SPDE limit of weakly inhomogeneous ASEP

We study ASEP in a spatially inhomogeneous environment on a torus T = Z/NZ of N sites. A given inhomogeneity ã(x) ∈ (0,∞), x ∈ T, perturbs the overall asymmetric jumping rates r < ` ∈ (0, 1) at bonds, so that particles jump from site x to x + 1 with rate rã(x) and from x + 1 to x with rate `ã(x) (subject to the exclusion rule in both cases). Under the limit N →∞, we suitably tune the asymmetry (`− r) to zero like N− 1 2 and the inhomogeneity ã to unity, so that the two compete on equal footing. At the level of the Gärtner (or microscopic Hopf–Cole) transform, we show convergence to a new SPDE — the Stochastic Heat Equation with a mix of spatial and spacetime multiplicative noise. Equivalently, at the level of the height function we show convergence to the Kardar–Parisi–Zhang equation with a mix of spatial and spacetime additive noise. Our method applies to a general class of ã(x), which, in particular, includes i.i.d., fractional-Brownian-motion like, and periodic inhomogeneities. The key technical component of our analysis consists of a host of estimates on the kernel of the semigroup Q(t) := etH for a Hill-type operator H := 1 2 ∂xx +A′(x), and its discrete analog, where A (and its discrete analog) is a generic Hölder continuous function.


Introduction
In this article we study the Asymmetric Simple Exclusion Process (ASEP) in a spatially inhomogeneous environment where the inhomogeneity perturbs the rate of jumps across bonds, while maintaining the asymmetry (i.e., the ratio of the left and right rates across the bond). Quenching the inhomogeneity, we run the ASEP and study its resulting Markov dynamics. Even without inhomogeneities, ASEP demonstrates an interesting scaling limit to the Kardar-Parisi-Zhang (KPZ) equation when the asymmetry is tuned weakly [BG97].
It is ultimately interesting to determine how the inhomogeneous rates modify the dynamics of such systems, and scaling limits thereof. In this work we tune the strengths of the asymmetry and the inhomogeneity to compete on equal levels, and we find that the latter introduces a new spatial noise into the limiting equation. At the level of the Gärtner (or microscopic Hopf-Cole) transform (see (1.1)), we obtain a new equation of Stochastic Heat Equation (SHE)-type, with a mix of spatial and spacetime multiplicative noise. At the level of the height function, we obtain a new equation of KPZ-type, with a mix of spatial and spacetime additive noise.
We now define the inhomogeneous ASEP. The process runs on a discrete N -site torus T (N ) := Z/N Z where we identify T (N ) with {0, 1, . . . , N − 1}, and, for x, y ∈ T (N ) , understand x + y to be mod N . To alleviate heavy notation, we will often omit dependence on N and write T in place of T (N ) , and similarly for notation to come. For fixed homogeneous jumping rates r < ∈ (0, 1) with r + = 1, and for fixed inhomogeneities a(x) ∈ (0, ∞), x ∈ T, the inhomogeneous ASEP consists of particles performing continuous time random walks on T. Jumping from x to x + 1 occurs at rate a(x)r, jumping from x + 1 to x currents at rate a(x) , and attempts to jump into occupied sites are forbidden. See Figure 1a. The particle at x jumps to x − 1 at rate a(x − 1) or to x + 1 at rate a(x)r; meanwhile the particle at 1 may not jump to the occupied site 2. (b): The particle dynamics are coupled with a height function as shown.
We will focus on the height function (also known as integrated current), denoted h(t, x). To avoid technical difficulties, throughout this article we assume the particle system to be half-filled so that N is even, and there are exactly N

Weakly inhomogeneous ASEP
We will also be considering the continuum torus T := R/Z [0, 1), which is to be viewed as the N → ∞ limit of 1 N T. Similarly for the continuum torus T , we let [x, x ] ⊂ T denote the interval going from x to x counterclockwise, let |[x, x ]| denote the length of the interval, and let dist T (x, x ), x, x ∈ T denote the analogous geodesic distance on T . For u ∈ [0, 1], let C u [0, 1] denote the space of u-Hölder continuous functions f : [0, 1] → R, equipped with the norm (1.6) [f ] C u (T ) := sup df (y) , (1.7) where the integral is in the Riemann-Stieltjes sense. The integral excludes 0 so that the possible jump of f there will not be picked up.
We now define the type of inhomogeneities to be studied. Throughout this article, we will consider possibly random ( a (N ) (x)) x∈T that may depend on N . Set a (N ) (x) := a (N ) (x) − 1, and put x, x ∈ T. (1.8) As announced previously, we will often write a (N ) = a, a (N ) = a, A (N ) = A, etc., to simplify notation. When x = 0, we will write A(x) := A(0, x). Consider also the scaled partial sums A N (x, x ) := A(xN, x N ) and A N (x) := A(xN ), which are linearly interpolated to be functions on T 2 and [0, 1), respectively. For f : T 2 → R, we define a seminorm that is analogous to [ · ] C u (T ) in (1.7): [f ] u,N := sup where → P denotes convergence in probability.

Remark 1.3.
(a) Assumption 1.2(a) ensures the rate a(x) is always nonnegative so that the process is well-defined.
(b) Note that we do not assume (a(0) + . . . a(N − 1)) = 0 or A(1) = 0. Weakly inhomogeneous ASEP Theorem 1.8. Consider a half-filled inhomogeneous ASEP on T (N ) , with deterministic, near stationary initial condition described as in the preceding. If, for some Z ic ∈ C(T ), then, under the scaling (1.2), for each T < ∞, where Z is the mild solution of (1.10) with initial condition Z ic . Remark 1.9. Though we formulate all of our results at the level of SHE-type equations, they can also be interpreted in terms of convergence of the ASEP height function (under suitable centering and scaling) to a KPZ-type equation which formally is written as (1.13) The solution to this equation should be (as in the case where A (x) ≡ 0) defined via H(t, x) = − log Z(t, x). One could also try to prove well-posedness of this inhomogeneous KPZ equation directly, though this is outside the scope of our present investigation and unnecessary for our aim.
Steps in the proof of Theorem 1.8 Given that Theorem 1.8 concerns convergence at the level of Z, our proof naturally goes through the microscopic transform (1.1). As mentioned earlier, for homogeneous ASEP, Z solves the microscopic SHE (1.3). On the other hand, with the presence of inhomogeneities, it was not clear at all that Gärtner's transform applies. As noted in [BCS14,Remark 4.5], transforms of the type (1.1) are tied up with the Markov duality. The inhomogeneous ASEP considered here lacks a certain type of Markov duality so that one cannot infer a useful transform from Markov duality. For specifically, referring to the notation in Remark 1.3 and [BCS14], the inhomogeneous ASEP does enjoy a Markov duality for the observable Q(t, x) (which is essentially η(t, x)Z(t, x) in our notation), but not for Q(t, x) (which is essentially Z(t, x) in our notation). The latter is crucial for inferring a transform of the type (1.1). The first step of the proof is to observe that, despite the (partial) lost of Markov duality, Z still solves an SHE-type equation ((2.6) in the following), with two significant changes compared to (1.3).
i) First the discrete Laplacian is now replaced by the generator of an inhomogeneous random walk. Interestingly, this walk is exactly Bouchaud's model [Bou92], which is often studied with heavy-tail a(x) (as opposed Assumption 1.2) in the context of randomly trapped walks.
ii) Additionally, a potential term (the term νa(x)Z(t, x)dt in (2.6)) appears due to the unevenness of quenched expected growth. For homogeneous ASEP with near stationary initial condition, the height function grows at a constant expected speed, and the term e νt in (1.1) is in place to balance such a constant growth.
Due to the presence of the inhomogeneity, in our case the quenched expected growth is no longer a constant and varies among sites. This results in a fluctuating potential that acts on Z(t, x).
The two terms in i)-ii) together make up an operator H (defined in (2.6)) of Hill-type that governs the microscopic equation. Correspondingly, the semigroup Q(t) := e tH now plays the role of standard heat kernel in the case of homogeneous ASEP. We EJP 25 (2020), paper 156. refer to Q(t) := e tH and its continuum analog Q(t) as Parabolic Anderson Model (PAM) semigroups.
Much of our analysis consists of estimating the kernels of Q(t) and Q(t). These estimates are crucial in order to adapt and significantly extend the core argument of [BG97]. To prepare for the analysis of Q(t), in Section 3, we establish bounds on the transition kernel of Bouchaud's walk, which was described in i), and show that the kernel is well-approximated by that of the simple random walk. The transition kernel of Bouchaud's walk has been studied with heavy-tail inhomogeneities in the context of trapping models -see the beginning of Section 3 for more discussions of this literature. Here we consider bounded and vanishing homogeneity, which is technically much simpler to deal with than heavy-tail ones. Given the vast literature (some of which is surveyed at the beginning of Section 3) on heat kernel estimates, it is quite possible that some of the bounds (Proposition 3.2(a)-(g)) obtained in this paper could be derived from, or follow similarly from existing techniques. However, for the sake of being self-contained, we provide with an elementary and short derivation via Picard iteration of the heat kernel bounds we use.
Based on the bounds in Section 3 on the kernel of Bouchaud's walk, in Section 4, we express the PAM semigroup Q(t) using the Feynman-Kac formula, with Bouchaud's walk being the underlying measure. We then expand the Feynman-Kac formula, and develop techniques to bound the resulting expansion to obtain estimates on the kernel of Q(t) and its continuum analog. For the operator 1 2 ∆ − V with a singular potential V , accessing the corresponding semigroup has been a classic subject of study in mathematical physics.
In particular, semigroup kernels and the Feynman-Kac formulas have been studied in [McK77,Sim82], and the expansion we use is similar to the one considered in [Sim79, Section 14, Chapter V]. The major difference is that our potential A has negative regularity, and hence is not function-valued.
Armed with the heat kernel bounds from Section 3 and the semigroup kernel expansion from Section 4, the final two sections adapt the key ideas from the work of Bertini and Giacomin [BG97] into the inhomogeneous setting to prove tightness (Section 5) and identify the limiting SPDE (Section 6).

Further directions
There are a number of directions involving inhomogeneous ASEP which could warrant further investigation. At a very basic level, in this article we limit our scope to half-filled systems on the torus with near stationary initial conditions so as to simplify the analysis. In particular, at half-filling, our height function is immediately periodic. With other filling fractions this is easily resolved by introducing a height shift to make the height function periodic. A much more significant concern is that with other filling fractions we need to work in a moving frame. We expect to see KPZ-type fluctuations when viewed relative to the characteristic velocity. Assuming half-filling and near stationary initial data ensures that the characteristic velocity is 0, hence we do not need to work in a moving frame. We do expect that it should be possible to relax both the half-filled condition and the near stationarity, for instance to consider initial data where there are particles at sites 1 through aN and then holes elsewhere (here a ∈ (0, 1) is the filling fraction). Relaxing the near stationary assumption should be doable as in [ACQ11] via a priori estimates which show that after a short time the initial data looks near stationary. Working in a moving frame is harder and the key self-averaging method used in [BG97] (see our Proposition 6.3) has not been developed in the context of a moving frame. The duality method from [CGST20] does work in a moving frame, but may not be readily applicable here (see Remark 6.3). We believe that these questions warrant further investigation.
Putting aside the weak asymmetry scaling, it is compelling to consider the nature of the long-time hydrodynamic limit (i.e., functional law of large numbers) or fluctuations (i.e., central limit type theorems) for inhomogeneous ASEP. For the homogeneous ASEP the hydrodynamic limit is dictated by Hamilton-Jacobi PDEs and the fluctuations characterized by the KPZ universality class -does any of this survive the introduction of inhomogeneities? These questions are complicated by the lack of an explicit invariant measure for our inhomogeneous ASEP, as well as a lack of any apparent exact solvability.
There are other types of inhomogeneities which can be introduced into ASEP and it is natural to consider whether different choices lead to similar long-time scaling limits or demonstrate different behaviors. Our choice of inhomogeneities stemmed from the fact that upon applying Gärtner's transform, it results in an SHE-type equation. On the other hand, our methods seem not to apply to site (instead of bond) inhomogeneities (so out of x we have a(x) and r a(x) as rates). It also does not seem to extend to non-nearest neighbor systems. A more direct approach (e.g. regularity structures [Hai13]; energy solutions [GJ14,GP18]; paracontrolled distributions [GIP15,GP17]; or renormalization group [Kup16]) at the level of the height function may eventually prove useful in dealing with these generalizations. For the homogeneous non-nearest neighbor ASEP (or other homogeneous generalizations which maintain a product invariant measure), the energy solution method has proved quite useful for demonstrating KPZ equation limit resultssee, for example, [GJ14].
Another type of inhomogeneities would involve jump out of x given by rates a(x) + b to the left and a(x) − b to the right. A special case of this type of inhomogeneities is studied in [FGS16] where they consider a single slow bond (i.e., a(x) ≡ a * for x = 0 and a(0) < a * ). In that case, they show that the inhomogeneities preserves the product Bernoulli invariant measure (note that the inhomogeneities we consider do not preserve this property). (In fact, the argument in [FGS16] for this preservation of the invariant measure may be generalized to more than just a single-site inhomogeneity.) Using energy solution methods, [FGS16] shows that depending on the strength of the asymmetry and the slow bond, one either obtains a Gaussian limit with a possible effect of the slow bond, or the KPZ equation without the effect of the slow bond. It would be interesting to see if this type of inhomogeneities (at every bond, not just restricted to a single site) could lead to a similar sort of KPZ equation with inhomogeneous spatial noise such as derived herein.
[CR97, RT08, Cal15] characterized the hydrodynamic limit for ASEP and TASEP with inhomogeneities that varies at a macroscopic scale. Those methods do not seem amenable to rough or rapidly varying parameters (such as the i.i.d. or other examples considered herein) and it would be interesting to determine their effect. A special case of spatial inhomogeneities is to have a slow bond at the origin. The slow bond problem is traditionally considered for the TASEP, with particular interest in how the strength of slow-down affects the hydrodynamic limit of the flux, see [JL92,BSS14] and the reference therein. As mentioned previously, this problem has been further considered in the context of weakly asymmetric ASEP in [FGS16]. There are other studies of TASEP (or equivalently last passage percolation) with inhomogeneities in [GTW02b,GTW02a,LS12,EJ17,Emr16,BP17]. The type of inhomogeneities in those works are of a rather different nature than those considered here. In terms of TASEP, their inhomogeneities mean that the i th jump of the j th particle occur at rate π i + π j for the inhomogeneity parameters {π i } and { π j }. Such inhomogeneities do not seem to result in a temporally constant (but spatially varying) noise in the limit. Thus, the exact methods which are applicable in those works do not seem likely to grant access to the fluctuations or phenomena surrounding our inhomogeneous process or limiting equation.
As mentioned previously, upon applying Gärtner's transform we obtain an SHEtype equation with the generator of Bouchaud's walk. Our particular result involves tuning the waiting time rate near unity, and under such scaling the inhomogeneous walk approximates the standard random walk. On the other hand, Bouchaud's model (introduced in [Bou92] in relation to aging in disordered systems; see also [BAČ06,BACČR15] for example) is often studied under the assumption of heavy-tailed waiting parameters. In such a regime, one expects to see the effect of trapping, and in particular the FIN diffusion [FIN99] is a scaling limit that exhibits the trapping effect. It would be interesting to consider a scaling limit of inhomogeneous ASEP in which the FIN diffusion arises. See the beginning of Section 3 for further discussion and references related to Bouchaud's model.
For the case A (x) = B (x) (spatial white noise), the operator H (in (1.10)) that goes into the SPDE (1.10) is known as Hill's operator. There has been much interest in the spectral properties of this and similar random Schrödinger type operator. In particular, [FL60,Hal65,FN77,McK94,CM99,CRR06] studied the ground state energy in great depth, and recently, [DL17] proved results on the point process for lowest few energies, as well as the localization of the eigenfunctions. On the other hand, the semigroup Q(t) := e tH is the solution operator of the (continuum) PAM (see [CM94,Kön16] and the references therein for extensive discussion on the discrete and continuum PAM). A compelling challenge is to understand how this spectral information translates into the long-time behavior of our SPDE. For instance, what can be said about the intermittency of this SPDE?

Outline
In Section 2, we derive the microscopic (SHE-type) equation for Z(t, x). As seen therein, the equation is governed by a Hill-type operator H that involves the generator of an (Bouchaud-type) inhomogeneous walk. Subsequently, in Sections 3 and 4 we develop the necessary estimates on the transition kernel of the inhomogeneous walk and Hill-type operator. Given these estimates, we proceed to prove Theorem 1.8 in two steps: by first establishing tightness of {Z N } N and then characterizing its limit point. Tightness is settled in Section 5 via moment bounds. To characterize the limit point, in Section 6, we develop the corresponding martingale problem, and prove that the process Z N (t, x) solves the martingale problem.

Notation
We use c(u, v, . . .) ∈ (0, ∞) to denote a generic, positive, finite, deterministic constant, that may change from line to line (or even within a line), but depends only on the designated variables u, v, . . .. We use subscript N to denote scaled processes/spaces/functions, e.g., Z N in (1.2). Intrinsic (not from scaling) dependence on N will be designated by superscript (N ), e.g., T (N ) , though, to alleviate heavy notation, we will often omit such dependence, e.g., T := T (N ) . For processes/spaces/functions that have discrete and continuum versions, we often use the same letter but in different fonts: math backboard for the discrete and math calligraphy for the continuum, e.g., A(x) and A(x). We list some of the reoccurring notation below for convenience of the readers. The left column is for discrete notation and the right is for analogous continuum notion. Some notation only occurs in one of the two contexts. In this section we derive the microscopic equation for Z(t, x). In doing so, we view { a(x) : x ∈ T} as being fixed (quenched), and consider only the randomness due to the dynamics of our process. In deriving this equation for Z(t, x) we will also treat N as fixed and hence drop it from the notation. The inhomogeneous ASEP can be constructed as a continuous time Markov process with a finite state space {0, 1} T , where {0, 1} indicates whether a given site is empty or occupied. Here we build the inhomogeneous ASEP out of graphical construction (see [Lig12, Cor12, Section 2.1.1]). For each x ∈ T, let {P → (t, x)} t≥0 and {P ← (t, x)} t≥0 be independent Poisson processes of rates r a(x) and a(x) respectively. A particle attempts a jump across the bond (x, x + 1) to the right (resp. left) whenever P → ( · , x) (resp. P ← ( · , x)) increases. The jump is executed if the destination is empty, otherwise the particle stays put. Let Recall from (1.1) that τ := r . Consider when a particle jumps from x to x + 1. Such a jump occurs only if η(t, x)(1 − η(t, x + 1)) = 1, and, with Z(t, x) defined in (1.1), such a jump changes Z(t, x) by (τ −1 − 1)Z(t, x). Likewise, a jump from x + 1 to x occurs only if η(t, x + 1)(1 − η(t, x)) = 1, and changes Z(t, x) by (τ − 1)Z(t, x). Taking into account the continuous growth due to the term e νt in (1.1), we have that (2.2) The differential in dZ(t, x) acts on the t variable. We may extract the expected growth a(x)rt and a(x) t from the Poisson processes P → ( · , x) and P ← ( · , x), so that the processes are martingales. We then rewrite (2.2) as EJP 25 (2020), paper 156.
where 1 A ( · ) denotes the indicator function of a given set A. Under the weak asymmetry scaling (1.2), (r − ) 2 = 1 N + O(N −2 ) acts as the relevant scaling factor for the quadratic variation. In addition to this scaling factor, we should also consider the quantities that involve η(t, x) and η(t, x + 1). Informally speaking, since the system is half-filled (i.e., having N/2 particles), we expect η(t, x) and η(t, x + 1) to self-average (in t) to 1 2 , and expect η(t, x)η(t, x + 1) to self-average to 1 4 . With r, → 1 2 and a(x) → 1, we expect d M (t, x), M (t, x) to behave like N −1 1 {x= x} Z 2 (t, x)dt, and hence dM (t, x) to behaves like ξZ, as N → ∞. Equation (2.6) gives the microscopic equation in differential form. For subsequent analysis, it is more convenient to work with the integrated equation. Consider the semigroup Q(t) := e tH , which is well-defined and has kernel Q(t; x, x) because H acts on the space {f : T → R} of finite dimensions. Using Duhamel's principle in (2.6) gives (2.9) More generally, initiating the process from time t * ≥ 0 instead of 0, we have (2.10) The semigroup Q(t) admits an expression by the Feynman-Kac formula as (2.11) Hereafter E x [ · ] (and similarly P x [ · ]) denotes expectation with respect to a reference process starting at x. Here the reference process X a (t) is a walk on T that attempts jumps from X a (t) to X a (t) ± 1 in continuous time (each) at rate √ r a(X a (t)).
Remark 2.1. It is natural to wonder whether it is possible to directly (without appealing to the Gärtner transform and SHE-type equation) see convergence of the height function to the KPZ-type equation (1.13). While a direct proof is certainly beyond the scope of this paper, we will briefly explain at a very heuristic level where the structure of the KPZ-type equation (1.13) arises from the microscopic evolution of the height function.
There are two changes that can occur for the ASEP height function h(t, x). Using the notation ∇f (x) = f (x + 1) − f (x), we have that h(t, x) can increase by 2 at rate a(x) provided that ∇h(t, x) = 1 and ∇h(t, x − 1) = −1; and h(t, x) can decrease by 2 at rate r a(x) provided that ∇h(t, x) = −1 and ∇h(t, x − 1) = 1. Since ∇h only takes values in {−1, 1} we can encode the indicator functions as linear functions of ∇h. This gives rise to the following evolution equation: can rewrite the above in the suggestive form While it is still highly non-trivial to prove convergence (with appropriate renormalization) of these terms to those in (1.13), it is now apparent that the new spatial noise A (x) comes from the term EJP 25 (2020), paper 156.

Transition probability of the inhomogeneous walk X a (t)
The focus of this section and Section 4 is to control the semigroup Q(t) and its continuum counterpart Q(t). As the first step, in this section we establish estimates on the transition kernel of the inhomogeneous walk X a (t). Note that X a (t) and p a depend on N through a(x) = a (N ) (x) and through the underlying torus T = T (N ) , but we omit such dependence in the notation.
The kernel p a has been studied with heavy-tailed a(x) in the context of trapped models. For heavy-tailed a(x), [ deviations of X a (x), which confirmed a prediction [BB03] based on the trapping nature of Bouchaud's walk. Estimates on the analogous continuum kernel (i.e., for the FIN diffusion) have been obtained [CHK19]. As mentioned previously, here we consider bounded and vanishing a(x), which is technically much simpler than heavy-tailed a(x).
These bounds mentioned above essentially imply Proposition 3.2(a).
Due to the technical nature of how p a enters our subsequent analysis, we need detailed bounds. Specifically, we will derive in Proposition 3.2 bounds on p a (t, · ), its Hölder continuity, its gradients, and its difference between the homogeneous walk kernel.
Put in a broader context, the type of bounds we seek to obtain on p a (t, · ) and its Hölder continuity go under the name of Nash-Aronson bounds [Nas58,Aro67] in elliptic PDEs (we thank one of the anonymous referees for pointing us to this literature). These type of bounds have since been pursued and generalized in various contexts such as Riemannian manifolds, metric measure spaces, and fractals. We point to [Gri92,SC02] and the references therein. On Riemannian manifolds, bounds on the gradient of heat kernels have been derived in, for example, [CLY81,LY86]. Modern works in probability have been investigating the analogous heat kernel in discrete settings. For the random conductance model, heat kernel estimates have been derived at various generality: for bounded below conductance in [BD10], and for general conductance with certain integrability assumptions in [Fol11,ADS15,AN19]. It was shown in [BBHK08,Bis11] that anomalies in heat-kernel decay may occur without integrability assumptions. The works [DF19, DF20] consider layered random conductance models and establish kernel estimates. For an overview on the random conductance model we point to [Bis11]. We also point out that a gradient estimate on Green's function of percolation clusters has been obtained in [BDCKY15,Theorem 6].
The starting point of our analysis is the backward Kolmogorov equation (3.2') As announced previously, we use c(u, v, . . .) ∈ (0, ∞) to denote a generic, positive, finite, deterministic constant, that may change from line to line (or even within a line), but depends only on the designated variables u, v, . . .. Recall that a(x) = 1+a(x). Our strategy of analyzing p a is perturbative. We solve (3.2') iteratively, viewing a(x) as a perturbation. Such an iteration scheme begins with the EJP 25 (2020), paper 156. unperturbed equation which is solved by the transition probability p(t; x, x) = P x [X(t) = x] of the continuous time symmetric simple random walk X(t) on T. Here, we record some useful bounds on p. Let ∇f (x) := f (x + 1) − f (x) denote the forward discrete gradient. When needed we write ∇ x or ∆ x to highlight which variable the operator acts on. Given any u ∈ (0, 1] and T < ∞, for all x, x , x ∈ T and t ≤ N 2 T . The bounds (3.4a)-(3.4i) follow directly from known results on the analogous kernel on Z. Indeed, with p Z (t; x− x) := P x [X Z (t) = x] denoting the transition kernel of continuous time symmetric simple random walk X Z (t) on the full-line Z, we have x, x ∈ {0, . . . , N − 1}. (3.5) The full-line kernel p Z can be analyzed by standard Fourier analysis, as in, e.g., [DT16, Equation (A.11)-(A.14)]. Relating these known bounds on p Z to p gives (3.4a)-(3.4i). Now we will start to study the perturbations around the solution to (3.3). Let Γ(v) denote the Gamma function, and let Σ n (t) := (s 0 , . . . , s n ) ∈ (0, ∞) n+1 : s 0 + . . . + s n = t .
To bound the last expression, for the given v ∈ (0, 1), we write 1 . (3.14) Referring back to (3.13), we see that |r n (t; x, x)| is bounded by (c a L ∞ (T) ) n when t ≤ 1, uniformly over x, x ∈ T. This being the case, by making the constant c(v) larger in (3.14), (b) Given the expansion (3.11), our goal is to bound x∈T |r n (t; x, x)|, for n = 1, 2, . . .. To this end, sum both sides of (3.12) over x ∈ T. Under the convention x 0 := x and the relabeling x n+1 = x, we bound In (3.15), sum over x n+1 , . . . , x 2 , x 1 in order, using the bound (3.4d) for the sum over x n+1 , . . . , x 2 and using x1 p(s 0 ; x 0 , x 1 ) = 1 for the sum over x 1 . We then obtain To bound the last integral, performing a change of variable s i := s i /t, we see that With t ≤ N 2 T , summing both sides over n ≥ 1 gives the desired result.
and apply the Dirichlet formula (3.7) with .
We now incorporate Lemma 3.1 with the assumed properties of a(x) from Assumption 1.2. To simplify notation, we will say that events {Ω Λ,N } Λ,N hold 'with probability Proposition 3.2. For given T < ∞, u ∈ (0, 1] and v ∈ (0, u A ), the following hold with probability → Λ,N 1: for all x, x , x ∈ T and t ∈ [0, N 2 T ], Proof. Recall the definition of A(x, x ) from (1.8) and recall the seminorm (3.23) In particular, under Assumption 1.

The semigroups Q(t) and Q(t)
Our goal in this section is to establish the relevant properties of the semigroups Q(t) = e tH and Q(t) = e tH . In particular, in Section 4.1, for any given potential A , we will construct Q(t) = e tH and establish bounds using integration by parts techniques. Then, in Section 4.2, we generalize these techniques to the microscopic setting to establish bounds on Q(t).
As mentioned in the introduction, we will utilize an expansion of the Feynman-Kac formula, which is similar to the expansion considered in [Sim79, Section 14, Chapter V] for singular but function-valued potentials. With the potential A (x) being non-functionvalued, we need to extract the smoothing effect of the heat semigroup to compensate the roughness of A (x). This is done by integration by parts in Lemmas 4.2 and 4.8.

Macroscopic
Recall that H = 1 2 ∂ xx + A (x). As previously explained in Remark 1.7, for the analysis within this subsection (that pertains to the limiting SPDE), the randomness of A plays no role, and we will assume without loss of generality A is a deterministic function in We begin by recalling the classical construction of H from [FN77]. Note that, even though [FN77] treats H on the closed interval [0, 1] with Dirichlet boundary condition, the (relevant) argument carries through for T as well.
Consider the symmetric quadratic form We now appeal to [Gru08, Definition 12.14] to define H to be the operator associated to F A . More precisely, let For each f ∈ D(H), the corresponding f is unique, and we define Hf := f . To summarize, (4.1) On T we have the following elementary bound To see this, for x, y ∈ T , write |f (x)| ≤ |f (y)| + [x,y] |f ( y)|d y, and integrate in y ∈ T to for some constant c = c(A) depending only on A. Given these properties, and that F A is symmetric, it then follows that (see [Gru08, Theorem 12.18, Corollary 12.19]) H is a self-adjoint, closed operator, with D(H) being dense in L 2 (T ).
Having constructed H, we now turn to the semigroup Q(t) = e tH . Heuristically, the semigroup should be given by the Feynman-Kac formula where B denotes a Brownian motion on T starting from x. The issue with this formula is that, under our assumptions, A ∈ C u A [0, 1] is not necessarily differentiable, namely the potential A may not be function-valued. As mentioned previously, for function- where Q is defined as follows. With the notation Σ n (t) from (3.6), d n s from (3.8), the convention x 0 := x, x := x n+1 , and with denoting the standard heat kernel on T , we define dA(x i ). (4.6) Note that, for each fixed (s 0 , . . . , s n ) ∈ Σ n (t), the function is actually a well-defined Riemann-Stieltjes integral. Namely, despite the heuristic nature of the preceding calculations, the function Q(t; x, x) in (4.4) is well-defined as long as the summations and integrals in (4.4)-(4.5) converge absolutely. This construction will be carried out in Proposition 4.6: there we define Q(t) via (4.4)-(4. 6), check that the result defines a bounded operator for each t ∈ [0, ∞), and verify that the result is indeed the semigroup generated by H.
Remark 4.1. In the case when A is equal to a Brownian motion B, one can also consider the chaos expansion of Q(t; x, x) (see, e.g., [Jan97]). That is, for each t, x, x, one views Q(t; x, x) as a random variable (with randomness over B), and decompose it into terms that belongs to n-th order Wiener chaoses of B. Such an expansion has been carried out in [GH18] for PAM in two dimensions, and it is conceivable that their method carries over in one dimension. We clarify here that our expansion (4.4)-(4.6) here is not the chaos expansion. For example, it is readily checked that E[R 1 (t; x, x)R 2 (t; x, x)] = 0, where the expectation is taken with respect to B.
Recall that [x 1 , x 2 ] ⊂ T denotes the interval going from x 1 to x 2 counterclockwise. We define the macroscopic analog of A(x 1 , x 2 ) (see (1.8)) as Note that the integral excludes 0 so that the possible jump of A(x) there will not be picked up by A(x 1 , x 2 ). Hereafter we adopt the standard notation f (x)| x=b Lemma 4.2 gives an integration-by-parts formula.
where, by convention, we set Mid 3 ( y, y) := Mid 1 ( y, y). Then we have Remark 4.3. The value of U(s, s ; y 1 , y 2 ) at y = y in (4.10) is irrelevant since the set has zero Lebesgue measure.
Proof. In (4.6), use the semigroup property P(s i ; (4.12) The integral in (4.12) is over x ∈ T in the Riemann-Stieltjes sense. Split the integral into integrals over T 1 (y i , y i+1 ) and T 2 (y i , y i+1 ). This gives where the equality in (4.13) follows since y i and y i+1 are fixed (the integral is in x). Then, in (4.13), integrate by parts (in x), and add the results for j = 1, 2 together. This gives U i = U( si−1 2 , si 2 ; y i , y i+1 ). Inserting this back into (4.11) completes the proof.
Equation (4.10) expresses K n in terms of U. We proceed to establish bounds on the latter. Here we list a few bounds on P(t; x, x ) that will be used in the subsequent analysis. They are readily checked from the explicit expression (4.3) of P (in the spirit of (3.4)). Given any u ∈ (0, 1] and T < ∞, for all x, x , x ∈ T and s ∈ [0, T ], T P(s; x, x)d x = 1,  1] , ∀x ∈ T j (y 1 , y 2 ), j = 1, 2.
Inserting this bound into (4.9) gives |U(s, s ; (4.17) In (4.16) use (4.7) to bound dist T (y j , x) by dist T (y 1 , x), and in (4.17) use (4.7) to bound dist T (y j , x) by dist T (y 2 , x). This way dist T has the same y variable as ∂ x P. We now have Integrate (4.18) over y 2 ∈ T , use (4.14a), (4.14e)-(4.14f) to bound the terms in (4.18), and use (4.14a)-(4.14c) to bound the terms in (4.18). We then conclude the desired result Based on Lemmas 4.2-4.4, we now establish bounds on K n . Recall the notation Σ n (t) from (3.6) and d n s from (3.8).
As for (b)-(d), integrating (4.21b)-(4.21d) over s ∈ Σ n (t), with the aid of the Dirichlet formula (3.7), one obtains the desired results via the same procedure as in the preceding. We do not repeat the argument.
Given Lemma 4.5, we now construct the semigroup Q(t), thereby verifying the heuristic given earlier. Recall R n from (4.5).
EJP 25 (2020), paper 156. Proposition 4.6. Fix u ∈ (0, 1] and T < ∞. The series R(t; x, x) := ∞ n=1 R n (t; x, x) converges uniformly over x, x ∈ T and t ∈ [0, T ], and satisfies the bounds (a) for all x, x ∈ T and t ∈ [0, T ]. Furthermore, with Q(t; x, x) := P(t; x, x) + R(t; x, x) (as in (4.4)), It remains to check that the so defined operators Q(t), t ≥ 0, form the semigroup of H. Fixing t, s ∈ [0, ∞), we begin by checking the semigroup property. Writing R 0 (t; x, y) := P(t; x, y) to streamline notation, we have  Here we rearranged the product of two infinite sums into iterated sums, which is permitted granted the bounds from Lemma 4.5.
We now turn to showing that lim t↓0 for all f, g ∈ H 1 (T ). The operator Q(t), by definition, is given by the series (4.4). This being the case, we consider separately the contribution from each term in the series.
First, for the heat kernel, with f, g ∈ H 1 (T ), it is standard to show that (4.26) Next we turn to the n = 1 term. Recall the given expressions (4.5)-(4.6) for R 1 . With the notation φ(t; x) := T P(t; x, y)φ(y)dy for a given function φ, we write Integrate by parts in x gives (4.27) with T having unit (and hence finite) Lebesgue measure, L 2 -norms and L ∞ -norms are controlled by the H 1 -norms:  Finally we consider the n ≥ 2 terms. Given the expressions (4.5)-(4.6) for R n , we (4.29) With f, g ∈ H 1 (T ), we have f L ∞ (T ) , g L ∞ (T ) < ∞. Thus, in (4.29), bound f (x) and g(x ) by their supremum, and use Lemma 4.5(a) for fixed v ∈ (0, u A ). This gives .
for each (t, x). Since each Z n is a function of Z ic and ξ, uniqueness of Z(t, x) follows.

Microscopic
Our goal is to bound the kernel Q(t; x, x) of the microscopic semigroup. Recall the operator H defined in (2.7) and the definition of ν from (1.1). Various operators and parameters (e.g., H, ν) depends on N , but, to alleviate heavy notation, we often omit the dependence in notation. Under weak asymmetry scaling (1.2), where X a (t) denotes the inhomogeneous walk defined in Section 2. Taylor-expanding the exponential function, and exchanging the expectation with the sums and integrals yields νa(x i ). (4.34) Note that, unlike in the continuum case in Section 4.1, the expansions (4.32) here is rigorous, provided that the sums and integrals in (4.33)-(4.34) converge. We now proceed to establish bounds that will guarantee the convergence of the sums and integrals. Our treatment here parallels Section 4.1, starting with a summation-bypart formula.
Given the summation-by-parts formula (4.36), we proceed to establish bounds on U. Unlike in the macroscopic case, where we assume A to be deterministic, the treatment of microscopic semigroup needs to address the randomness of a. Recall the terminology 'with probability → Λ,N 1' from (3.22).
Using this bound in (4.36), together with |ν| ≤ c N (from (4.31)), we obtain |U(s, s ; For the kernel p(t, x, x) of the homogeneous walk, we indeed have x∈T p(t, x, x) = 1. Since p a = p + r, the preceding identity together with the bound from Proposition 3.2(i) yields, with probability → Λ,N 1, x∈T p a (t, x, x) ≤ 2. Based on Lemmas 4.8-4.9, we now establish bounds on K n .

(a)
x∈T Σn(t) Proof. The proof follows by the same line of calculation as in the proof of Lemma 4.5, with p a , U, K n replacing P, U, K, and with sums replacing integrals accordingly. In particular, in place of (4.21a)-(4.21d), here we have, with probability → Λ,N 1, (4.45d) Given (4.45a)-(4.45d), the rest of the proof follows by applying the Dirichlet formula (3.7). We omit repeating the argument.
We now proceed to establish bounds on R. In the following, we will often decompose R into the sum of p, the kernel of the homogeneous walk on T, and a remainder term R := Q − p. Recall from (3.9) that p a (t; x, x) = p(t; x, x) + r(t; x, x). Referring to (4.32), we see that Proposition 4.11. Fix u ∈ (0, 1], v ∈ (0, u A ) and T < ∞. The following hold with probability → Λ,N 1: EJP 25 (2020), paper 156.
• Finally, to show (h), we fix t ∈ [t, t + 1], and set δ :  We conclude this section by establishing the convergence of the microscopic semigroup Q(t) to its macroscopic counterpart Q(t). Recall from Assumption 1.2(c) that A and A are coupled. The semigroups Q(t) and Q(t) being constructed from A and A, the coupling in Assumption 1.2(c) induces a coupling of Q(t) and Q(t).
x, x) := N r(N 2 t; N x, N x), and R n,N (t; x, x) := N R n (tN 2 ; N x, N x), and linearly interpolate these kernels in x and x. Recall from (4.4) and (4.32) that Q(t) and Q(t) are given in terms of R n (t) and P(t), and R n (t) and p a , respectively. Finally, recall that p a (t) = p(t) + r(t). We may write Given that f ∈ C(T ), with the aid of Lemma 3.1, it is standard to check that:  Given this, it suffices to check, for each fixed n ≥ 1, Such a statement is straightforwardly checked (though tedious) from the given expressions (4.5)-(4.6), (4.10) and (4.33)-(4.34), (4.37) of R n and R n , together with the aid of Lemmas 3.1, 4.5, and 4.10. We omit the details here. is shown by establishing moment bounds on Z N through iterating the microscopic equation (analogous to (2.9)); see [BG97,Section 4] and also [CST18,Section 3]. Here we proceed under the same general strategy. A major difference here is that the kernel Q(t; x, x ) (that governs the microscopic equation (2.9)) is itself random. We hence proceed by conditioning. For given u ∈ (0, 1], v ∈ (0, u A )

Moment bounds and tightness
Fix x ∈ T. Assumption 1.2(a) asserts that the Poisson clocks P ← (t, x) and P → (t, x) that dictate jumps between x and x + 1 have bounded rates. Each jump changes Z(t, x) by a factor of τ ±1 (see (1.1)). Under the weak asymmetry scaling (1.2) we have τ −1 ≤ exp(c/ √ N ). This being the case, we have for some X(j, x), X(j, x) that are stochastically dominated by Poisson(c), and are independent of the sigma algebra F (j) defined in (2.1). Now, use (5.5) in (5.4) to get Inserting this back into (5.2) concludes the desired result.
This is exactly the first desired bound (5.7a) for t = i ∈ Z >0 , and hence the label (5.7a').
We now turn to the gradient moment estimates (5.7b). Set (5.14) Following the same procedure leading to (5.9), but starting with Z(i, To bound the term J(x, x ), in (5.14), use Then, sum over x ∈ T, using the bound (5.7a') on Z(j, x) 2 a * ,k and the bounds from We now proceed to bound I(x) − I(x ). Recall that Q(t) = p(t) + R(t). Decompose I(x) = I p (x) + I R (x) into the corresponding contributions of p(t) and Q(t): I p (x) := x∈T p(i; x, x)Z ic ( x) and I R (x) := x∈T R(i; x, x)Z ic ( x). For I p , using translation invariance of p (i.e., p(t; x, x) = p(t; x+i, x+i)), we have I p (x)−I p (x ) = x∈T p(t; x, x)(Z ic ( x)− Z ic ( x + (x − x))). Given this expression, together with the Hölder continuity of Z ic ( · ) from our assumption (1.12), we have  As for I R , using the bound from Proposition 4.11(f) for u = v and the boundedness of Z ic (x) gives Finally we turn to the gradient moment estimate (5.7c). Fix i < i ∈ Z ∩ [0, N 2 T ], x ∈ T, and set x, x) 2 Z(j, x) a * ,k .
To alleviate heavy notation, hereafter we omit dependence on (i, i , x) and write I and J in places of I(i, i , x) and J(i, i , x). Following the same procedure leading to (5.9), starting from t = i instead of t = 0, here we have Using the bound (5.7a') on Z(i, x) a * ,k and the bounds from Proposition 4.11(a) and (b) As for I, decompose it into I = I p + I R , where Taking · a * ,k of I p , with the aid of (5.7b'), we have taking · a * ,k using (5.7a') and the bound from Proposition 4.11(e) gives I p a * ,k ≤ Inserting the preceding bounds on J, I p , and I R into (5.19), we obtain for all x = x and t, t ≥ 0, we may use (5.20) to approximate Z( t , x) with Z(t, x), and hence infer (5.7a)-(5.7c) from (5.7a')-  To relate Z N to Z N , we proceed to bound the difference Z N − Z N . Fix u ∈ (0, 1), v ∈ (0, u A ) and set I j : The r.h.s. of (5.21) is deterministic (i.e., not depending on a). This being the case, take E[ · ] in (5.21), and apply Markov inequality Setting k = 5 and take union bounds over j = 0, 1, . . . , T N 2 yields We achieve this via a martingale problem.

Martingale problem
Recall that, even though H and its semigroup Q(t) := e tH are possibly random, they are independent of the driving noise ξ. This being the case, conditioning on a generic realization of A, throughout this subsection, we assume Q(t) and H are deterministic, (constructed from a deterministic A ∈ C u A [0, 1]).
We say that a C([0, ∞) × T )-valued process Z solves the martingale problem corresponding to (1.10) if, for any f ∈ {ϕ n } , Hf, Z(s) ds, are local martingales in t, with respect the canonical filtration σ(Z(s) : s ≤ t). Equivalently, consider a measure space (µ, C([0, ∞) × T ), B), where B denotes the Borel sigma algebra of C([0, ∞) × T ). Let Z(s) denote the canonical process, and, for a given f ∈ {ϕ n } , view (6.1)-(6.2) as functionals of the canonical process. We say the measure space solves the martingale problem if, for any f ∈ {ϕ n } , (6.1)-(6.2) are local martingales in t, with respect the canonical filtration. Proof. Fix Z ∈ C([0, ∞) × T ) that solves the martingale problem. The first step is to show that Z is a weak solution. That is, extending the probability space if necessary, there exists a white noise measure ξ(t, x)dtdx such that, for any given f ∈ {ϕ n } , Next, for given n ≥ 1, consider the process F (t) := e −λnt ϕ n , Z(t) . Using Itô calculus, with the aid of (6.3) (for f = ϕ n ), we have Further, write e tλn ϕ n = Q(t)ϕ n and e λn(t−s) ϕ n ( x) = T Q(t − s; x, x)ϕ n (x)dx, and use the fact that Q(t − s; x, x) = Q(t − s; x, x), we now have  EJP 25 (2020), paper 156.
As stated previously, to prove Theorem 1.8, it now suffices to identify limit points of {Z N } N . This being the case, after passing to a subsequence, hereafter we assume Z N ⇒ Z, for some C([0, ∞), C(T))-valued process Z. By Skorokhod's representation theorem, extending the probability space if necessary, we further assume Z and Z N inhabit the same probability space, with for each given T < ∞. Our goal is to show that Z solves the martingale problem (6.1')-(6.2'). We further refer to (6.1') and (6.2') as the linear and quadratic martingale problems, respectively.

Linear martingale problem
Here we show that Z solves the linear martingale problem (6.1'). Let ) denote the scaled discrete Laplacian, and H N := 1 2 ∆ N + N 2 νa(N x) denote the scaled operator. Multiply both sides of (2.6) by ϕ n (N x), integrate over t ∈ [0, N 2 t] and sum over x ∈ T.
We have that M n (N 2 t) := is a martingale.
Indeed, the r.h.s. of (6.6) resemble the r.h.s. of (6.1'), and one would hope to show convergence of the former to the latter in order to establish M n (t) being a local martingale.
For the case of homogeneous ASEP, we have 1 2 ∆ N in place of H N , and the eigenfunctions ϕ n are C 2 . In this case, using Taylor expansion it is straightforward to show that t 0 1 2 ∆ N ϕ n , Z N (s) N ds converges to its continuum counterpart t 0 T 1 2 ϕ n , Z(s) ds. Here, on the other hand, we only have ϕ n ∈ H 1 (T ), and a(x) and Z N (t, x) lack differentiability in x. Given the situation, a direct proof of t 0 H N ϕ n , Z N (s) N ds converging to its continuum counterpart seems challenging.
To circumvent the aforementioned issue, we route through the integrated (i.e., mild) equation (2.10). For a given t ≥ 0 and k ∈ Z >0 , put t i := i k t, set (t * , t) = (N 2 t i−1 , N 2 t i ) in (2.10), and subtract Z(N 2 t i−1 , x) from both sides. This gives Z(s, x) where 'Id' denotes the identity operator. Multiply both sides by ϕ n (N x), and sum over x ∈ T and i = 1, . . . , k. After appropriate scaling, we obtain Further taking the k → ∞ limit using the continuity of Z(t) gives Let N → ∞ in (6.15). Given that ϕ n ∈ H 1 (T ) ⊂ L 1 (T ), with the aid of Proposition 4.12, (e sλn − 1) 2 ds ≤ k −2 (u, v, T, Λ).  Knowing that M n (t) is an F -martingale, we conclude that M n (t) is a local martingale.

Quadratic martingale problem
Our goal here is to show that Z solves the quadratic martingale problem (6.2). With M n (t) given in (6.6), the first step is to calculate the cross variation of M n (t)M n (t): Given (2.8), the r.h.s. of (6.19) permits an explicit expression in terms of η(s, x) and Z(s, x). Relevant to our purpose here is an expansion of the expression that exposes the N → ∞ asymptotics. To this end, with Z(t, x) defined in (1.1), note that Recall the filtration F (t) from (2.1). In the following we use B(t, x) = B (N ) (t, x) to denote a generic F -adopted process that may change from line to line (or even within a line), but is bounded uniformly in t, x, N . Set x))(∇Z(t, x − 1)).
Indeed, the r.h.s. of (6.26a) is the discrete analog of t 0 ϕ n ϕ n , Z 2 (s) ds that appears in (6.2'). By (3.23), a L ∞ (T) ≤ N −u A with probability → Λ,N 1. With the aid of moment bounds from Proposition 5.2, it is conceivable L 1 (t) converges in C[0, T ] to zero in probability. On the other hand, W (s, x) does not converge to zero for fixed (s, x). In order to show that L 2 (t) converges to zero, we capitalize on the spacetime averaging in (6.26c). The main step toward showing such an averaging is a decorrelation estimate on W (s, x), stated in Proposition 6.6.
To prove the decorrelation estimate, we follow the general strategy of [BG97] x, x), which we now establish. These are considerably more difficult to demonstrate in the inhomogeneous case (versus the homogeneous case). Remark 6.3. Self-averaging properties like Proposition 6.6 are often encountered in the context of convergence of particle systems to SPDEs. In particular, in addition to the approach of [BG97] that we are following, alternative approaches have been developed in different contexts. This includes hydrodynamic replacement [Qua11] and the Markov duality method [CGST20]. The last two approaches do not seem to apply in the current context. For hydrodynamic replacement, one needs two-block estimates to relate the fluctuation of h(t, x) to the quantity W (t, x). Inhomogeneous ASEP under Assumption 1.2 sits beyond the scope of existing proofs of two-block estimates. As for the duality method, it is known [BCS14] that inhomogeneous ASEP enjoys a duality via the function Q(t, x) := n i=1 η(t, x i )τ h(t,xi) . (Even though [BCS14] treats ASEP in the full-line Z, duality being a local property, readily generalizes to T.) For the method in [CGST20] to apply, however, one also needs a duality via the function Q(t, x) := n i=1 τ h(t,xi) , which is lacking for the inhomogeneous ASEP.
In what follow, for f, g ∈ [0, ∞) × T 2 → R, we write x, x)|. (6.27) Recall also that R(t) := Q(t) − p(t) denotes the difference of Q(t) = e tH to p(t), the kernel of the homogeneous walk. Proof. Throughout this proof we assume that s, t ≤ T N 2 , and, to simplify notation, write c = c(u, v, Λ, T ) and Ω = Ω(u, v, Λ, T, N ). At times below we will apply earlier lemmas or propositions wherein variables were labeled x or u. We will not, however, always apply them with the values of x and u specified in our proof (for instance, we may want to apply a result with u = 1). In order to avoid confusion, when we specify the value · of x or u (or other variables) used in that application of an earlier result we will write x → · or u → · .
EJP 25 (2020), paper 156. Given Lemma 6.4, we now proceed to establish an integral inequality of the conditional expectation of W (t + s, x).
Proof. Throughout this proof we assume s, t ≤ T N 2 , and, to simplify notation, we write c = c(u, v, Λ, T ). Recall from (6.22) that W (t + s, x) involves x-gradients of Z. The idea is to derive equations for ∇ x Z(t, x). To this end, set (t * , t) → (s, s + t) in (2.10) and take ∇ x on both sides to get  To evaluate the last term in (6.39), recall that B(t, x) denotes a generic F -adopted uniformly bounded process, and note that, from (3.23), we have |a(x)| ≤ ΛN −u A under Ω.
Recall the notation K f,g , k f,g from (6.27). Using  To proceed, we bound the residual terms in (6.42) that involves D, F 1 , and F 3 . We begin with the term N E a * [ |D(x)D(x − 1)| ]. Using the expression (6.38) for D(x), we take · a * ,2 on both sides, and write Q(t) = p(t) + R(t). With the aid of the moment EJP 25 (2020), paper 156.
As for F 3 (x). Recall that B(t, x) denotes a (generic) uniformly bounded process.
It now remains only to show that L 1 and L 2 converge in C[0, T ] to zero in probability. Given the moment bounds Proposition 5.2, it is not hard to check that L 1 , L 2 is tight in C[0, T ]. This being the case, it suffices to establish one point convergence: Lemma 6.7. For a fixed t ∈ R ≥0 , we have that L 1 (t), L 2 (t) → P 0.
We begin with L 1 . Recall that B denotes a generic uniformly bounded process.