . (4.24)Combining the estimates (4.19), (4.22), and (4.24) completes the proof. (cid:3) In spirit of Lemma 4.7, towards the proof for Proposition 4.5 it suffices to estimate C N , I ,2 . This is exactly the content ofthe following result, whose proof follows the dynamical strategy developed in [ ] and [ ] . emma 4.8. Consider the following events parameterized by κ , β ∈ R > and ǫ ∈ R > arbitrarily small but universal, in whichthe implied constant is universal: (cid:2) G N , κ , β , ǫ (cid:3) • = ” N − β (cid:13)(cid:13) Z N (cid:13)(cid:13) − ∞ ; κ (cid:13)(cid:13) C N , I ,2 (cid:13)(cid:13) ∞ ; κ & κ N , ǫ — (4.25) where we have introduced the quantity κ N , ǫ • = N − ǫ + ǫ ⋆ ,2 + N + ǫ ⋆ ,2 + β ⋆ + ǫ τ N , ⋆ + N + ǫ ⋆ ,2 + β ⋆ + ǫ τ N , ⋆ + N − + ǫ ⋆ ,2 + β ⋆ + ǫ ℓ N + N − + β ⋆ + ǫ ⋆ ,2 τ − N , ⋆ ℓ N . (4.26) Provided any β ∈ R > , any κ ∈ R > arbitrarily large but universal, and any ǫ ∈ R > arbitrarily small but universal, we have P (cid:2) G N , κ , β , ǫ (cid:3) . κ , ǫ N − β . (4.27) Proof.
Again, we will assume κ = T ∈ R > , we define T N • = T − N − β ⋆ − ǫ with ǫ ∈ R > arbitrarily small but universal. We proceed with the following elementary deterministic estimate via Proposition 3.12: (cid:12)(cid:12)(cid:12) C N , I ,2 T , x (cid:12)(cid:12)(cid:12) . N β ⋆ Z TT N ̺ − S , T X y ∈ I ⋆ ,2 Z NS , y d S + Ψ (4.28) . N − ǫ + ǫ ⋆ ,2 (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 + Ψ ; (4.29)to introduce and analyze the quantity Ψ , we first define it below and record elementary estimates: Ψ • = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T N X y ∈ I ⋆ ,2 P NS , T , x , y · N ” A τ N , ⋆ ; T DA ℓ N q NS , y — Z NS , y d S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.30) . N β ⋆ X y ∈ I ⋆ ,2 Z T N ̺ − S , T (cid:12)(cid:12)(cid:12) A τ N , ⋆ ; T DA ℓ N q NS , y (cid:12)(cid:12)(cid:12) d S · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 (4.31) . N β ⋆ + ǫ ⋆ ,2 + ǫ Ψ · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 , (4.32)where we have additionally introduced the following factor Ψ defined by the integral formula Ψ • = ÝX y ∈ I ⋆ ,2 Z (cid:12)(cid:12)(cid:12) A τ N , ⋆ ; T DA ℓ N q NS , y (cid:12)(cid:12)(cid:12) d S ; (4.33)the bound (4.32) follows from observing the integrand in Ψ is non-negative. With (4.29) and (4.32), it suffices to estimate Ψ . Moreover, these estimates within (4.29) and (4.32) are deterministic, and at this point, we will now take expectations.To this end, we proceed like in the proof of Proposition 5.3 in [ ] and obtain the following: E Ψ = ÝX y ∈ I ⋆ ,2 E –Z A NS , y d S ™ ; (4.34) A NS , y • = E F S h(cid:12)(cid:12)(cid:12) A τ N , ⋆ ; T DA ℓ N q NS , y (cid:12)(cid:12)(cid:12)i . (4.35)To clarify what the second quantity A N • , y actually is, it is the expectation of | A τ N , ⋆ ; T DA ℓ N q NS , y | when viewed as a functionalof the path-space, namely a functional D ( R > S , Ω ) → R , after conditioning upon the particle system at the time S ∈ R > ,or equivalently on the σ -algebra of F S . With speed-of-propagation-type estimates as in Lemma 5.7 in [ ] , we have E Ψ . sup y ∈ I ⋆ ,2 E –Z e A N ,loc S , y d S ™ + e − log N ; (4.36) e A N ,loc S , y • = E F S ,loc h(cid:12)(cid:12)(cid:12) A loc τ N , ⋆ ; T DA ℓ N q NS , y (cid:12)(cid:12)(cid:12)i , (4.37)where we clarify the object e A N ,loc S , y as follows, in which the parameter ǫ ∈ R > is arbitrarily small but universal and ⋆ − y , N • = y − N + ǫ τ N , ⋆ + N + ǫ τ N , ⋆ + N ǫ ℓ N ; (4.38a) ⋆ + y , N • = y + N + ǫ τ N , ⋆ + N + ǫ τ N , ⋆ + N ǫ ℓ N . (4.38b) • the operator A loc τ N , ⋆ ; T is the time-averaging operator, although upon replacing the global particle dynamic with thelocal periodic dynamic on I y ,loc • = J ⋆ − y , N , ⋆ + y , N K ⊆ Z , and E F S ,loc is the expectation with respect to this local particledynamic upon conditioning upon the particle configuration at time S ∈ R > restricted to the sub-lattice I y ,loc ⊆ Z .In particular, e A N ,loc S , y is a functional of particle configurations on the local sub-lattice I y ,loc ⊆ Z at time S ∈ R > . oughly speaking, and this is made precise within the proof of Lemma 5.7 and Proposition 5.3 in [ ] , information at theinitial time of S ∈ R > outside of the neighborhood I y ,loc ⊆ Z propagates into the support of the functional DA ℓ N q NS , y onan event of exponentially small probability courtesy of large-deviations estimates for random walks.Ultimately, applying Lemma 4.4 to the remaining expectation on the RHS of (4.36) like in the proof for Proposition 5.3in [ ] , we obtain the following estimate in which f N denotes the Radon-Nikodym derivative of the initial measure µ N with respect to stationary grand-canonical ensemble µ gc Z :sup y ∈ I ⋆ ,2 E –Z e A N ,loc S , y d S ™ = E ”e f I y ,loc N e A N ,loc0, y — (4.39) . ǫ Ψ + Ψ , , (4.40)where we have introduced the quantities Ψ • = N − + β ⋆ + ǫ | ⋆ + N − ⋆ − N | (4.41a) . N + β ⋆ + ǫ τ N , ⋆ + N + β ⋆ + ǫ τ N , ⋆ + N − + β ⋆ + ǫ ℓ N ; (4.41b) Ψ • = sup ̺ ∈ [ − ] log E µ can ̺ , I exp h E F S ,loc h(cid:12)(cid:12)(cid:12) A loc τ N , ⋆ ; T DA ℓ N q NS ,0 (cid:12)(cid:12)(cid:12)ii . (4.41c)To estimate Ψ , we observe that the quantity within the inner expectation is uniformly bounded. Upon applying elementaryconvexity inequalities for the logarithm and exponential functions, we see Ψ . sup ̺ ∈ [ − ] E µ can ̺ , I h(cid:12)(cid:12)(cid:12) A loc τ N , ⋆ ; T DA ℓ N q NS , ± ⋆ (cid:12)(cid:12)(cid:12)i (4.42) . N − + β ⋆ τ − N , ⋆ ℓ N , (4.43)where the final estimate (4.43) is a consequence of the Kipnis-Varadhan inequality, for example in Proposition 6.1 withinAppendix 1 of [ ] , and an estimate on the spectral gap of the local particle dynamic of N − + β ⋆ ℓ N up to universal constants.To conclude the proof, we begin with the Chebyshev inequality; this gives the following estimate with universal impliedconstant courtesy of (4.29), (4.32), (4.36), (4.40), (4.41b), and (4.43): P (cid:2) G N ,0, β (cid:3) . N − β κ − N , ǫ E ”(cid:13)(cid:13) Z N (cid:13)(cid:13) − ∞ ;0 (cid:13)(cid:13) C N , I ,2 (cid:13)(cid:13) ∞ ;0 — (4.44) . ǫ κ − N , ǫ N − ǫ − β + ǫ ⋆ ,2 + κ − N , ǫ N β ⋆ + ǫ ⋆ ,2 + ǫ − β E Ψ (4.45) . ǫ N − β + ǫ ⋆ ,2 κ − N , ǫ · h N − ǫ + N + β ⋆ + ǫ τ N , ⋆ + N + β ⋆ + ǫ τ N , ⋆ + N − + β ⋆ + ǫ ℓ N + N − + β ⋆ τ − N , ⋆ ℓ N i (4.46) = N − β . (4.47)This completes the proof. (cid:3) Proof of
Proposition 4.5 . For what follows, the constant κ ∈ R > is sufficiently large though universal; the dependences ofconstants on κ ∈ R > in the forthcoming argument are thus omitted. We first record the following estimates with universalimplied constants upon recalling notation of Lemma 4.8: N + ǫ τ N , ⋆ . N − + β ⋆ + ǫ ; (4.48a) N β ⋆ + ǫ τ N , ⋆ + N + β ⋆ + ǫ τ N , ⋆ . N − + β ⋆ + ǫ + N − + β ⋆ + ǫ ; (4.48b) κ N , ǫ . N − ǫ + β ⋆ + N − + β ⋆ + ǫ + N − + β ⋆ + ǫ + N − + β ⋆ + ǫ + N − β ⋆ + ǫ . (4.48c)In particular, provided β ⋆ ∈ R > sufficiently small although still universal and then afterwards choosing ǫ ∈ R > sufficientlysmall, each quantity is bounded above by N − β , where β ∈ R > is some universal exponent. Courtesy of Lemma 4.6, Lemma4.7, and Lemma 4.8, we have the following estimate with probability at least 1 − c N − β ′ provided β ′ ∈ R > another constantdepending only on β ∈ R > for our choosing, and c ∈ R > is another universal constant: (cid:13)(cid:13) C N , I ,1 (cid:13)(cid:13) ∞ ; κ (cid:13)(cid:13) C N , I ,5 (cid:13)(cid:13) + (cid:13)(cid:13) C N , I ,2 (cid:13)(cid:13) ∞ ;0 (4.49) . N − β (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ + N − β + β ′ (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ . (4.50)Choosing β ′ = β , for example, completes the proof. (cid:3) Two-Blocks Estimate.
For the current section, we implement the replacement of local averages A ℓ N ; X q N • , • A m N ; X q N • , • ,which we emphasize, by definition, amounts to an estimate for DA ℓ N ; m N q N • , • . This is made precise as follows. roposition 4.9. Provided κ ∈ R > sufficiently large and β ∈ R > , consider the following event: (cid:2) H κ , β (cid:3) • = ”(cid:13)(cid:13) C N , I ,3 (cid:13)(cid:13) ∞ ; κ & N − β (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ — . (4.51) Provided β ⋆ ∈ R > is sufficiently small, there exists a pair of universal constants β u ,1 , β u ,2 ∈ R > such that provided κ ∈ R > arbitrarily large but universal, we have P ” H κ , β u ,1 — . κ N − β u ,2 . (4.52)The first technical estimate we require is a variation of Lemma 4.4 engineered for unions of two connected but disjointsub-lattices; it is often referred to as the moving-particle lemma in the theory of hydrodynamic limits; see [ ] or [ ] . Lemma 4.10.
Consider a sub-lattice I • = I ∪ I ⊆ Z , where I , I ⊆ Z are possibly disjoint although connected sub-latticessatisfying the following constraints. • For ℓ ∈ Z > , we have | I | . ℓ and | I | . ℓ , both with universal implied constants. • For m ∈ Z > , we have inf x ∈ I inf y ∈ I | x − y | . m, again with universal implied constant.If ϕ : Ω → R has support in I ⊆ Z , we have, with universal implied constant and ǫ ∈ R > arbitrarily small but universal, E µ gc I ” ϕ · e f I N — . ǫ N − + β ⋆ + ǫ m ℓ + N − + β ⋆ + ǫ ℓ + sup ̺ ∈ [ − ] log E µ can ̺ , I e ϕ . (4.53) Above, we have introduced the probability densities f I • , N • = d µ I • , N d µ gc I and e f I • , N • = • − R • f I S , N d S, where the initial condition µ N , asa probability measure on Ω , is generic.Proof. Following the proof for Lemma 3.12 in [ ] , and namely the proof for Lemma 4.4 accounting for the factor of N β ⋆ ,it suffices to establish the following estimate since | I | . ℓ , which actually follows from the proof of Proposition 4.4 in [ ] : D I ( e f I N ) . ǫ N − + β ⋆ + ǫ m + N − + β ⋆ + ǫ ℓ . (4.54)Indeed, the beginning of such argument begins with an application of the entropy inequality from Section 8 in Appendix1 of [ ] , for example, and the log-Sobolev inequality for the symmetric simple exclusion process without a slow bond on I ⊆ Z from Theorem A of [ ] with constant | I | . ℓ . This completes the proof. (cid:3) Remark . We briefly comment on the estimate from Lemma 4.10 concerning one relatively subtle aspect of the classicaltwo-blocks estimate within [ ] that is brought to light in its quantitative version which begins with Lemma 4.10. Observethe constraints on the size of the sub-lattices I , I ⊆ Z ; we are concerned only with those statistics supported on this pairof sub-lattices which are therefore still mesoscopic with respect to the scale of ℓ ∈ Z > , not the scale of m ∈ Z > . Withoutsuch a restriction, we must implement the replacement of the prefactors m ℓ m . Because the first scale ℓ ∈ Z > is closeto microscopic and the latter scale m ∈ Z > is as close to macroscopic as possible, this alternative estimate is much worse. Proof of
Proposition 4.9 . Again, we assume κ = T ∈ R > , we againemploy the time T N • = T − N − β ⋆ − ǫ given ǫ ∈ R > arbitrarily small although still universal, and afterwards we establish thefollowing via Proposition 3.12: (cid:12)(cid:12) C N , I ,3 (cid:12)(cid:12) . N β ⋆ Z TT N ̺ − S , T X y ∈ I ⋆ ,2 Z NS , y d S + Ψ (4.55) . N − ǫ + ǫ ⋆ ,2 (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 + Ψ , (4.56)where Ψ is a new quantity satisfying the following estimate: Ψ . N β ⋆ Z T N ̺ − S , T X y ∈ I ⋆ ,2 (cid:12)(cid:12)(cid:12) DA ℓ N ; m N q NS , y (cid:12)(cid:12)(cid:12) Z NS , ± ⋆ d S (4.57) . N β ⋆ + ǫ ⋆ ,2 + ǫ Ψ · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 ; (4.58)above, we have introduced the quantity Ψ defined by the following integral, similar to the proof of Lemma 4.8: Ψ • = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T ÝX y ∈ I ⋆ ,2 (cid:12)(cid:12)(cid:12) DA ℓ N ; m N q NS , y d S (cid:12)(cid:12)(cid:12) d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 (4.59) = ÝX y ∈ I ⋆ ,2 Z (cid:12)(cid:12)(cid:12) DA ℓ N ; m N q NS , y (cid:12)(cid:12)(cid:12) d S . (4.60) he argument now branches from the proof of Lemma 4.8 for now. By definition and an elementary calculation, we have (cid:12)(cid:12)(cid:12) DA ℓ N ; m N q NS , y (cid:12)(cid:12)(cid:12) . ÝX m N w = − m N e A w ℓ N ; X q NS , y ; (4.61) e A w ℓ N ; X q NS , y • = ℓ − N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ N X z = ¯ η N ; w y + z ; (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.62)¯ η N ; w y + z • = η N y + z − η N y + w + z , (4.63)so that we have the following preliminary estimate for Ψ : Ψ . ÝX y ∈ I ⋆ ,2 ÝX m N w = − m N Z e A w ℓ N ; X q NS , y d S . (4.64)We emphasize that all estimates thus far are deterministic, and at this point we will take expectations. To this end, let usfirst introduce the sub-lattice of I y , w • = J w + y − ℓ N , w + y + ℓ N K ∪ J ± ⋆ − ℓ N , ± ⋆ + ℓ N K ⊆ Z associated to some w ∈ Z .Additionally, rather than completely reproduce the proof for Lemma 4.8 in detail, we instead provide the following stepswhich are justified with following the proof of Lemma 4.8. • Courtesy of (4.64), we have E Ψ . sup y ∈ I ⋆ ,2 sup w ∈ J − m N , m N K E ”e f I y , w N e A w ℓ N ; X q N y — . (4.65)Courtesy of Lemma 4.10, we thus establish, for ǫ ∈ R > arbitrarily small though universal, the following entropyinequality estimate: E ”e f I y , w N e A w ℓ N ; X q N y — . N − + β ⋆ m N | I y , w | + sup ̺ ∈ [ − ] log E µ can ̺ , I y , w exp ”e A w ℓ N ; X q N y — (4.66) . N − β ⋆ + ǫ + sup ̺ ∈ [ − ] E µ can ̺ , I y , w (cid:12)(cid:12)(cid:12)e A w ℓ N ; X q N y (cid:12)(cid:12)(cid:12) ; (4.67)this final estimate (4.67) is consequence of a direct calculation combined with the previous convexity inequalitiesfor the exponential and logarithm functions. • Towards estimating the remaining expectation, we claim the following upper bound with universal implied con-stants, and in which ǫ ∈ R > is arbitrarily small but universal:sup ̺ ∈ [ − ] E µ can ̺ , I y , w (cid:12)(cid:12)(cid:12)e A w ℓ N ; X q N y (cid:12)(cid:12)(cid:12) . ℓ − N . (4.68)To prove the estimate (4.68), we first fix any canonical-ensemble-parameter ̺ ∈ [ −
1, 1 ] . Second, we observe thatwith respect to the canonical ensemble µ can ̺ , I y , w , the following discrete-time process is a discrete-time martingalewith respect to the filtration generated by the increments: ℓ ℓ − N ℓ X z = ¯ η N ; w y + z . (4.69)Moreover, the increments of this discrete-time martingale are uniformly bounded, thus the estimate (4.68) followsfrom the Azuma martingale inequality, for example.We combine (4.65) with (4.67) and (4.68) while recalling ℓ N = N β ⋆ + ǫ to establish the following given ǫ ∈ R > arbitrarilysmall but universal: E Ψ . N − β ⋆ + ǫ + N − β ⋆ + ǫ . (4.70)Combining (4.56) and (4.58) with this previous estimate in (4.70), we deduce, again assuming ǫ ∈ R > is arbitrarily smallthough universal, E ”(cid:13)(cid:13) Z N (cid:13)(cid:13) − ∞ ;0 (cid:13)(cid:13) C N , I ,3 (cid:13)(cid:13) ∞ ;0 — . ǫ N − ǫ + ǫ ⋆ ,2 + N − β ⋆ + ǫ ⋆ ,2 + ǫ . (4.71)The desired estimate follows from the Chebyshev inequality combined with (4.71) upon choosing β u ,1 , ǫ ∈ R > sufficientlysmall but universal, depending only on β ⋆ ∈ R > ; this completes the proof. (cid:3) Mesoscopic Fluctuations.
The final preliminary ingredient towards the proof for Proposition 4.2 is precisely statedin the following result, which the current subsection is dedicated towards. roposition 4.12. Provided any κ ∈ R > sufficiently large, β ∈ R > , and ǫ ∈ R > arbitrarily small though universal, considerthe event (cid:2) I κ , β , ǫ (cid:3) • = ”(cid:13)(cid:13) C N , I ,4 (cid:13)(cid:13) ∞ ; κ & N − β + N − β (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ; κ — . (4.72) Provided β ⋆ ∈ R > is sufficiently small, there exists a pair of universal constants β u ,1 , β u ,2 ∈ R > such that provided κ ∈ R > arbitrarily large but universal and ǫ ∈ R > arbitrarily small but universal, we have P ” I κ , β u ,1 , ǫ — . κ , ǫ N − β u ,2 . (4.73)We prepare a couple of preliminary ingredients towards the proof of Proposition 4.12; the crucial backbone is a spatialregularity estimate for Z N , which is therefore stochastic, for which the main input is Proposition 3.14. Lemma 4.13.
Consider the following event provided κ ∈ R > arbitrarily large though universal, provided ǫ ∈ R > arbitrarilysmall but universal, and provided k ∈ Z : (cid:2) I κ , ǫ , k (cid:3) • = •(cid:13)(cid:13)(cid:13) ̺ / T ∇ k Z N (cid:13)(cid:13)(cid:13) ∞ ; κ & N ǫ c N , k + N ǫ c N , k (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ; κ ˜ ; (4.74) in the above, we have introduced the quantity c N , k • = N − + β ⋆ + N − + β ⋆ + ǫ m N + N − β ⋆ + ǫ + N − + β ⋆ . (4.75) Provided D , κ ∈ R > arbitrarily large but universal and any ǫ ∈ R > arbitrarily small but universal, for | k | . m N we establishthe following high-probability estimate: P (cid:2) I κ , ǫ , k (cid:3) . κ , ǫ , D N − D . (4.76) Proof.
Again, we will assume that κ = ∇ k , x Z NT , x = Ψ + Ψ + Ψ + Ψ + Ψ ; (4.77)above, we have introduced the following integral quantities of Ψ , Ψ , Ψ and Ψ , within which for any T ∈ R > , we define T N • = T − N − + β ⋆ : Ψ • = X y ∈ Z ∇ k , x P N T , x , y Z N y ; (4.78a) Ψ • = Z TT N X y ∈ Z ∇ k , x P NS , T , x , y · Z NS , y d ξ NS , y ; (4.78b) Ψ • = Z T N X y ∈ Z ∇ k , x P NS , T , x , y · Z NS , y d ξ NS , y ; (4.78c) Ψ • = Z TT N X y ∈ I ⋆ ,2 ∇ k , x P NS , T , x , y · N q NS , y Z NS , y d S ; (4.78d) Ψ • = Z T N X y ∈ I ⋆ ,2 ∇ k , x P NS , T , x , y · N q NS , y Z NS , y d S . (4.78e)We first estimate the quantities Ψ and Ψ as their analysis is more straightforward. Precisely, consequence of Proposition3.12, we have the following bound upon replacing the gradient by two copies of the heat kernel at different points: k Ψ k ∞ ;0 . N β ⋆ + ǫ ⋆ ,2 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z TT N ̺ − S , T d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 (4.79) . N − + β ⋆ + ǫ ⋆ ,2 (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 . (4.80)More interestingly, we first observe that | k | . m N . N ̺ / S , T for any S T N . Courtesy of Proposition 3.14, we deduce (cid:13)(cid:13) Ψ (cid:13)(cid:13) ∞ ;0 . N − + β ⋆ + ǫ ⋆ ,2 + ǫ · | k | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T N ̺ − + ǫ S , T d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 + N − β ⋆ + ǫ ⋆ ,2 + ǫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T N ̺ − + ǫ S , T d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ;0 · (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 (4.81) . ǫ N − + β ⋆ + ǫ ⋆ ,2 + ǫ m N (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 + N − β ⋆ + ǫ ⋆ ,2 + ǫ (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 . (4.82) t remains to estimate the quantities of Ψ and Ψ ; control of these quantities will amount to high-probability estimates incontrast to the deterministic natures of (4.80) and (4.82), and the procedure we will employ follows the idea of the proofbehind Proposition 6.1 in [ ] . In particular, consider the decomposition P ” | Ψ | & N ǫ c N , k + N ǫ c N , k (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 — ∞ X ℓ = P (cid:2) I ℓ ,1 (cid:3) ; (4.83) P ” | Ψ | & N ǫ c N , k + N ǫ c N , k (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 — ∞ X ℓ = P (cid:2) I ℓ ,2 (cid:3) ; (4.84)where we have introduced the following list of events parameterized by ℓ ∈ Z > : I ℓ ,1 • = ¦ | Ψ | & N ǫ c N , k + N ǫ c N , k (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 © ∩ I ℓ ,0 ; (4.85a) I ℓ ,2 • = ¦ | Ψ | & N ǫ c N , k + N ǫ c N , k (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 © ∩ I ℓ ,0 ; (4.85b) I ℓ ,0 • = ¦ ℓ + (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 ℓ © ; (4.85c)we emphasize the Ψ and Ψ -quantities appearing in the bounds (4.83) and (4.84) are actually evaluated at one particularpoint in space-time. Courtesy of the Chebyshev inequality, for any p ∈ R > arbitrarily large although universal, we deducethe following elementary calculation, in which Z N ; ℓ • = Z N ∧ ℓ : P ” | Ψ | & N ǫ c N , k + N ǫ c N , k (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 — . p N − p ǫ c − pN , k ∞ X ℓ = E h€ + (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 Š − p | Ψ | p I ℓ ,0 i (4.86) . N − p ǫ c − pN , k ∞ X ℓ = ℓ − p − p ǫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T N X y ∈ Z ∇ k , x P NS , T , x , y · Z N ; ℓ S , y d ξ NS , y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p L p ω (4.87)Courtesy of the BDG-type inequality of Lemma 3.1 in [ ] and Proposition 3.14, we have ℓ − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T N X y ∈ Z ∇ k , x P NS , T , x , y · Z N ; ℓ S , y d ξ NS , y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ω . p ℓ − N Z T N X y ∈ Z (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) · (cid:13)(cid:13)(cid:13) Z N ; ℓ S , y (cid:13)(cid:13)(cid:13) L p ω d S (4.88) . Ψ + Ψ , (4.89)where the additional quantities Ψ and Ψ are defined and estimated as follows: Ψ • = N − + β ⋆ + ǫ | k | Z T N ̺ − + ǫ S , T X y ∈ Z (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) d S (4.90a) . ǫ N − + β ⋆ + ǫ m N ; (4.90b) Ψ • = N − β ⋆ + ǫ Z T N ̺ − + ǫ S , T X y ∈ Z (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) d S (4.90c) . ǫ N − β ⋆ + ǫ . (4.90d)We combine (4.87), (4.89), (4.90b), and (4.90d) along with the definition of c N , k ∈ R > to deduce P ” | Ψ | & N ǫ c N , k + N ǫ c N , k (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 — . p , ǫ N − p ǫ c − pN , k (cid:0) N − p + p β ⋆ + p ǫ m pN + N − p β ⋆ + p ǫ (cid:1) ∞ X ℓ = ℓ − p ǫ (4.91) . p , ǫ N − p ǫ . (4.92)Proceeding along an identical strategy, from the following estimate, which we deduce through Proposition 3.12, ℓ − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z TT N X y ∈ Z ∇ k , x P NS , T , x , y · Z N ; ℓ S , y d ξ NS , y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ω . p ℓ − N Z TT N X y ∈ Z (cid:12)(cid:12)(cid:12) ∇ k , x P NS , T , x , y (cid:12)(cid:12)(cid:12) · (cid:13)(cid:13)(cid:13) Z N ; ℓ S , y (cid:13)(cid:13)(cid:13) L p ω d S (4.93) . N β ⋆ Z TT N ̺ − S , T X y ∈ Z (cid:12)(cid:12)(cid:12) P NS , T , x , y (cid:12)(cid:12)(cid:12) d S (4.94) . N − + β ⋆ , (4.95) e deduce P ” | Ψ | & N ǫ c N , k + N ǫ c N , k (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 — . p , ǫ N − p ǫ c − pN , k N − p + p β ⋆ ∞ X ℓ = ℓ − p ǫ (4.96) . p , ǫ N − p ǫ . (4.97)Finally, let us estimate the first quantity on the RHS of (4.77); to this end, we consider two distinct regimes. • We first assume that m N . N ̺ / T ; in this case, we have N ̺ / T &
1, so may employ Proposition 3.14 to deduce (cid:13)(cid:13) Z N (cid:13)(cid:13) − ∞ ;0 k Ψ k ∞ ;0 . ǫ N − + β ⋆ + ǫ ̺ − + ǫ T m N X y ∈ Z E N , − T , x , y + N − − β ⋆ + ǫ ̺ − + ǫ T X y ∈ Z E N , − T , x , y (4.98) . N − + β ⋆ + ǫ ̺ − + ǫ T + N − β ⋆ + ǫ ̺ − + ǫ T . (4.99) • We now assume m N & N ̺ / T ; in this case, observe we have ̺ T . N − m N = N − − β ⋆ + ǫ . For starters, we have | Ψ | Ψ + Ψ + Ψ , (4.100)where the new quantities are defined via Ψ • = X y ∈ Z P N T , x , y · (cid:12)(cid:12)(cid:12) Z N y − Z N x (cid:12)(cid:12)(cid:12) ; (4.101a) Ψ • = X y ∈ Z P N T , x + k , y · (cid:12)(cid:12)(cid:12) Z N y − Z N x + k (cid:12)(cid:12)(cid:12) ; (4.101b) Ψ • = (cid:12)(cid:12)(cid:12) ∇ k Z N x (cid:12)(cid:12)(cid:12) . (4.101c)We now employ the off-diagonal estimates from Proposition 3.12 to establish the following estimate within which Q x , ⋆ • = { y ∈ Z : | y − x | N − β ⋆ } ⊆ Z : Ψ . X y ∈ Q x , ⋆ P N T , x , y · (cid:12)(cid:12)(cid:12) Z N y − Z N x (cid:12)(cid:12)(cid:12) + (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 X y Q x , ⋆ P N T , x , y (4.102) . X y ∈ Q x , ⋆ P N T , x , y · (cid:12)(cid:12)(cid:12) Z N y − Z N x (cid:12)(cid:12)(cid:12) + e − log N (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 . (4.103)We estimate the remaining summation from the RHS of (4.103) with the following moment estimate consequenceof the a priori estimates on near-stationary initial data: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X y ∈ Q x , ⋆ P N T , x , y · (cid:12)(cid:12)(cid:12) Z N y − Z N x (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ω X y ∈ Q x , ⋆ P N T , x , y · (cid:13)(cid:13)(cid:13) Z N y − Z N x (cid:13)(cid:13)(cid:13) L p ω (4.104) . p , ǫ N − − β ⋆ + ǫ . (4.105)Similarly, we have Ψ . X y ∈ Q x + k , ⋆ P N T , x + k , y · (cid:12)(cid:12)(cid:12) Z N y − Z N x + k (cid:12)(cid:12)(cid:12) + e − log N (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ;0 (4.106)along with the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X y ∈ Q x + k , ⋆ P N T , x + k , y · (cid:12)(cid:12)(cid:12) Z N y − Z N x + k (cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ω . p , ǫ N − − β ⋆ + ǫ . (4.107)Finally, we have E (cid:13)(cid:13) Ψ (cid:13)(cid:13) L p ω . p , ǫ N − + ǫ m N . (4.108)We conclude this second bullet point by turning the high-moment estimates in (4.105), (4.107), and (4.108) intohigh-probability estimates via the Chebyshev inequality. In particular, recalling the definition of c N , k ∈ R > in thestatement of the current result, we deduce the following provided any p ∈ R > arbitrarily large but universal and ǫ ∈ R > arbitrarily large but universal: X j = P ”(cid:12)(cid:12) Ψ j (cid:12)(cid:12) & N ǫ c N , k + N ǫ (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 — . p , ǫ N − p ǫ . (4.109) n the spirit of Proposition 6.1 in [ ] , we may employ the stochastic continuity strategy within Lemma A.1 to deduce thefollowing estimate from (4.92), (4.97), (4.99), (4.100), and (4.109): X j = P ”(cid:13)(cid:13) Ψ j (cid:13)(cid:13) ∞ ;0 & N ǫ c N , k + N ǫ (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 — . D , ǫ N − D . (4.110)Combining this estimate in (4.110) along with the deterministic estimates of (4.80) and (4.82) completes the proof uponrecalling the definition of c N , k ∈ R > . (cid:3) The second preliminary ingredient that we require is the following mesoscopic fluctuation estimate for the single con-served quantity of the interacting particle system, or precisely the particle number.
Corollary 4.14.
Provided κ ∈ R > and ǫ ∈ R > and retaining the notation of Lemma 4.13 , consider the following event: (cid:2) I κ , ǫ (cid:3) • = •(cid:13)(cid:13)(cid:13) ̺ • ¯ A m N η N • , • Z N • , • (cid:13)(cid:13)(cid:13) ∞ ; κ & N + ǫ m − N c N , κ + N + ǫ m − N c N , κ (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 ˜ , (4.111) where we have introduced above the following quantity of interest: ¯ A m N η NT , y Z NT , y • = ÝX m N w = η NT , y + w · Z NT , y . (4.112) Provided any D ∈ R > , we have the following estimate for any κ ∈ R > arbitrarily large and any ǫ ∈ R > arbitrarily small: P (cid:2) I κ , ǫ (cid:3) . κ , ǫ , D N − D . (4.113) Proof.
We will again assume κ = A m N η N • , • Z N • , • . N m − N · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log ‚ + ∇ m N Z N • , • Z N • , • Œ(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · Z N • , • . (4.114)Because m N . N − β ⋆ by definition, upon Taylor expansion of the exponential function defining Z N , we have the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ m N Z N • , • Z N • , • (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . N − β ⋆ . (4.115)Because we have | log ( + x ) | . | x | uniformly over the set { x ∈ R : | x | } , from (4.114) and (4.115) we deduce¯ A m N η N • , • Z N • , • . N m − N · (cid:12)(cid:12)(cid:12) ∇ m N Z N • , • (cid:12)(cid:12)(cid:12) . (4.116)For any D ∈ R > and ǫ ∈ R > , with probability at least 1 − c D , ǫ N − D , where c D , ǫ ∈ R > depends only on D , ǫ ∈ R > , we havethe following estimate courtesy of Lemma 4.13: N m − N · (cid:12)(cid:12)(cid:12) ∇ m N Z N • , • (cid:12)(cid:12)(cid:12) . N + ǫ m − N c N ,0 + N + ǫ m − N c N ,0 (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ;0 . (4.117)The result then follows after combining (4.116) and (4.117). (cid:3) Proof of
Proposition 4.12 . By definition of C N , I ,4 , courtesy of Proposition 3.12 and Corollary 4.14, for any D ∈ R > arbi-trarily large and any ǫ ∈ R > arbitrarily small, we have the following estimate with probability at least 1 − c D , ǫ N − D , where c D , ǫ ∈ R > depends only on D , ǫ ∈ R > : (cid:13)(cid:13) C N , I ,4 (cid:13)(cid:13) ∞ ; κ . N β ⋆ + ǫ ⋆ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T ̺ − S , T ̺ − S d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ; κ · (cid:13)(cid:13)(cid:13) ¯ A m N η N • , ± ⋆ Z N • , ± ⋆ (cid:13)(cid:13)(cid:13) ∞ ; κ (4.118) . N + β ⋆ + ǫ ⋆ ,2 + ǫ m − N c N , κ + N + ǫ ⋆ ,2 + ǫ m − N c N , κ (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ; κ . (4.119)It remains to compute the N -dependent prefactor: N + β ⋆ + ǫ ⋆ ,2 + ǫ m − N c N , κ . N β ⋆ + ǫ c N , κ (4.120) = N − + β ⋆ + ǫ + N − + β ⋆ + ǫ + N − β ⋆ + ǫ + N − + β ⋆ + ǫ . (4.121)The desired estimate now follows from (4.119) and (4.121) provided β ⋆ ∈ R > is sufficiently small although still universaland ǫ ∈ R > is sufficiently small depending only on β ⋆ ∈ R > . (cid:3) Proof of Proposition 4.2.
We observe that k C N , I k ∞ ; κ k C N , I ,1 k ∞ ; κ + k C N , I ,3 k ∞ ; κ + k C N , I ,4 k ∞ ; κ . Moreover, to analyze C N , II , courtesy of Proposition 3.12, we establish the following estimate for C N , II upon realizing the additional factor of N − ttached to e q N • , • in the total quantity Q N • , • : Ψ . κ N + ǫ ⋆ ,2 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T X y ∈ Z P NS , T , x , y E N , κ S , T , x , − ⋆ d S (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ; κ (4.122) . κ N − + ǫ ⋆ ,2 + β ⋆ . (4.123)The desired estimate now follows from combining the previous estimates with those from Proposition 4.5, Proposition 4.9,and Proposition 4.12, respectively, assuming that β ⋆ ∈ R > is sufficiently small although still universal. This completes theproof. (cid:3)
5. P
ATHWISE C OMPARISON
This current section is dedicated towards an appropriate comparison between the microscopic Cole-Hopf transform Z N and the following auxiliary space-time functions defined through the following characterizing stochastic integral equations;we emphasize these aforementioned comparison estimates are in the pathwise topology with high-probability, as indicatedin the title of this section. Definition 5.1.
We first define Y N • , • as the unique solution to the following space-time stochastic integral equation, wherethe allowed space-time coordinates are exactly those for Z N : Y NT , x • = X y ∈ Z P N T , x , y Z N y + Z T X y ∈ Z P NS , T , x , y Y NS , y d ξ NS , y . (5.1)Second, let us define X N • , • as the unique solution to the following space-time stochastic integral equation, again where theallowed space-time coordinates are exactly those for Z N and Y N : X NT , x • = X y ∈ Z ¯ P N T , x , y Z N y + Z T X y ∈ Z ¯ P NS , T , x , y X NS , y d ξ NS , y . (5.2)Lastly, we define Φ N ,1 • = Z N − Y N and Φ N ,2 • = Y N − X N . Remark . Intuitively, the above object Y N is obtained by simply forgetting the quantity corresponding to the statistic of Q N within Proposition 2.1, and X N is obtained via afterwards replacing this heat kernel P N by the heat kernel ¯ P N . Indeed,the comparison we achieve in this section amounts to showing each of these "forgetful" steps contributes negligible error.To be precise, the comparison estimates we require in this section for the proof of Theorem 1.4 are presented as follows.Applying Lemma B.1 and Lemma 5.4 below, such result additionally provides tightness of Z N , and every limit point is equalto that of X N ; as noted before Proposition 6.1, it then suffices to prove Theorem 1.4 for X N which is standard via [ ] . Proposition 5.3.
Provided constants κ ∈ R > and β ∈ R > , consider the following pair of events, where the implied constantsare universal: ” E κ , β — • = ”(cid:13)(cid:13) Φ N ,1 (cid:13)(cid:13) ∞ ; κ & N − β — ; (5.3a) ” E κ , β — • = ”(cid:13)(cid:13) Φ N ,2 (cid:13)(cid:13) ∞ ; κ & N − β — . (5.3b) There exist universal constants β u ,1 , β u ,2 ∈ R > such that provided any κ ∈ R > sufficiently large, we have P ” E κ , β u ,1 — + P ” E κ , β u,1 — . κ N − β u ,2 . (5.4)Roughly speaking, the proof for Proposition 5.3 is given through the following strategy; we focus upon the estimate for Φ N ,1 , in particular the replacement Z N Y N just for this illustration. In this case, the estimate in Proposition 4.2 suggeststhat provided the a priori estimate k Z N k ∞ ; κ . N ǫ given ǫ ∈ R > sufficiently small, then via some elementary calculationswe deduce (5.3a). Moreover, the estimate (5.3a) in turn implies that Z N ≈ Y N , and some parabolic estimate via Gronwallinequalities implies that the required a priori estimate for Z N just presented is satisfied for Y N ; therefore, Z N inherits thisestimate, and so we are in business.To circumvent whatever potentially circular nature of this argument, we employ a technical pathwise argument in spiritof standard stochastic analysis via suitable stopping times.5.1. A Priori Estimates.
As mentioned in the prior brief outline of the proof for Proposition 5.3, we require the followinga priori kk ∞ ; κ -estimates for the gadgets Y N and X N . emma 5.4. Provided any constants κ , δ ∈ R > , consider the following events in which the implied constants are universal: G Y κ , δ • = ”(cid:13)(cid:13) Y N (cid:13)(cid:13) ∞ ; κ & N δ — ; (5.5a) G X κ , δ • = ”(cid:13)(cid:13) X N (cid:13)(cid:13) ∞ ; κ & N δ — . (5.5b) Provided any κ ∈ R > sufficiently large but universal, any δ ∈ R > arbitrarily small but universal, and any D ∈ R > arbitrarilylarge but universal, we have the following assuming β ⋆ ∈ R > is sufficiently small but still universal: P ” G Y κ , δ — + P ” G X κ , δ — . κ , δ , D N − D . (5.6) Proof.
We record the proof for the event G Y κ , δ ; the proof for the event G X κ , δ follows from almost identical procedure, becauseany heat kernel estimates we employ for P N are certainly valid for ¯ P N . Moreover, we introduce the following reduction. • It suffices to establish the following estimate for one single κ ∈ R > uniformly over exponents p ∈ R > sufficientlylarge but universal: kk Y N k L p ω k ∞ ; κ . κ , p
1. (5.7)Indeed, uniformly in T ∈ R > satisfying T x ∈ Z , by the estimate within (5.7), we deducethe following high-probability estimate for any c ∈ R > via the Chebyshev inequality: P ” c · e − κ | x | N Y NT , x & N δ — . κ , p N − p δ c p . (5.8)We define κ ′ • = κ , and consider the decomposition Z = ∪ j ∈ Z I j , N , in which I j , N • = jN + J N K . As the consequenceof a union bound and a subsequent elementary calculation, we establish the following estimate from the estimate(5.8) uniformly in T ∈ R > satisfying T P • sup x ∈ Z e − κ ′ | x | N Y NT , • & N δ ˜ X j ∈ Z P – sup x ∈ I j , N e − κ ′ | x | N Y NT , • & N δ ™ (5.9) . κ , p X j ∈ Z N − p δ e − κ p | j | (5.10) . κ , p N − p δ . (5.11)To establish the proposed estimate, we then combine (5.11) with the stochastic continuity estimate within LemmaA.1, at the cost of a slightly larger parameter κ ∈ R > .In view of this preceding bullet point, it remains to establish that one-point moment estimate (5.7). To this end, as before,we again first assume κ = [ ] , forexample, while treating the slightly less robust heat kernel estimates for P N . More precisely, we first obtain the followingpreliminary moment estimate courtesy of Lemma 3.1 in [ ] , for example: (cid:13)(cid:13)(cid:13) Y NT , x (cid:13)(cid:13)(cid:13) L p ω . p Ψ + Ψ , (5.12)where we have introduced this following set of quantities along with their corresponding estimates, within which ǫ ∈ R > is arbitrarily small but universal; we briefly note that the second term in the definition of the upper-bound Ψ below comesfrom the consideration of time-scales satisfying T . N − , in which such a term in Ψ is obtained via noting, by Proposition2.1, every jump in Z N contributes a net change in Z N of order at most N − / up to universal constants: Ψ • = X y ∈ Z P N T , x , y (cid:13)(cid:13)(cid:13) Z N y (cid:13)(cid:13)(cid:13) L p ω (5.13a) . p
1; (5.13b) Ψ • = Z T N X y ∈ Z (cid:12)(cid:12)(cid:12) P NS , T , x , y (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) Z NS , y (cid:13)(cid:13)(cid:13) L p ω d S + N − + ǫ (cid:13)(cid:13)(cid:13) Z N x (cid:13)(cid:13)(cid:13) L p ω ; (5.13c)indeed, this estimate (5.13b) is a consequence of definition of near-stationary initial data for Z N . To estimate Ψ , we willemploy the assumption that β ⋆ ∈ R > is sufficiently small although still universal. For starters, we first recall the sub-lattice I ∂ ,1 • = J − N β ∂ ,1 , N β ∂ ,1 K ⊆ Z , where β ∂ ,1 • = − ǫ ∂ ,1 given ǫ ∂ ,1 • = β ⋆ + ǫ and ǫ ∈ R > is arbitrarily small though still universal.Now, courtesy of Proposition 3.12, we deduce, again as consequence of the near-stationary initial data assumption, Ψ . Ψ + Ψ + Ψ + N − + ǫ , (5.14) here we have introduced the following triple of quantities with their corresponding estimates, for which ǫ , ǫ , ǫ ∈ R > are arbitrarily small but universal constants: Ψ • = Z T N X y ∈ Z (cid:12)(cid:12)(cid:12) ¯ P NS , T , x , y (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) Y NS , y (cid:13)(cid:13)(cid:13) L p ω d S ; (5.15a) Ψ • = Z T N X y I ∂ ,1 (cid:12)(cid:12)(cid:12) P NS , T , x , y − ¯ P NS , T , x , y (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) Y NS , y (cid:13)(cid:13)(cid:13) L p ω d S ; (5.15b) . ǫ , ǫ N − ǫ Z T ̺ − + ǫ S , T · sup x ∈ Z (cid:13)(cid:13)(cid:13) Y NS , x (cid:13)(cid:13)(cid:13) L p ω d S ; (5.15c) Ψ • = Z T N X y ∈ I ∂ ,1 (cid:12)(cid:12)(cid:12) P NS , T , x , y (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) Y NS , y (cid:13)(cid:13)(cid:13) L p ω d S (5.15d) . N − + β ⋆ + ǫ | I ∂ ,1 | Z T ̺ − + ǫ S , T · sup x ∈ Z (cid:13)(cid:13)(cid:13) Y NS , x (cid:13)(cid:13)(cid:13) L p ω d S (5.15e) . N − β ⋆ + ǫ Z T ̺ − + ǫ S , T · sup x ∈ Z (cid:13)(cid:13)(cid:13) Y NS , x (cid:13)(cid:13)(cid:13) L p ω d S . (5.15f)To estimate Ψ , we employ the heat kernel estimate for ¯ P N within Proposition A.1 within [ ] , for example, and similarto the proof of Proposition 3.2 in [ ] , we establish Ψ . Z T ̺ − S , T · sup x ∈ Z (cid:13)(cid:13)(cid:13) Y NS , x (cid:13)(cid:13)(cid:13) L p ω d S . (5.16)We combine (5.12), (5.13b), (5.14), (5.15c), (5.15f), and (5.16) to deducesup x ∈ Z (cid:13)(cid:13)(cid:13) Y NT , x (cid:13)(cid:13)(cid:13) L p ω . ǫ , ǫ + Z T ̺ − S , T · sup x ∈ Z (cid:13)(cid:13)(cid:13) Y NS , x (cid:13)(cid:13)(cid:13) L p ω d S + N − β ∂ ,3 Z T ̺ − + ǫ S , T · sup x ∈ Z (cid:13)(cid:13)(cid:13) Y NS , x (cid:13)(cid:13)(cid:13) L p ω d S . (5.17)The estimate (5.7) thereby follows via the singular Gronwall inequality; this completes the proof. (cid:3) We additionally record a useful a priori estimate for Z N ; although the following estimate appears severely sub-optimalin view of the main result in Theorem 1.4, it will serve convenient to deal with technical issues. Lemma 5.5.
Provided κ ∈ R > , consider the following event with universal implied constant: (cid:2) G Z κ (cid:3) • = ”(cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ & e N — . (5.18) Provided any κ ∈ R > arbitrarily large but universal, if β ⋆ ∈ R > is sufficiently small though still universal, we have P (cid:2) G Z κ (cid:3) . κ e − N / . (5.19) Proof.
Similar to the proof of Lemma 5.4, it suffices to establish the following uniformly in exponents p ∈ R > :sup T ∈ [ ] sup x ∈ Z e − κ | x | N (cid:13)(cid:13)(cid:13) Z NT , x (cid:13)(cid:13)(cid:13) L p ω . p e N / . (5.20)The strategy we employ to obtain (5.20) is effectively an identical though sub-optimal replica of the proof for Lemma 5.4;similar to that proof, we assume κ = (cid:13)(cid:13)(cid:13) Z NT , x (cid:13)(cid:13)(cid:13) L p ω . Ψ + Ψ + Ψ ; (5.21)the newly introduced quantities Ψ , Ψ , and Ψ from the estimate (5.21) above are defined via Ψ • = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X y ∈ Z P N T , x , y Z N y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ω ; (5.22a) Ψ • = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T X y ∈ Z P NS , T , x , y · Z NS , y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ω ; (5.22b) Ψ • = (cid:13)(cid:13)(cid:13) C N , I T , x (cid:13)(cid:13)(cid:13) L p ω + (cid:13)(cid:13)(cid:13) C N , II T , x (cid:13)(cid:13)(cid:13) L p ω . (5.22c) e first estimate Ψ through the Cauchy-Schwarz inequality, the fact that P N is a probability measure with respect to theforward spatial coordinate, and the assumption of near-stationary initial data and the resulting a priori pointwise momentbounds: Ψ . X y ∈ Z P N T , x , y (cid:13)(cid:13)(cid:13) Z N y (cid:13)(cid:13)(cid:13) L p ω (5.23) . p
1. (5.24)To estimate Ψ , we again employ Lemma 3.1 from [ ] and Proposition 3.12 similar to the proof for Lemma 5.4 to establishthe next pair of estimates: Ψ . p N Z T X y ∈ Z (cid:12)(cid:12)(cid:12) P NS , T , x , y (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) Z NS , y (cid:13)(cid:13)(cid:13) L p ω d S + N − (cid:13)(cid:13)(cid:13) Z N x (cid:13)(cid:13)(cid:13) L p ω (5.25) . N β ⋆ Z T ̺ − S , T X y ∈ Z P NS , T , x , y (cid:13)(cid:13)(cid:13) Z NS , y (cid:13)(cid:13)(cid:13) L p ω d S + N − . (5.26)Finally, we estimate Ψ again via Proposition 3.12 and that by definition, we have | I ⋆ ,2 | . N ǫ ⋆ ,2 : Ψ . N β ⋆ + ǫ ⋆ ,2 Z T ̺ − S , T · sup x ∈ Z (cid:13)(cid:13)(cid:13) Z NS , x (cid:13)(cid:13)(cid:13) L p ω . (5.27)We deduce (5.20) via (5.21), (5.24), (5.26), (5.27) and the singular Gronwall inequality. This completes the proof. (cid:3) Proof of Proposition 5.3.
We first analyze the quantity Φ N ,1 and the corresponding event E κ , β u,1 . To this end, we firstobserve the following integral equation courtesy of unfolding definitions and (4.1): Φ N ,1 T , x = Z T X y ∈ Z P NS , T , x , y Φ N ,1 S , y d ξ NS , y + E N , − T , x + E N , + T , x . (5.28)We follow the approach within Section 8 from [ ] , beginning with the following stopping times. Definition 5.6.
Provided ǫ ST ∈ R > arbitrarily small although universal, we consider the following random stopping times,within which κ ∈ R > is arbitrary large but universal and β u ,1 ∈ R > and ǫ ∈ R > are the parameters in Proposition 4.2: τ κ , β u,1 , ǫ • = inf ¦ T (cid:13)(cid:13) E N , I (cid:13)(cid:13) ∞ ; κ ; T + (cid:13)(cid:13) E N , II (cid:13)(cid:13) ∞ ; κ ; T & N − β u ,1 + N − β u ,1 (cid:13)(cid:13) Z N (cid:13)(cid:13) + ǫ ∞ ; κ © ∧
1; (5.29a) τ ( ) κ , ǫ ST • = inf ¦ T (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ ; T & N ǫ ST © ∧
1; (5.29b) τ ( ) κ , ǫ ST • = inf ¦ T (cid:13)(cid:13) Φ N ,1 (cid:13)(cid:13) ∞ ; κ ; T & N − ǫ ST © ∧
1; (5.29c) τ κ • = τ κ , β u,1 , ǫ ∧ τ ( ) κ , ǫ ST ∧ τ ( ) κ , ǫ ST ∧
1; (5.29d)above, we have introduced the family of norms: k Ψ k ∞ ; κ ; T • = sup S ∈ [ T ] sup x ∈ Z e − κ | x | N | Ψ S , x | . (5.30)Moreover, we define e Φ N ,1 as the solution to the following stochastic integral equation with those usual allowed space-timecoordinates: e Φ N ,1 T , x = Z T X y ∈ Z P NS , T , x , y e Φ N ,1 S , y d ξ NS , y + e E N , I T , x + e E N , II T , x ; (5.31) e E N , • T , x • = X w ∈ Z P NT ∧ τ κ , T , x , w · E N , • T ∧ τ κ , w . (5.32)Provided these stopping times, the proof for the estimate for Φ N ,1 in Proposition 5.3 follows immediately from the nextresult, which we thus dedicate the remainder of this first half of the subsection towards. Lemma 5.7.
We have P [ τ κ = ] . κ N − ǫ for any sufficiently large κ ∈ R > , where ǫ ∈ R > is some universal constant. To relate e Φ N ,1 with Φ N ,1 , we appeal to the following elementary lemma. Lemma 5.8.
For any T f ∈ [
0, 1 ] , we have the containment of events { τ κ = T f } ⊆ ∩ T ∈ [ T f ] ∩ x ∈ Z { e Φ N ,1 T , x = Φ N ,1 T , x } .Proof. Conditioning on { τ κ = T f } , we observe e C N , ± = C N , ± globally on [ T f ] × Z . Thus, globally on this space-time block,we obtain the following stochastic integral equation for the dynamics of the difference e Φ N ,1 − Φ N ,1 via a straightforward alculation: e Φ N ,1 T , x − Φ N ,1 T , x • = Z T X y ∈ Z P NS , T , x , y · ”e Φ N ,1 S , y − Φ N ,1 S , y — d ξ NS , y . (5.33)This quantity e Φ N ,1 − Φ N ,1 then solves a microscopic version of the stochastic heat equation provided vanishing initial dataon the space-time block [ T f ] × Z ; any one of a number of results and / or estimates thus implies e Φ N ,1 − Φ N ,1 = (cid:3) To illustrate the utility behind this proxy e Φ N ,1 , we introduce the following high-probability estimate which is very muchin the spirit of Lemma 5.4 and its proof. Lemma 5.9.
Provided κ , β ∈ R > , consider the event (cid:2) G κ , β (cid:3) • = ”(cid:13)(cid:13)e Φ N ,1 (cid:13)(cid:13) ∞ ; κ & N − β — . (5.34) Provided β = β u ,1 , where β u ,1 ∈ R > is the constant from Proposition 4.2 , we have the following estimate for any κ , D ∈ R > upon choosing ǫ ST ∈ R > sufficiently small depending only on β u ,1 ∈ R > : P (cid:2) G κ , β (cid:3) . κ , D N − D . (5.35) Proof.
We will assume κ = L p -estimate for all p ∈ R > :sup T ∈ [ ] sup x ∈ Z (cid:13)(cid:13)(cid:13)e Φ N ,1 T , x (cid:13)(cid:13)(cid:13) L p ω . p N − β u ,1 . (5.36)To this end, by definition of e Φ N ,1 we have, also courtesy of the BDG-type inequality of Lemma 3.1 in [ ] , (cid:13)(cid:13)(cid:13)e Φ N ,1 T , x (cid:13)(cid:13)(cid:13) L p ω . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T X y ∈ Z P NS , T , x , y e Φ N ,1 S , y d ξ NS , y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ω + N − β u ,1 + ǫ ST (5.37) . p Ψ + Ψ + Ψ + N − β u ,1 + ǫ ST (5.38)where, courtesy of Proposition 3.12 and Proposition A.1 of [ ] , for ǫ , ǫ , ǫ ∈ R > sufficiently small universal constants,we have the following list of definitions and corresponding estimates: Ψ • = N Z T X y ∈ Z (cid:12)(cid:12)(cid:12) ¯ P NS , T , x , y (cid:12)(cid:12)(cid:12) · (cid:13)(cid:13)(cid:13)e Φ N ,1 S , y (cid:13)(cid:13)(cid:13) L p ω d S (5.39a) . Z T ̺ − S , T X y ∈ Z ¯ P NS , T , x , y · (cid:13)(cid:13)(cid:13)e Φ N ,1 S , y (cid:13)(cid:13)(cid:13) L p ω d S ; (5.39b) Ψ • = N Z T X y I ∂ ,1 (cid:12)(cid:12)(cid:12) P NS , T , x , y − ¯ P NS , T , x , y (cid:12)(cid:12)(cid:12) · (cid:13)(cid:13)(cid:13)e Φ N ,1 S , y (cid:13)(cid:13)(cid:13) L p ω d S (5.39c) . ǫ , ǫ N − ǫ Z T ̺ − + ǫ S , T X y ∈ Z (cid:12)(cid:12)(cid:12) P NS , T , x , y − ¯ P NS , T , x , y (cid:12)(cid:12)(cid:12) · (cid:13)(cid:13)(cid:13)e Φ N ,1 S , y (cid:13)(cid:13)(cid:13) L p ω d S ; (5.39d) Ψ • = N Z T X y ∈ I ∂ ,1 (cid:12)(cid:12)(cid:12) P NS , T , x , y (cid:12)(cid:12)(cid:12) · (cid:13)(cid:13)(cid:13)e Φ N ,1 S , y (cid:13)(cid:13)(cid:13) L p ω d S (5.39e) . N − β ⋆ Z T ̺ − + ǫ S , T · sup x ∈ Z (cid:13)(cid:13)(cid:13)e Φ N ,1 S , x (cid:13)(cid:13)(cid:13) L p ω d S . (5.39f)We then combine (5.38), (5.39b), (5.39d), and (5.39f) along with the singular Gronwall inequality to deduce the desiredestimate (5.36). This completes the proof. (cid:3) We proceed to establish the estimate of (5.3a); we choose ǫ ST • = β u ,1 and then introduce the following reductions. • We first consider the event G κ ,1 • = { τ κ , β u ,1 , ǫ = } . Courtesy of Proposition 4.2, we know P [ G C κ ,1 ] . ǫ N − β u ,2 , where β u ,2 ∈ R > is a universal constant. • Second, we consider the event of G κ ,2 • = {k e Φ N ,1 k ∞ ; κ . N − β u ,1 } , which, in turn, provides k e Φ N ,1 k ∞ ; κ . N − ǫ ST tobe totally transparent. Courtesy of Lemma 5.9, we have P [ G C κ ,2 ] . D N − D for any D ∈ R > . We proceed to define events corresponding to a priori estimates from Lemma 5.4 and Lemma 5.5. First, we define G κ ,3 • = {k Y N k ∞ ; κ . N ǫ ST } ; courtesy of Lemma 5.4, we have P [ G C κ ,3 ] . D N − D for any D ∈ R > . Moreover, we nowdefine G κ ,4 • = {k Z N k ∞ ; κ . e N } , so that P [ G C κ ,4 ] . D N − D also for any D ∈ R > . Observe that G κ ,4 ⊆ G ′ κ ,4 , where ” G ′ κ ,4 — • = –(cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ . e − N + sup T ∈ [ ] sup | x | . N e − κ | x | N Z NT , x ™ . (5.40) • The last event we introduce requires the following discretization. First, let us decompose [
0, 1 ] = ∪ j = I j , where I j • = [ jN − , ( j + ) N − ] ⊆ [
0, 1 ] . The last event is then defined as follows, in which we recall ¯ ξ [ S , T ] , x denotesthe sum of the contributions of all Poisson clocks at x ∈ Z in the time-block [ S , T ] ⊆ R > : (cid:2) G κ ,5 (cid:3) • = – sup j ∈ J ,0,200 K sup | x | . N ¯ ξ I j , x . N ǫ ™ . (5.41)A standard application of concentration or large-deviation inequalities implies that P [ G κ ,5 ] . ǫ e − log N . • Finally, let us define the total intersection of these events G κ ,total • = T j = G κ , j , so that P ” G C κ ,total — . N − β u ,2 . (5.42)The constraints defining G κ ,total are those that we condition on and employ for the upcoming pathwise argument.To prove (5.3a), courtesy of (5.42), it suffices to condition on the previous event G κ ,total . Now, suppose that on such event G κ ,total , we have τ κ =
1. Via definition of the event G κ ,total , we must have either that τ κ = τ ( ) κ , ǫ ST or that τ κ = τ ( ) κ , ǫ ST is true.Within the following paragraph, we argue that τ κ = τ ( ) κ , ǫ ST , so that the latter possibility is impossible on G κ ,total . • Suppose that τ κ = τ ( ) κ , ǫ ST for the sake of establishing contradiction. As consequence of Lemma 5.8, for all times inthe time-block S ∈ [ τ κ ] = [ τ ( ) κ , ǫ ST ] , we have e Φ N ,1 S , • = Φ N ,1 S , • . However, on the event G κ ,total , we have the pathwiseestimate of k e Φ N ,1 S , • k ∞ ; κ . N − ǫ ST , from which we deduce k Φ N ,1 k ∞ ; κ ; τ κ . N − ǫ ST . Moreover, within the event G κ ,total ,we may extend this estimate to k Φ N ,1 k ∞ ; κ ; τ + κ . N − ǫ ST , where τ + κ • = τ κ + e − N . However, this is a contradiction tothe definition of τ ( ) κ , ǫ ST , so that we necessarily have τ κ = τ ( ) κ , ǫ ST on the event G κ ,total .Thus, we must necessarily have τ κ = τ ( ) κ , ǫ ST on this event G κ ,total ; recall as well our assumption τ κ =
1, from which we aimto deduce another contradiction. The first step that we will take consists of the following inequality, for which we employconstraints satisfied on the event G κ ,total as well as the identification e Φ N ,1 = Φ N ,1 on [ τ κ ] × Z as courtesy of Lemma 5.8: (cid:13)(cid:13) Z N (cid:13)(cid:13) ∞ ; κ ; τ κ (cid:13)(cid:13) Y N (cid:13)(cid:13) ∞ ; κ + (cid:13)(cid:13) Φ N ,1 (cid:13)(cid:13) ∞ ; κ ; τ κ (5.43) = (cid:13)(cid:13) Y N (cid:13)(cid:13) ∞ ; κ + (cid:13)(cid:13)e Φ N ,1 (cid:13)(cid:13) ∞ ; κ ; τ κ (5.44) (cid:13)(cid:13) Y N (cid:13)(cid:13) ∞ ; κ + (cid:13)(cid:13)e Φ N ,1 (cid:13)(cid:13) ∞ ; κ (5.45) . N ǫ ST + N − ǫ ST (5.46) . N ǫ ST . (5.47)Again, courtesy of the definition of the event G κ ,total , we may extend the estimate (5.47) to the estimate k Z N k ∞ ; κ ; τ + κ . N ǫ ST ,where we recall τ + κ • = τ κ + e − N . Because τ κ = τ ( ) κ , ǫ ST on the event G κ ,total , we obtain a contradiction to the definition of thelatter time τ ( ) κ , ǫ ST . This completes the proof of (5.3a).Thus, it remains to establish (5.3b) for which we employ a different, more direct approach. To be more precise, followingthe proof behind Lemma 5.4, it suffices to establish the following moment estimate provided any κ ∈ R > sufficiently largeand p ∈ R > arbitrarily large but universal:sup T ∈ [ ] sup x ∈ Z e − κ | x | N (cid:13)(cid:13)(cid:13) Φ N ,2 T , x (cid:13)(cid:13)(cid:13) L p ω . κ , p N − β u ,1 . (5.48)We consider two regimes towards establishing (5.3a). Roughly speaking, the first regime is the short-time regime and thesecond regime is what is left, namely times for which we may exploit the smoothing property of the heat kernel P N .First consider T ∈ R > satisfying T N − δ for δ ∈ R > arbitrarily small but universal. To this end, it suffices to obtainthe following estimate with δ ∈ R > a universal constant, in which Φ N ,2,1 • = Y N − Z N • and Φ N ,2,2 • = X N − Z N • :sup x ∈ Z (cid:13)(cid:13)(cid:13) Φ N ,2,1 T , x (cid:13)(cid:13)(cid:13) L p ω + sup x ∈ Z (cid:13)(cid:13)(cid:13) Φ N ,2,2 T , • (cid:13)(cid:13)(cid:13) L p ω . p N − δ . (5.49) e establish the estimate for Φ N ,2,1 , since the estimate for Φ N ,2,2 follows from an identical approach. First observe (cid:13)(cid:13)(cid:13) Φ N ,2,1 T , x (cid:13)(cid:13)(cid:13) L p ω . Ψ + Ψ , (5.50)where, at least courtesy of Proposition 3.12, we have Ψ • = X y ∈ Z P N T , x , y (cid:13)(cid:13)(cid:13) Z N y − Z N x (cid:13)(cid:13)(cid:13) L p ω (5.51a) . p N − δ ; (5.51b) Ψ • = Z T N X y ∈ Z P NS , T , x , y · (cid:13)(cid:13)(cid:13) Y NS , y (cid:13)(cid:13)(cid:13) L p ω d S + N − (cid:13)(cid:13)(cid:13) Z N x (cid:13)(cid:13)(cid:13) L p ω (5.51c) . p N − δ + N − ; (5.51d)above, the estimate in (5.51b) is a consequence of the a priori regularity estimates for the near-stationary initial data, andthe estimate in (5.51d) follows from the proof for Lemma 5.4 along with the moment estimate therein. Combining (5.50),(5.51b), and (5.51d) completes the proof of the estimate (5.49) for Φ N ,2,1 .Second, it suffices to assume T ∈ R > satisfies N − δ T
1. Unlike the previous bullet point, within this time-regime,we may not necessarily be able to compare Y N and X N both to Z N • directly. Alternatively, we observe the following integralequation courtesy of unfolding definition: Φ N ,2 T , x = Ψ + Ψ + Ψ (5.52)where above, we have introduced the following quantities within which we additionally establish some convenient notationin the difference of relevant heat kernels e P N • = P N − ¯ P N : Ψ • = X y ∈ Z e P N T , x , y Z N y ; (5.53a) Ψ • = Z T X y ∈ Z e P NS , T , x , y · Y NS , y d ξ NS , y ; (5.53b) Ψ • = Z T X y ∈ Z ¯ P NS , T , x , y · Φ N ,2 S , y d ξ NS , y . (5.53c)To estimate Ψ , assuming β ⋆ ∈ R > sufficiently small though universal, for a sufficiently small constant β ∂ ∈ R > , we havethe following upper bound courtesy of Proposition 3.12, for which ǫ , ǫ ∈ R > is some pair of universal constants whoseparticular values are not important: k Ψ k L p ω . ǫ , ǫ N − ǫ ̺ − + ǫ T · X y ∈ Z N − ̺ − T E N , − T , x , y · (cid:13)(cid:13)(cid:13) Z N y (cid:13)(cid:13)(cid:13) L p ω + N − + β ⋆ ̺ − T X y ∈ I ∂ ,1 (cid:13)(cid:13)(cid:13) Z N y (cid:13)(cid:13)(cid:13) L p ω (5.54) . p N − ǫ + δ + N − ǫ ∂ ,1 + β ⋆ + δ . (5.55)Indeed, this estimate in (5.55) is a consequence of the a priori initial data upper bounds. To then estimate Ψ , we employan approach similar to the estimate (5.14); in particular, we obtain k Ψ k L p ω . p Z T N X y ∈ Z (cid:12)(cid:12)(cid:12) e P NS , T , x , y (cid:12)(cid:12)(cid:12) · (cid:13)(cid:13)(cid:13) Y NS , y (cid:13)(cid:13)(cid:13) L p ω d S + N − (cid:13)(cid:13)(cid:13) Z N x (cid:13)(cid:13)(cid:13) L p ω (5.56) . p , ǫ , ǫ N − ǫ Z T ̺ − + ǫ S , T d S + N − (5.57) . ǫ N − ǫ + N − . (5.58)Combining (5.52), (5.55), and (5.58), we have (cid:13)(cid:13)(cid:13) Φ N ,2 T , x (cid:13)(cid:13)(cid:13) L p ω . p , ǫ , ǫ N − ǫ + δ + N − ǫ ∂ ,1 + β ⋆ + δ + N − + Z T X y ∈ Z ¯ P NS , T , x , y (cid:13)(cid:13)(cid:13) Φ N ,2 S , y (cid:13)(cid:13)(cid:13) L p ω d S . (5.59)Employing the singular Gronwall inequality to (5.59) completes the proof of the estimate (5.49) for Φ N ,2,2 . This completesthe proof of (5.48), and thus of Proposition 5.3. . P ROOF OF T HEOREM
Proposition 6.1.
The sequence { X N } ∞ N = converges to the solution of SHE with initial data Z ∞ • in the Skorokhod space D . Following those proofs for both Theorem 3.3 within [ ] and Theorem 1.1 within [ ] , we establish the required tightnessfor { X N } ∞ N = . To establish the convergence, we again follow the proofs for Theorem 3.3 within [ ] and Theorem 1.1 within [ ] via the martingale problem uniquely identifying the SHE; ultimately, it suffices to obtain the following estimate. Lemma 6.2.
Consider any functional w N of the particle system of the following type: w NS , y • = m Y j = η NS , y + i j , (6.1) where m ∈ Z > is uniformly bounded, and those indices { i j } mj = are both mutually distinct and uniformly bounded. Provided any ϕ ∈ C ∞ c ( R ) and any T ∈ R > uniformly bounded, we have the following convergence in probability, in which ϕ N , • • = ϕ N − • : lim N →∞ Z T N − X x ∈ Z ϕ N , x · w NS , x ” X NS , x — =
0. (6.2)
Proof.
We follow the proof of Lemma 2.5 within [ ] identically because we have the Dirichlet form estimate from Lemma4.3 along with Lemma 4.13, Proposition 5.3, and Lemma 5.4; this completes the proof. (cid:3) A PPENDIX
A. S
TOCHASTIC C ONTINUITY L EMMA
The current appendix section is dedicated towards the following result employed throughout this article; roughly speak-ing, the result asserts that the random fields Z N , Y N , and X N cannot exhibit any real variation on remarkably small time-blocks uniformly on the space-time block [
0, 1 ] × Z with very high probability.To state the result, we consider the decomposition [
0, 1 ] = ∪ j = I j , where I j • = [ τ j , τ j + ] ⊆ [
0, 1 ] and τ j • = jN − . Lemma A.1.
Provided κ ∈ R > arbitrarily large although universal and ǫ ∈ R > arbitrarily small although universal, considerthe following events with a universal implied constant: (cid:2) G Z N ; κ (cid:3) • = – sup j ∈ J K sup T ∈ I j sup | x | N e − κ | x | N Z NT , x & sup w ∈ Z e − κ | w | N Z N τ j , w ™ ; (A.1a) (cid:2) H Z N ; κ , ǫ (cid:3) • = – sup j ∈ J K sup T ∈ I j sup | x | N e − κ | x | N (cid:12)(cid:12)(cid:12) D T − τ j Z N τ j , x (cid:12)(cid:12)(cid:12) & N − + ǫ sup w ∈ Z e − κ | w | N Z N τ j , w ™ (A.1b) Moreover, define G Y N and G X N and G e Φ N ,1 analogously via the replacement Z N Y N and Z N X N and Z N e Φ N ,1 , respectively,and for each of these three new random fields also the replacement J − N , N K Z .Provided any D ∈ R > , we have the estimates P (cid:2) G Z N ; κ (cid:3) + P (cid:2) G Y N ; κ (cid:3) + P (cid:2) G X N ; κ (cid:3) + P (cid:2) G e Φ N ,1 ; κ (cid:3) + P (cid:2) H Z N ; κ , ǫ (cid:3) . D , κ , ǫ N − D . (A.2) Proof.
The high-probability estimates for each of the events G Y N ; κ and G X N ; κ and G e Φ N ,1 ; κ follow from the proof of tightnessin Proposition 1.4 of [ ] . The high-probability estimate for G Z N ; κ follows from that of H Z N ; κ ; ǫ , so it remains to estimate theprobability of this final event. To this end, we appeal to Proposition 2.1 along with the observation that ∆ !! x Z NT , x = N f NT , x Z NT , x derived via Taylor expansion, where f N is a uniformly bounded functional of the particle system.Thus, writing ξ N = ¯ ξ N + N ¯ f N with ¯ ξ N a Poisson process of rate at most N up to uniformly bounded constants and ¯ f N another uniformly bounded functional of the particle system, we haved Z NT , x = N e f NT , x Z NT , x d T + Z NT , x d ¯ ξ NT , x , (A.3)where again e f N is another uniformly bounded functional of the particle system.We now introduce the following event, for which we recall ¯ ξ [ S , T ] , x denotes the total sum of contributions from all thePoisson clocks at x ∈ Z in the time-block [ S , T ] ⊆ R > , and in which the implied constant is large but universal: [ I ǫ ] • = – sup j ∈ J K sup | x | . N ¯ ξ I j , x . N ǫ ™ . (A.4) onsequence of large-deviations estimates for the Poisson distribution, we obtain the following estimate via union bound: P (cid:2) I C ǫ (cid:3) X j = X | x | . N P ” ¯ ξ I j , x & N ǫ — (A.5) . ǫ X j = X x ∈ Z e − log N (A.6) . e − log N . (A.7)The high-probability estimate (A.7) thus allows us to condition on the event I ǫ . Because jump-sizes in the d ¯ ξ N -quantityon the RHS of (A.3) are of size at most N − up to uniformly bounded factors, and because the drift quantity from that RHSof (A.3) contributes at most N − up to uniformly bounded factors on the time-block I j , the estimate for P [ H Z N ; κ ; ǫ ] thusfollows from a direct analysis of (A.3), since courtesy of (A.7), with high probability the total contribution of jumps on thetime-block I j to the RHS of (A.3) is at most N − + ǫ up to uniformly bounded constants. This completes the proof. (cid:3) A PPENDIX
B. T
ECHNICAL L EMMA FOR T IGHTNESS
Lemma B.1.
Suppose { A N • } N ∈ Z > is a family of stochastic processes with sample paths in D . Moreover, suppose we additionallyhave another family { C N • } N ∈ Z > of stochastic processes with sample paths in D such that the following conditions hold: • The family { C N • } N ∈ Z > is tight in D . • We have the following high-probability estimate for which β ∈ R > and β ′ ∈ R > are universal, and for which κ ∈ R > is arbitrarily large but universal: sup N ∈ Z > P ”(cid:13)(cid:13) A NT − C NT (cid:13)(cid:13) ∞ ; κ . κ N − β ′ — . κ N − β . (B.1) The family { A N • } N ∈ Z > is tight with respect to the Skorokhod topology on D . Moreover, all the limit points are the correspondinglimit points of { C N } N ∈ Z > along the same subsequence. The same is true for D ◦ in place of D as well.Proof. We begin with the following elementary observation in which C ∈ R > is a large constant that will be taken C → ∞ independently of N ∈ Z > : P ”(cid:13)(cid:13) A NT (cid:13)(cid:13) ∞ ; κ > C — P ”(cid:13)(cid:13) C NT (cid:13)(cid:13) ∞ ; κ + (cid:13)(cid:13) A NT − C NT (cid:13)(cid:13) ∞ ; κ > C — . (B.2)Recalling the assumption in (B.1), from the previous upper bound we deduce the following within which c κ ∈ R > dependsonly on its subscript: P ”(cid:13)(cid:13) A NT (cid:13)(cid:13) ∞ ; κ > C — P •(cid:13)(cid:13) C NT (cid:13)(cid:13) ∞ ; κ & C ˜ + c κ N − β , (B.3)from which we havelim sup C →∞ lim sup N ∈ Z > P ”(cid:13)(cid:13) A NT (cid:13)(cid:13) ∞ ; κ > C — . κ lim sup C →∞ lim sup N ∈ Z > P ”(cid:13)(cid:13) C NT (cid:13)(cid:13) ∞ ; κ > C — + lim sup C →∞ lim sup N ∈ Z > N − β (B.4) =
0; (B.5)the last identity (B.5) follows from the assumed tightness of { C N • } N ∈ Z > and Theorem 16.8 in [ ] .Before we proceed, we employ the family of regularity metrics W ( • , δ ) : D ([
0, 1 ] , R ) → R parameterized by δ ∈ R > from Section 16 in [ ] without defining it explicitly in this article. To use this metric, we observe the following consequenceof our assumption (B.1) combined with the definition of this regularity metric. P (cid:2) W ( A N , δ ) > W ( C N , δ ) + c κ N − β ′ (cid:3) . κ N − β . (B.6)We then deduce the following estimates for any ǫ ∈ R > :lim sup δ → lim sup N →∞ P (cid:2) W ( A N , δ ) > ǫ (cid:3) . κ lim sup δ → lim sup N →∞ P (cid:2) W ( C N , δ ) + c κ N − β ′ > ǫ (cid:3) + lim sup δ → lim sup N →∞ N − β (B.7) lim sup δ → lim sup N →∞ P • W ( C N , δ ) > ǫ ˜ (B.8) =
0, (B.9)where the last identity (B.9) follows from Theorem 16.8 within [ ] , again. Combining (B.5) and (B.9) with Theorem 16.8in [ ] now completes the proof. (cid:3) PPENDIX
C. I
NDEX FOR N OTATION
C.1.
Expectation Operators.
Provided a probability measure µ on some probability space, we denote by E µ expectationwith respect to this probability measure. Moreover, provided any σ -algebra F , we denote by E µ F the conditional expectationwith respect to the probability measure µ conditioning on F .C.2. Lattice Differential Operators and N -dependent Scaling. Provided k ∈ Z , define the discrete differential operators ∇ k , ∆ k acting on any function ϕ : Z → R as follows: ∇ k , x ϕ x • = ϕ x + k − ϕ x ; ∆ k , x ϕ x • = ϕ x + k + ϕ x − k − ϕ x . (C.1)Moreover, define the appropriately rescaled operators ∇ ! k , x • = N ∇ k , x and ∆ !! k , x • = N ∆ k , x , and these should be interpreted asapproximations to their continuum differential counterparts. More generally, provided a generic bounded linear operator T acting on any linear space, each additional ! in the superscript denotes another scaling factor of N ∈ Z > . For example,we define T ! • = N T and T !! • = N T .C.3. Averaging.
Consider any domain I cont ⊂ R and any suitable function ϕ : I cont → R . Moreover, we consider any finiteset I disc and any suitable function ψ : I disc → R . We now establish the following notation: − Z I cont ϕ x d x • = | I cont | Z I cont ϕ x d x ; (C.2a) ÝX x ∈ I disc ψ x • = | I disc | X x ∈ I disc ψ x . (C.2b)C.4. Landau Notation for Asymptotics.
Provided any generic set I , the notation a . I b is equivalent to a c I b for anyreal numbers a , b ∈ R , where the constant c I is allowed to depend on every element of I .C.5. Miscellaneous Space-Time Objects.
We introduce a list of space-time notation used throughout this article. • Consider T f ∈ R > and κ ∈ R > . We define a norm as follows provided any ϕ T , x : [ T f ] × Z → R : (cid:13)(cid:13) ϕ T , x (cid:13)(cid:13) ∞ ; κ , T f • = sup T ∈ [ T f ] sup x ∈ Z e − κ | x | N | ϕ T , x | . (C.3) • Simply for notational convenience and compact presentation, provided any S , T ∈ R > , we define ̺ S , T • = | T − S | . • Provided ( T , X ) ∈ R > × R , we define the space-time shift-operator τ T , X acting on possibly random fields as follows: τ T , X f s , y ( η Nr , z ) • = f T + s , X + y , ( η NT + r , X + z ) . (C.4) • Lastly, provided any a , b ∈ R , we define the discretization J a , b K • = [ a , b ] ∩ Z .R EFERENCES [ ] G. Amir, I. Corwin and J. Quastel, "Probability distribution of the free energy of the continuum directed polymer model in ( + ) -dimensions". Communications in Pure and Applied Math , 64:466-537 (2011). [ ] R. Basu, S. Sarkar, and A. Sly, "Invariant Measures for TASEP with a Slow Bond". arXiv, 2017. [ ] R. Basu, V. Sidoravicius, and A. Sly, "Last passage percolation with a defect line and the solution of the Slow Bond Problem". arXiv, 2014. [ ] L. Bertini and G. Giacomin, "Stochastic Burgers and KPZ Equations from Particle Systems".
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