---
abstract: 'We prove that under a particular *weak scaling*, the 4-parameter interacting particle system introduced by Corwin and Petrov in [@corwin15] converges to the equation. This expands the relatively small number of systems for which *weak universality* of the equation has been demonstrated.'
address:
- 'I. Corwin, Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA, and Clay Mathematics Institute, 10 Memorial Blvd. Suite 902, Providence, RI 02903, USA, and Institut Henri Poincaré, 11 Rue Pierre et Marie Curie, 75005 Paris, France'
- 'L.-C. Tsai, Stanford University, Building 380, Sloan Mathematical Center, Stanford, CA 94305'
author:
- Ivan Corwin
- 'Li-Cheng Tsai'
bibliography:
- 'CT-vertexKPZ.bib'
title: 'KPZ equation limit of higher-spin exclusion processes'
---

Introduction
============

This paper demonstrates how the equation [@kardar86] arises as a scaling limit of a 4-parameter interacting particle system introduced in [@corwin15] (called here the ) under fairly general choices of three parameters ($\nu\in [0,1)$, $\alpha>0$, $J\in{ \mathbb{Z} }_{>0}$) and special tuning of the remaining paremeter ($q\to 1$). This system, through various specializations, and limit procedures includes all known integrable models in the universality class. It is closely connected to the study of higher-spin vertex models within quantum integrable systems and hence enjoys a number of nice algebraic properties, some of which play important roles in our convergence proof.

The equation is a paradigmatic continuum model for a randomly growing interface with local dynamics subject to smoothing, lateral growth and space-time noise (for more background, see the review [@corwin12]). Its spatial derivative solves the stochastic Burgers equation with conservative noise, and its exponential (Hopf-Cole transform) satisfies the with multiplicative white-noise. The connection to stochastic Burgers equation suggests a relation to interacting particle systems while the connection to the suggests a relation to directed polymer models (whose partition functions satisfy discrete versions of the ).

The equation is written as $$\begin{aligned}
\label{eq:KPZeq}
    \partial_{\tau}{\mathcal{H}}(\tau,r)
    =
    \tfrac{1}{2}\delta \,\partial_r^2 {\mathcal{H}}(\tau,r)
    + \tfrac{1}{2} \kappa\, \big(\partial_r {\mathcal{H}}(\tau,r)\big)^2
    + \sqrt{D}\, \eta(\tau,r),\end{aligned}$$ where $\eta$ is space-time white noise, $\delta,\kappa\in { \mathbb{R} }$, and $D>0$. Care is needed in making sense of the above equation, and the proper notion of solution is that of the [*Hopf-Cole solution to the equation*]{} which is defined by setting ${\mathcal{H}}(\tau,r) =\tfrac{\delta}{\kappa} \log {\mathcal{Z}}(\tau,r)$ where ${\mathcal{Z}}$ solves the well-posed $$\begin{aligned}
\label{eq:SHE}
    \partial_{\tau}{\mathcal{Z}}(\tau,r)
    =
    \tfrac{1}{2}\delta \,\partial_r^2 {\mathcal{Z}}(\tau,r)
    + \tfrac{\kappa}{\delta} \sqrt{D} \,{\mathcal{Z}}(\tau,r) \eta(\tau,r).\end{aligned}$$

To understand how a microscopic system might scale to the equation, it helps to understand how the equation itself scales. For real $b,z$ define ${\mathcal{H}}^{{ \varepsilon }}(\tau,r):= { \varepsilon }^b {\mathcal{H}}({ \varepsilon }^{-z}\tau, { \varepsilon }^{-1}r)$. Then ${\mathcal{H}}^{ \varepsilon }$ satisfies the scaled equation $$\begin{aligned}
    \partial_{\tau}{\mathcal{H}}^{ \varepsilon }(\tau,r)
    =
    { \varepsilon }^{2-z}\tfrac{1}{2}\delta \,\partial_r^2 {\mathcal{H}}^{ \varepsilon }(\tau,r)
    + { \varepsilon }^{2-z-b} \tfrac{1}{2} \kappa \,\big(\partial_r {\mathcal{H}}^{ \varepsilon }(\tau,r)\big)^2
    + { \varepsilon }^{b-\frac{z}{2}+\frac{1}{2}} \sqrt{D} \, \eta(\tau,r).\end{aligned}$$ There exists no choice of $b,z$ for which the coefficients of the scaled equation remain unchanged. However, if one simultaneously changes the values of some of the $\delta,\kappa,D$ parameters as ${ \varepsilon }$ changes, the equation may scale to itself. If the equation remains invariant under such a scaling, it stands to reason that a microscopic model with similar properties may converge to the equation under a similar type of scaling and tuning of parameters. Such scalings are generally called *weak scalings* since they involve taking some of the $\delta,\kappa,D$ parameters to zero with ${ \varepsilon }$. It is thus a goal to show the *weak universality* of the equation by demonstrating how under these scalings, the equation arises from a variety of different microscopic models. Weak universality should be distinguished from *universality* which holds that without any tuning of parameters, a variety of different systems will converge under the choice of $b=1/2$ and $z=3/2$ to a universal limit called the *fixed-point* [@Corwin15a].

There are very few proved instances of weak universality of the equation. The first result was in the context of the [@bertini97] for near equilibrium initial condition (see also [@amir11] for step initial condition). The result came under *weak asymmetry* scaling through which $b=1/2, z=2$ and $\kappa\mapsto { \varepsilon }^{1/2} \kappa$ ($\delta$ and $D$ remain unscaled). This result was extend in [@dembo13] to certain non-nearest neighbor (and non-exactly solvable) exclusion processes. The only other weak universality result [@alberts14] was in the context of discrete directed polymers with arbitrary disordered distributions. This result came under *weak noise* scaling through which $b=0, z=2$ and $D\mapsto { \varepsilon }D$ ($\delta$ and $\kappa$ remain unscaled).

Owing to the round-about Hopf-Cole definition of the equation, in order to prove that a system converges to the equation, one must transform it microscopically into an approximate . The work of [@Hairer] provides direct meaning to the equation (though for $r$ on the torus, not the full real line). As of yet, this approach has not yielded weak universality results for the equation. The work of [@JaraGoncalves] defines an *energy solution* for the equation and shows tightness of a certain class of exclusion processes and that all limit points are energy solutions. However, it remains open to determine whether the notion of an energy solution is sufficiently strong to pick out the unique Hopf-Cole solution.

The partition function for a directed polymer model naturally solves a microscopic with a simple noise. On the other hand, the ASEP result relies heavily on the existence of a microscopic Hopf-Cole transform (known as the Gärtner transform [@gaertner87]), and the resulting has a much more complicated noise. This renders the associated analysis quite challenging. The work of [@dembo13] also relies on an approximate form of the Gaertner transform.

Microscopic Hopf-Cole transforms are hard to come by. For the model considered herein, this transform is achieved in Proposition \[prop:dSHE\]. The first indication that such a transform should exist came from the $k=1$ version of the Markov duality enjoyed by the model, given in [@corwin15 Theorem 2.19]. This result shows that $\operatorname{\mathbf{E}}(q^{y_n(t)+n})$ satisfies the Kolmogorov backward equation in the $t$ and $n$ variables for a very simple one-particle random walk. The existence of a nice martingale in the evolution equation for $q^{y_n(t)+n}$ requires further work given here.

The particular choice of weak scalings present in our result is new. It corresponds to the equation scaling given by $b=1, z=3, \delta\mapsto { \varepsilon }\delta, \kappa \mapsto { \varepsilon }^2 \kappa$ and $D$ remaining unchanged. In terms of the scaling of the microscopic model, we have $b=1, z=3$ and $q=e^{-{ \varepsilon }}$, while $\nu\in [0,1)$, $\alpha>0$ and $J\in{ \mathbb{Z} }_{>0}$ remain fixed. One sees that microscopically, these choices of parameters corresponds to the above weak scaling. As there are many parameters at play, it is likely that there exist other weak scalings of the system which realized the same equation limit.

There are a number of degenerations of the , including the discrete time Bernoulli $q$-TASEP [@borodin13] and (through a limit transition) the continuous time $q$-TASEP [@BigMac; @BCS]. Strictly speaking, our results do not immediately apply to the continuous time $q$-TASEP. In fact, this case has been studied in [@morenoflores] and similar results have been proved there in (though that paper has not yet appeared in print). The restriction on parameters $\nu\in [0,1)$, $\alpha>0$ and $J\in { \mathbb{Z} }_{>0}$ does not allow us to probe all of the degenerations of the system introduced in [@corwin15]. For instance, the stochastic six-vertex model [@BCG](a discrete time version of ASEP) arises through a different choice of specialization, as does the $q$-Hahn TASEP [@CHahn]. These systems likewise enjoy dualities and one may hope to prove their weak universality. We leave this for future work.

### Outline {#outline .unnumbered}

In Section \[sect:main\] we introduce the 4-parameter particle system and then proceed to state our main results. These are stated in terms of the as Theorems \[thm:main\] and \[thm:step\] (though Corollary \[cor:KPZ\] provides the equivalent statements in terms of the equation). Section \[sect:dSHE\] provides the discrete Hopf-Cole transform satisfied by the system. Section \[sect:mom\] provides moment estimates necessary to show tightness as ${ \varepsilon }\to 0$. Section \[sect:mg\] demonstrates how the limit points satisfy the martingale problem for the .

### Acknowledgements {#acknowledgements .unnumbered}

IC wishes to thank Jeremy Quastel for discussions regarding his work on the convergence of $q$-TASEP to the equation. IC was partially supported by the NSF through DMS-1208998 as well as by the Clay Mathematics Institute through the Clay Research Fellowship, by the Institute Henri Poincare through the Poincare Chair, and by the Packard Foundation through a Packard Foundation Fellowship. LCT was partially supported by the NSF through DMS-0709248.

Definition of the Model and Results {#sect:main}
===================================

We begin by recalling the definition of the . Let $ { \mathbb{N} }:={ \mathbb{Z} }_{\geq 0} $, $ { \mathbb{N} }^* := { \mathbb{N} }\cup\{\infty\} $ and $ { \mathbb{Z} }^*:={ \mathbb{Z} }\cup\{\infty\} $. Define the space of right-finite particle configurations $$\begin{aligned}
    { \mathbb{X} }_m
    :=
    { {\left\{ 
        {\vec{x}}= (\infty =\ldots = x_{m-2}=x_{m-1}>x_{m}>x_{m+1}>\ldots) \in { \mathbb{Z} }^*  
     \right\}} },\end{aligned}$$ where imaginary particles are placed at $ \infty $ for the convenience of notations, and define the space of infinite particle configurations $ { \mathbb{X} }_{\infty} := \{{\vec{x}}=(\ldots>x_{-1}>x_{0}>x_{1}>\ldots)\in{ \mathbb{Z} }^{{ \mathbb{Z} }}\} $, with the corresponding spaces of gap configurations $$\begin{aligned}
    { \mathbb{G} }_m :=
    { {\left\{  {\vec{g}}=(g_n) : g_n=\infty,\forall n< m; \quad g_n\in{ \mathbb{N} },\forall n\geq m  \right\}} },
    \quad
    { \mathbb{G} }_\infty := { \mathbb{N} }^{ \mathbb{Z} }.\end{aligned}$$ Fixing $ J\in{ \mathbb{Z} }_{>0} $, we let $ {\operatorname{mod}_J}(s) := s - { \lfloor s/J \rfloor }J $, or more explicitly, $ ({\operatorname{mod}_J}(0),{\operatorname{mod}_J}(1),\ldots) = (0,1,\ldots,J-1,0,\ldots,J-1,\ldots) $. Fixing $ q,\nu\in[0,1)$ and $\alpha>0$ we let $ \alpha_j := \alpha q^{j} $ and $ \alpha(s) := \alpha_{{\operatorname{mod}_J}(s)} $, and equip our probability space with independent Bernoulli random variables $$\begin{aligned}
    { B }_n(s,g) \sim \operatorname{Ber}{ {\left(  \frac{\alpha(s)(1-q^g)}{1+\alpha(s)}  \right)} },
    \quad
    { B' }_n(s,g) \sim \operatorname{Ber}{ {\left(  \frac{\alpha(s)+\nu q^g}{1+\alpha(s)}  \right)} },\end{aligned}$$ indexed by $ (s,g,n)\in({ \mathbb{N} },{ \mathbb{N} }^*,{ \mathbb{Z} }) $, with the corresponding filtration $ { \mathscr{F} }(t) := \sigma( { B }_n(s,g), { B' }_n(s,g): (n,g)\in{ \mathbb{N} }\times{ \mathbb{N} }^*, s=0,\ldots,t-1 ) $. Recall from [@corwin15] that the following definition of the .

\[def:EPseq\] Given $ {\vec{x}}(0)\in{ \mathbb{X} }_m $, a right-finite particle configuration, we define a $ { \mathbb{X} }_m $-valued Markov chain $ \{{\vec{y}}(s)\}_{s\in{ \mathbb{N} }} $ by setting $ {\vec{y}}(0):={\vec{x}}(0) $, and update $ {\vec{y}}(s) $ as follows. We update $ {\vec{y}}(s) $ sequentially, starting from $ m $, by letting $$\begin{aligned}
\label{eq:seqn}
    y_m(s+1) = y_m(s) + { B }_m(s,\infty),\end{aligned}$$ and letting, for $ n>m $, $$\begin{aligned}
    \label{eq:seqi}
    y_n(s+1) =
    \left\{ \begin{array}{l@{,}l}
        y_n(s) + { B' }_n(s,g_n(s))    &   \text{ if } y_{n-1}(s+1) > y_{n-1}(s),
        \\
        y_n(s) + { B }_n(s,g_n(s)) &   \text{ if } y_{n-1}(s+1) = y_{n-1}(s),  
    \end{array}
    \right.\end{aligned}$$ where $ g_n(s) := y_{n}(s) - y_{n-1}(s) -1 $ the $ n $-th gap of $ {\vec{y}}(s) $. Namely, we move $ y_n(s) $ one step to the right with probability $ \frac{\alpha(s)}{1+\alpha(s)} $, and subsequently, we move $ y_{n}(s) $ depending on how $ y_{n-1}(s) $ was updated: if $ y_{n-1}(s) $ moved we then move $ y_n(s) $ one step to the right with probability $ \frac{\alpha(s)(1-q^{g_n(s)})}{1+\alpha(s)} $, otherwise we move $ x_n(s) $ one step to the right with probability $ \frac{\alpha(s)+\nu q^{g_n(s)}}{1+\alpha(s)} $.

The $ \{{\vec{x}}(t)\}_{t\in{ \mathbb{N} }} $ is then defined as the $ { \mathbb{X} }_m $-valued Markov chain $ {\vec{x}}(t) := {\vec{y}}(Jt) $.

The Markov chain ${\vec{y}}(t)$ was defined through a local sequential update of particles. Taking $J>1$ and considering ${\vec{x}}(t)$, a priori one might think this property is lost. It was shown in [@corwin15 Section 3] that, in fact, ${\vec{x}}(t)$ can be updated through a local sequential update (just like for $J=1$). In this case, each particle may move a distance between $0$ to $J$ sites to the right, and the jump probabilities depend on the length of the previous particle’s jump as well as the length of the gap. The explicit form of this probability is somewhat more involved and given in [@corwin15 Theorem 3.15]. In particular, given a gap $g$ and a jump of the previous particle by $h$, a particle jumps by $h'$ according to the probability given by $R^{(J)}_{\alpha} (g,h;g+h-h',h')$ where a concise formulas for $R^{(J)}_\alpha$ is given in [@corwin15 Theorem 3.15] and there is also a dependence on $q,\nu$ which is suppressed in the notation. For the purposes of this paper, it suffices to study the ${\vec{y}}(t)$ process and prove convergence of it to the equation. It follows then immediately that the ${\vec{x}}(t)$ process likewise converges.

The process introduced in Definition \[def:EPseq\] is defined by the *sequential* update of –, which is inconvenient for our purpose. We now recast the definition as a *parallel* update, and, as a byproduct, extend Definition \[def:EPseq\] to the space $ { \mathbb{X} }:= \cup_{n\in{ \mathbb{Z} }^*} { \mathbb{X} }_n $ of possibly infinite particle configurations. To this end we require the following lemma, which we prove in Section \[sect:dSHE\].

\[lem:move\] For any fixed $ {\vec{g}}\in ({ \mathbb{N} }^*)^{ \mathbb{Z} }$, $ s\in{ \mathbb{N} }$, $ i\leq n\in{ \mathbb{Z} }$, letting $$\begin{aligned}
\label{eq:moveI}
    { I }_{n,m}(s,{\vec{g}})
    :=
    \Big(
        \prod_{n\geq i> m}
        { {\left(  { B' }_{i}(s,g_{i}) - { B }_{i}(s,g_{i})  \right)} }
    \Big)
    { B }_m(s,g_m),\end{aligned}$$ we have $$\begin{aligned}
\label{eq:move}
    { K }_n(s,{\vec{g}}) := \sum_{m\leq n} { I }_{n,m} \in\{0,1\},\end{aligned}$$ where the series converges in $ L^k $ for all $ k\geq 1 $ and hence almost surely. Further, $$\begin{aligned}
\label{eq:moveIter}
    { K }_{n}(s,{\vec{g}})
    =
    { K }_{n-1}(s,{\vec{g}}) { B' }_n(s,g_n) + (1-{ K }_{n-1}(s,{\vec{g}})) { B }_n(s,g_n).\end{aligned}$$

\[def:EPpar\] Fix $ m\in{ \mathbb{N} }^* $ and $ {\vec{x}}(0)\in{ \mathbb{X} }_m $. Letting $ {\vec{g}}({\vec{y}}) := (y_{n-1}-y_{n}-1)_{n\in{ \mathbb{Z} }} $ and $ {\vec{K}}(s,{\vec{g}}) := (K_n(s,{\vec{g}}))_{n\in{ \mathbb{Z} }} \in \{0,1\}^{ \mathbb{Z} }$ (by Lemma \[lem:move\]), we define a stochastic map $$\begin{aligned}
\label{eq:trans}
    { \mathcal{T} }(s):   { \mathbb{X} }_m \longrightarrow { \mathbb{X} }_m,
    \quad
    {\vec{y}}\longmapsto {\vec{y}}+ {\vec{K}}(s,{\vec{g}}({\vec{y}})),\end{aligned}$$ and define the $ { \mathbb{X} }_m $-valued Markov chain $ \{{\vec{x}}(t)\} $ and $ \{{\vec{y}}(t)\} $ by letting $ {\vec{y}}(s) := { \mathcal{T} }(s-1) \circ { \mathcal{T} }(s-2) \circ \cdots \circ { \mathcal{T} }(0)({\vec{x}}(0)) $ and $ {\vec{x}}(t) := {\vec{y}}(tJ) $.

\[rmk:EPequiv\] Under the map $ { \mathcal{T} }(s) $, we have that $
    { K }_{n}(s,{\vec{g}}({\vec{y}}(s))) = \operatorname{\mathbf{1}}_{ {\left\{ y_{n}(s+1)>y_n(s) \right\}} },
$ whereby becomes $$\begin{aligned}
    \operatorname{\mathbf{1}}_{ {\left\{  y_{n}(s+1)>y_n(s)  \right\}} }
    =
    \operatorname{\mathbf{1}}_{ {\left\{  y_{n-1}(s+1)>y_{n-1}(s)  \right\}} } { B' }_n(s,g_n(s))
    +
    \operatorname{\mathbf{1}}_{ {\left\{  y_{n-1}(s+1)=y_{n-1}(s)  \right\}} } { B }_n(s,g_n(s)).\end{aligned}$$ This is equivalent to , which reduces to when $ g_{m-1}=\infty $. It is thus easy to see that Definition \[def:EPpar\] is equivalent to Definition \[def:EPseq\] when restricting to $ {\vec{x}}(0)\in{ \mathbb{X} }_m $, $ m\in{ \mathbb{N} }$.

Our main result is the convergence to the of a certain exponential transform of the process $ {\vec{y}}(t) $. Recall that we say a process $ {\mathcal{Z}}$ on $ { \mathbb{R} }_+\times{ \mathbb{R} }$ is a mild solution of the starting from the initial condition $ {\mathcal{Z}^{\text{ic}}}$ if $$\begin{aligned}
\label{eq:SHEmild}
    {\mathcal{Z}}(\tau,r)
    = \int_{{ \mathbb{R} }} P_\tau(r-r') {\mathcal{Z}^{\text{ic}}}(r') dr'
        + \int_0^{\tau} \int_{{ \mathbb{R} }} P_{\tau-\tau'}(r-r') {\mathcal{Z}}(\tau',r') \eta(dr',d\tau'),\end{aligned}$$ where $ P_\tau(r) := \exp[-r^2/(2\tau)] (2\pi\tau)^{-1/2} $ denotes the standard heat kernel, and $\eta$ denotes the space-time white noise. For the existence, uniqueness, continuity, and positivity of solutions of , see [@corwin12 Proposition 2.5].

The key step of showing the convergence to the is finding a discrete . To state it, we fix a parameter $ { \rho }\in(0,1) $, measuring the limiting density (see Remark \[rmk:dens\]), set $$\begin{aligned}
    &
    \label{eq:abconst}  
    { \gamma }:= \frac{1-{ \rho }}{1-\nu { \rho }},
    \quad\quad
    a_j := \frac{\alpha_j { \gamma }}{1+\alpha_j { \gamma }},
    \quad\quad
    b := \frac{{ \gamma }}{1-{ \gamma }},
    \quad\quad
    b' := \frac{\nu{ \gamma }}{1-\nu{ \gamma }},    
\\  
    \label{eq:driftt}
    &
    { \mu }(t) := (a_{{\operatorname{mod}_J}(t)}-a_{{\operatorname{mod}_J}(t)+1})(b-b')^{-1},
\\
    &
    \label{eq:centt}
    { \lambda }(t) := \frac{1+\alpha(t) { \gamma }}{1+q\alpha(t) { \gamma }},
\\
    &
    \notag
    { \widehat{\mu} }(t) := \sum_{s=0}^{t-1} { \mu }(s),
    \quad
    { \widehat{\lambda} }(t) := \prod_{s=0}^{t-1} { \lambda }(s),&\end{aligned}$$ and, with $ Q_n(t) := q^{y_n(t)+n} $ denoting the one-particle duality function (see [@corwin15] for the definition of duality functions), we define the exponential transformation $$\begin{aligned}
\label{eq:Z}
    Z(t,\xi) := { \widehat{\lambda} }(t) { \rho }^{\xi+{ \widehat{\mu} }(t)} Q_{\xi+{ \widehat{\mu} }(t)}(t),\end{aligned}$$ for $ t\in{ \mathbb{N} }$ and $\xi \in \Xi(t) := ({ \mathbb{Z} }-{ \widehat{\mu} }(t)) $. The discrete is expressed in terms of a certain random walk $ { R }(0)+\ldots+{ R }(t-1) $ on $ { \mathbb{R} }$. Here, ${ R }(s)\in({ \mathbb{N} }- { \mu }(s))$, $ s\in{ \mathbb{N} }$, are independent random variables introduced in , with zero mean and variance as in . Let $ \Xi(t_2,t_1) := { \mathbb{N} }+({ \widehat{\mu} }(t_1)-{ \widehat{\mu} }(t_2)) $. For $ t_1\leq t_2 $, $ \zeta\in\Xi(t_2,t_1) $, let $$\begin{aligned}
    \label{eq:hk}
    {p}(t_2,t_1,\zeta) := \operatorname{\mathbf{P}}({ R }(t_1)+{ R }_{ \varepsilon }(t_1+1)+\ldots{ R }(t_2-1)=\zeta)\end{aligned}$$ denotes the corresponding semigroup. We use the shorthand notations $
    [{p}(t_2,t_1)*f(t_1)](\xi):=
    \sum_{\zeta\in\Xi(t_1)} {p}(t_2,t_1,\xi-\zeta) f(t_1,\zeta)
$ to denote convolution. Let $ { \overline{K} }_{n}(s) := { K }_n(s) - \operatorname{\mathbf{E}}({ K }_n(s)|{ \mathscr{F} }(s)) $ and let $$\begin{aligned}
\label{eq:mgg}
    { W }(t,\xi) := { \lambda }(t) (q-1) { \overline{K} }_{\xi+{ \widehat{\mu} }(t)}(t),\end{aligned}$$ representing the discrete analog of the space-time white noise.

\[prop:dSHE\] For all $ t_1\leq t_2\in{ \mathbb{N} }$ and $ \xi\in\Xi(t_2) $, we have the following discrete $$\begin{aligned}
\label{eq:dSHE}
    Z(t_2,\xi)
    =
    {Z_\text{dr}}(t_2,t_1,\xi)
    +
    {Z_\text{mg}}(t_2,t_1,\xi),\end{aligned}$$ where $$\begin{aligned}
    &
    \label{eq:Zdr}
    {Z_\text{dr}}(t_2,t_1,\xi) := [{p}(t_2,t_1)*Z(t_1) ](\xi),
\\
    &
    \label{eq:Zmg}
    {Z_\text{mg}}(t_2,t_1,\xi)
    :=
    \sum_{s=t_1}^{t_2-1} [{p}(t_2,s+1)*(Z(s)W(s)) ](\xi+{ \mu }(s)).\end{aligned}$$ Further, for all $ \xi_1,\xi_2\in\Xi(t) $, $$\begin{aligned}
\label{eq:qv}
\begin{split}
    &
    Z(t,\xi_1) Z(t,\xi_2)
    \operatorname{\mathbf{E}}\big[
        { W }(t,\xi_1) { W }(t,\xi_2) \big| { \mathscr{F} }(t)
        \big]
\\
    &
    \quad
    =
    { {\left( \frac{(\nu+\alpha(t)){ \rho }}{1+\alpha(t)} \right)} }^{|\xi_1-\xi_2|}
    { \Theta }_1(t,\xi_1\wedge\xi_2) { \Theta }_2(t,\xi_1\wedge\xi_2),
\end{split}\end{aligned}$$ where $$\begin{aligned}
    &
    \label{eq:qv1}
    { \Theta }_1(t,\xi) :=
    q { \lambda }(t) Z(t,\xi) - [{p}(t+1,t)*Z(t)](\xi-{ \mu }(t)),
\\
    &
    \label{eq:qv2}
    { \Theta }_2(t,\xi) :=
    - { \lambda }(t)  Z(t,\xi) + \big[{p}(t+1,t)*Z(t)](\xi-{ \mu }(t)).\end{aligned}$$

Proceeding to our main result, we consider the weak noise scaling $ q={ q_\varepsilon }:= e^{-{ \varepsilon }} $, $ { \varepsilon }\to 0 $. Hereafter throughout the paper, we *fix* $\alpha>0$, $\nu\in[0,1)$, and ${ \rho }\in (0,1)$, scaling only the parameter $ { q_\varepsilon }\to 1 $. To indicate this scaling, we denote *parameters* such as $ \alpha_j $ and $ \alpha(s) $ by $ {\alpha^\varepsilon_{j}} $ and $ {\alpha_\varepsilon}(s) $, but for *processes* such as $ {\vec{x}}(t) $, $ { B }_n(s,g) $, we often omit the dependence on $ { \varepsilon }$ to simplify notations. Under this scaling, to the first order – read $$\begin{aligned}
    &
    \label{eq:driftte}
    { \mu_\varepsilon }(t) =
    { \varepsilon }\alpha{ \gamma }(1+\alpha{ \gamma })^{-2}(b-b')^{-1} + O({ \varepsilon }^{2}),
\\
    &
    \label{eq:centte}
    { \lambda_\varepsilon }(t) = 1 + { \varepsilon }\alpha { \gamma }(1+\alpha{ \gamma })^{-1} + O({ \varepsilon }^{2}).\end{aligned}$$ Let $ { r_* }:= (b-b')^{-1} $, $$\begin{aligned}
    \label{eq:tcent}
    { \tau^{\varepsilon}_* }&:= { {\left[  (a_{0})^2 -({a^{\varepsilon}_{J}})^2 + (a_0-{a^{\varepsilon}_{J}})(b+b')  \right]} }^{-1}
\\
    \label{eq:tcentApprox}  
    &=
    (1+\alpha{ \gamma })^2(J\alpha{ \gamma })^{-1}(2 a_0+b+b')^{-1} + O({ \varepsilon }).\end{aligned}$$ Note that here $ a_0 = \alpha{ \gamma }(1+\alpha{ \gamma })^{-1} $ is independent of $ { \varepsilon }$. We extend the process $ Z(t,\xi) $, defined for $ t\in{ \mathbb{N} }$ and $ \xi\in\Xi(t) $, to a continuous process on $ { \mathbb{R} }_+\times{ \mathbb{R} }$ by first linearly interpolate in $ \xi $ and then linearly interpolate in $ t $, and then we introduce the scaled process $$\begin{aligned}
\label{eq:Ze}
    {Z_\varepsilon}(\tau,r) := Z({ \varepsilon }^{-3}{ \tau^{\varepsilon}_* }J \tau, { \varepsilon }^{-1} { r_* }r),\end{aligned}$$ or, equivalently $ {Z_\varepsilon}(\tau,r) = \exp(H_{ \varepsilon }(\tau,r)) $, where $$\begin{aligned}
\label{eq:He}
    H_{ \varepsilon }(\tau,r)
    :=&
    -{ \varepsilon }y_{{n_\varepsilon}(\tau,r)}({t_\varepsilon}(\tau))
    + { {\left( { \varepsilon }^{-1}\log\rho -1 \right)} } r_* r
    + { {\left(  \log{ \lambda_\varepsilon }({t_\varepsilon}(\tau)) + { \widehat{\mu}_\varepsilon }({t_\varepsilon}(\tau))  \right)} },
$$ $ {t_\varepsilon}(\tau) :=  { \varepsilon }^{-3}{ \tau^{\varepsilon}_* }J\tau $ and $ {n_\varepsilon}(\tau,r) := { \varepsilon }^{-1}{ r_* }r + { \widehat{\mu}_\varepsilon }({t_\varepsilon}(\tau)) $. Following [@bertini97], we consider *near equilibrium* initial conditions:

\[def:neareq\] Let $ {Z_\varepsilon}(0,\xi) $ be the exponential transform (given as in ) associated with $ \{{\vec{x}^{\varepsilon}}(0)\}_{ \varepsilon }\subset{ \mathbb{X} }$. We say $ \{{\vec{x}^{\varepsilon}}(0)\}_{ \varepsilon }\subset{ \mathbb{X} }$ is *near equilibrium* if, given any $ k\in{ \mathbb{Z} }_{>0} $ and $ v\in(0,1/2) $, there exists $ u=u(k,v), C=C(k,v)<\infty $ such that $$\begin{aligned}
    &
    \label{eq:neareq:mom}
    { {\left\Vert {Z_\varepsilon}(0,\xi) \right\Vert} }_{k} := { {\left(  \operatorname{\mathbf{E}}{ {\left( {Z_\varepsilon}(0,\xi)^k \right)} }  \right)} }^{1/k}
    \leq C e^{u|\xi|},
\\
    &
    \label{eq:neareq:momx}
    { {\left\Vert {Z_\varepsilon}(0,\xi)-{Z_\varepsilon}(0,\xi') \right\Vert} }_k \leq C|\xi-\xi'|^v e^{u(|\xi|+|\xi'|)},   \end{aligned}$$ for all $ \xi,\xi'\in { \varepsilon }({ r_* })^{-1}{ \mathbb{Z} }$ and $ { \varepsilon }>0 $ small enough.

Hereafter we endow the spaces $ C({ \mathbb{R} }) $, $ C({ \mathbb{R} }_{+}\times{ \mathbb{R} }) $ and $ C((0,\infty)\times{ \mathbb{R} }) $ the topology of uniform convergences on compact subsets, and use $ \Rightarrow $ to denote weak convergence of probability laws. The following is our main result.

\[thm:main\] Let $ {\mathcal{Z}}$ be the unique $ C({ \mathbb{R} }_+,{ \mathbb{R} }) $-valued solution of starting from a $ C({ \mathbb{R} }) $-valued process $ {\mathcal{Z}^{\text{ic}}}$, and let $ {Z_\varepsilon}(\tau,r)\in C({ \mathbb{R} }_+,{ \mathbb{R} }) $ be as in , with some near equilibrium initial condition $ \{{\vec{x}^{\varepsilon}}(0)\}_{ \varepsilon }$. If $ {Z_\varepsilon}(0,{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) \Rightarrow {\mathcal{Z}^{\text{ic}}}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}})  $, then $$\begin{aligned}
    {Z_\varepsilon}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}},{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) \Rightarrow {\mathcal{Z}}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}},{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}})
    \text{ on }
    C({ \mathbb{R} }_+\times{ \mathbb{R} }),\end{aligned}$$ as $ { \varepsilon }\to 0 $.

Definition \[def:neareq\] (and therefore Theorem \[thm:main\]) leaves out an important initial condition, e.g. the step initial condition: $ x_n(0) := {-n} $ for $ n\in{ \mathbb{N} }$ and $ x_{n}=\infty $ for $ n\in{ \mathbb{Z} }_{<0} $. Following [@amir11], we generalize Theorem \[thm:main\] to the following:

\[thm:step\] Let $ {\widetilde{\mathcal{Z}}}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}},{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $ be the unique solution of starting from the delta measure $ \delta({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $, let $ \{{\vec{y}}(t)\}_{t} \in { \mathbb{X} }_0 $ be the process starting from the step initial condition, and let $ {\widetilde{Z}_\varepsilon}(\tau,r) := { \varepsilon }^{-1}(1-\rho){ r_* }{Z_\varepsilon}(\tau,r) $. Then $$\begin{aligned}
    {\widetilde{Z}_\varepsilon}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}},{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) \Rightarrow {\widetilde{\mathcal{Z}}}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}},{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}})
    \text{ on }
    C((0,\infty)\times{ \mathbb{R} }),\end{aligned}$$ as $ { \varepsilon }\to 0 $.

\[rmk:dens\] By Theorem \[thm:main\] we have that $ H_{ \varepsilon }(\tau,r)-H_{ \varepsilon }(\tau,r+{ \varepsilon }/{ r_* }) \to 0 $ in probability. Plugging this in , a posteriori we find that, $ x_{{n_\varepsilon}(\tau,r)+1}({t_\varepsilon}(\tau)) - x_{{n_\varepsilon}(\tau,r)}({t_\varepsilon}(\tau)) \approx { \varepsilon }^{-1} \log(1/{ \rho }) $, or equivalently the limiting density is $ { \varepsilon }/\log(1/{ \rho }) $.

Hereafter we adapt the convention that $ m,n,i,j,k\in{ \mathbb{Z} }$; $ s,t\in{ \mathbb{N} }$; $ \tau,\tau'\in{ \mathbb{R} }_+ $; and $ r\in{ \mathbb{R} }$. To simplify notation, we let $ {\vec{g}}(t) := {\vec{g}}({\vec{y}}(t)) $, $ { B }_n(t):={ B }_n(t,{\vec{g}}(t)) $, $ { B' }_n(t):={ B' }_n(t,{\vec{g}}(t)) $, $ { K }_n(t):= { K }_n(t,{\vec{g}}(t)) $ and $ { I }_{n,m}(t):= { I }_{n,m}(t,{\vec{g}}(t)) $, with the consensus that an underlying process $ {\vec{y}}(t) $ has been fixed. We will specify explicitly when a result applies only for near equilibrium initial conditions or the step initial condition, and without specification the result holds for any initial condition $ {\vec{x}}(0)\in{ \mathbb{X} }$.

This is an immediate consequence of the following propositions, which we establish in Section \[sect:mom\] and \[sect:mg\], respectively.

\[prop:tight\] For near equilibrium initial conditions, the collection of processes $ \{{Z_\varepsilon}\}_{ \varepsilon }$ is tight in $ C({ \mathbb{R} }_+,{ \mathbb{R} }) $.

\[prop:unique\] For near equilibrium initial conditions, any limiting point $ {\mathcal{Z}}$ of $ \{{Z_\varepsilon}\}_{ \varepsilon }$ solves the .

We let $ {\widetilde{Z}}(\tau,r) := { r_* }{ \varepsilon }^{-1}(1-{ \rho }) Z(t,r) $ so that $ {\widetilde{Z}_\varepsilon}(\tau,r) = {\widetilde{Z}}({ \varepsilon }^{-3}{ \tau^{\varepsilon}_* }\tau, { \varepsilon }^{-1}{ r_* }r) $. The prefactor of $ {\widetilde{Z}}(\tau,r) $ is choose so that $$\begin{aligned}
\label{eq:Zdmass}
    { \varepsilon }{ r_* }^{-1} \sum_{\xi\in\Xi(0)} {\widetilde{Z}}(0,\xi) = 1.\end{aligned}$$ Further, using the exponential decay (in $ |\xi| $) of $ {\widetilde{Z}}(0,\xi) $, one easily obtains $ {\widetilde{Z}_\varepsilon}(0,{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) \Rightarrow \delta({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $. With this and Theorem \[thm:main\], following the argument of [@amir11 Section 3], Theorem \[thm:step\] is an immediate consequence of the following moment estimates of $ {\widetilde{Z}}(\tau,r) $, which we establish in Section \[sect:mom\].

\[prop:stepEst\] For the step initial condition, for any $ T>0 $, $ k\geq 1 $ and $ v\in(0,1/2) $, there exists $ C=C(T,k,v)<\infty $ such that $$\begin{aligned}
    &
    \label{eq:momd}
    { \Vert  {\widetilde{Z}}(\tau,r)  \Vert }_{2k} \leq C ({ \varepsilon }^{3}\tau)^{-1/2},
\\
    &
    \label{eq:momdx}
    { \Vert  {\widetilde{Z}}(\tau,r) - {\widetilde{\mathcal{Z}}}(\tau,r')  \Vert }_{2k} \leq C ({ \varepsilon }|r-r'|)^{v} ({ \varepsilon }^3\tau)^{-(1+v)/2},\end{aligned}$$ for all $ \tau\in(0,{ \varepsilon }^{-3}T] $ and $ r,r'\in{ \mathbb{R} }$.

With $ {\vec{x}}(t) = {\vec{y}}(tJ) $, from Theorems \[thm:main\] and \[thm:step\] we immediately obtain the following corollary on the convergence of $ {\vec{x}}(t) $. Recall $ H_{ \varepsilon }(\tau,r) $ is defined as in . We let $$\begin{aligned}
    {H^J_\varepsilon}(\tau,r) :=& H_{ \varepsilon }(J\tau,r)
    =
    -{ \varepsilon }x_{{ \varepsilon }^{-1} { r_* }r + { \varepsilon }^{-3}{ \mu_\varepsilon }{ \tau^{\varepsilon}_* }\tau }({ \varepsilon }^{-3}{ \tau^{\varepsilon}_* }\tau)
\\
    &+ { {\left( { \varepsilon }^{-1}\log\rho -1 \right)} } r_* r
    + { {\left(  \log({ \varepsilon }^{-3} { \lambda_\varepsilon }{ \tau^{\varepsilon}_* }\tau) + { \varepsilon }^{-3}{ \mu_\varepsilon }{ \tau^{\varepsilon}_* }\tau  \right)} },\end{aligned}$$ where $ { \mu_\varepsilon }:= \sum_{s=0}^{J-1} { \mu_\varepsilon }(s) $ and $ { \lambda_\varepsilon }:= \sum_{s=0}^{J-1} { \lambda_\varepsilon }(s) $.

\[cor:KPZ\]

1.  2.  Let $ {\mathcal{Z}}(\tau,r) $ and $ {\mathcal{Z}^{\text{ic}}}(r) $ be as in Theorem \[thm:main\] so that $ {\mathcal{H}}(\tau,r) := \log {\mathcal{Z}}(\tau,r) $ is the unique solution of the equation starting from $ \log {\mathcal{Z}^{\text{ic}}}$, and let $ \{{\vec{x}}^{ \varepsilon }(0)\}_{ \varepsilon }$ be a collection of near equilibrium initial conditions. If $ {Z_\varepsilon}(0,{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) \Rightarrow {\mathcal{Z}^{\text{ic}}}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $, we have $$\begin{aligned}
    \label{eq:KPZcnvg}
            {H^J_\varepsilon}\Rightarrow {\mathcal{H}}\text{ on } C((0,\infty)\times{ \mathbb{R} })
        \end{aligned}$$ as $ { \varepsilon }\to 0 $.

3.  Let $ {\widetilde{\mathcal{Z}}}(\tau,r) $ be as in Theorem \[thm:step\], let $ \{{\vec{x}}^{ \varepsilon }(t)\}_{ \varepsilon }$ be started from the step initial condition, and let $ {\widetilde{H}^J_\varepsilon}(\tau,x) := {H^J_\varepsilon}(\tau,x) + \log({ \varepsilon }^{-1}(1-{ \rho }){ r_* }) $. We have $$\begin{aligned}
            {\widetilde{H}^J_\varepsilon}\Rightarrow {\widetilde{\mathcal{H}}}\text{ on } C((0,\infty)\times{ \mathbb{R} })
        \end{aligned}$$ as $ { \varepsilon }\to 0 $.

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1.  2.  In the convergence does not include $ \tau=0 $ as we do not assume $ {\mathcal{Z}^{\text{ic}}}(r)>0 $.

3.  From Theorems \[thm:main\] and \[thm:step\], one also easily obtains corresponding convergence results for $ {Z_\varepsilon}(\tau J,r) $, the centered scaled exponential transform of $ {\vec{x}}(t) $, but we do not state the results explicitly here.

Discrete , Proof of Proposition \[prop:dSHE\] {#sect:dSHE}
=============================================

Fixing $ s\in{ \mathbb{N} }$ and $ {\vec{g}}\in({ \mathbb{N} }^*)^\infty $, we let $ { K }_n $ and $ { I }_{n,i} $ denote $ { K }_n(s,{\vec{g}}) $ and $ { I }_{n,i}(s,{\vec{g}}) $, respectively, and let $ { K }_{n,i} := \sum_{n \geq i' \geq i} { I }_{n,i'} $ denote the $ i $-th partial sum of . With $ { B }_{k}(s,g) $ and $ { B' }_{k}(s,g) $ defined as in the preceding, we have $$\begin{aligned}
    &
    \operatorname{\mathbf{E}}{ {\left|  { B' }_{k}(s,g) - { B }_{k}(s,g)  \right|} } \leq 1 - \frac{1}{1+\alpha}\frac{1-\nu}{1+\alpha} < 1,&
    &
    \operatorname{\mathbf{E}}{ B }_k(s,g) \leq \frac{\alpha}{1+\alpha} < 1.\end{aligned}$$ Consequently, $ { K }_{n,i} \to { K }_n $ (as $ i\to-\infty $) in $ L^k $ for all $ k\geq 1 $.

To show $ { K }_n\in\{0,1\} $, first we use the identity $ { I }_{n,i} = ({ B' }_n-{ B }_n){ I }_{n-1,i} $ (which follows from ) to obtain $$\begin{aligned}
\label{eq:moveIter:}
    { K }_{n,i-1}(s,{\vec{g}})
    =
    { K }_{n-1,i-1}(s,{\vec{g}}) { B' }_n(s,g_n) + (1-{ K }_{n-1,i-1}(s,{\vec{g}})) { B }_n(s,g_n).\end{aligned}$$ We now show that, in fact, $ { K }_{n,i}\in\{0,1\} $ for all $ n\geq i $. Indeed, $ K_{n,n} = B_n \in\{0,1\} $. The general case then follows by induction on $ n-i\in{ \mathbb{N} }$ using . Consequently, $ { K }_{n,i} \to { K }_n \in\{0,1\} $.

The identity follows directly by letting $ i\to-\infty $ in .

Turning to proving Proposition \[prop:dSHE\], as this does not involve the scaling $ { \varepsilon }\to 0 $, throughout this section we *suppress* the dependence of parameters on $ { \varepsilon }$. We begin by deriving an equation for $ Q_n(t) $. Consider the time-inhomogeneous random walk $ { R' }(0) + { R' }(1) + \ldots + { R' }(t) $, where $ { R' }(s)$, $ s\in{ \mathbb{N} }$, are $ { \mathbb{N} }$-valued, independent, with distribution $$\begin{aligned}
\label{eq:RWt}
    \operatorname{\mathbf{P}}({ R' }(t) =n )
    :=
    { p' }_n(t+1,t)
    :=
    \left\{\begin{array}{l@{,}l}
            \frac{\alpha(t)(1-q)}{1+\alpha(t)}
            { {\left(  \frac{\nu+\alpha(t)}{1+\alpha(t)}  \right)} }^{n}
            { {\left(  1 - \frac{\alpha(t)+\nu}{1+\alpha(t)}  \right)} }
            &
            \text{ when } n>0,
            \\
            1 - \frac{\alpha(t)(1-q)}{1+\alpha(t)}
            &
            \text{ when } n=0,
            \\
            0
            &
            \text{ otherwise}.
    \end{array}\right.\end{aligned}$$ Let $ [{ p' }(t+1,t)*Q(t)]_n := \sum_{m\in{ \mathbb{Z} }} { p' }(t+1,t,n-m) Q_{m}(t) $ denote convolution.

For any $ t\in{ \mathbb{N} }$ and $ n \in{ \mathbb{Z} }$, we have $$\begin{aligned}
    \label{eq:duality}
    Q_{n}(t+1) = [{ p' }(t+1,t) Q(t)]_n + Q_n(t) { W }'_n(t),\end{aligned}$$ where $ { W' }_n(t) := (q-1) { \overline{K} }_m(t) $. Further, for any $n_1, n_2\in{ \mathbb{Z} }$, $$\begin{aligned}
\label{eq:qvraw}
\begin{split}
    &
    Q_{n_1}(t)Q_{n_2}(t) \operatorname{\mathbf{E}}{ {\left(  { W' }_{n_1}(t){ W' }_{n_2}(t) \middle| { \mathscr{F} }(t)  \right)} }
\\
    &
    \quad
    =
    { {\left( \frac{\nu+\alpha(t)}{1+\alpha(t)} \right)} }^{|n_1-n_2|}
    { \Theta' }_1(t,n_1\wedge n_2) { \Theta' }_2(t,n_1\wedge n_2),
\end{split}\end{aligned}$$ where $ { \Theta' }_{1}(t,n) := qQ_{n}(t) - [{ p' }(t+1,t)Q(t)]_{n} $ and $ { \Theta' }_{2}(t,n) := [{ p' }(t+1,t)Q(t)]_{n} - Q_{n}(t) $.

Fixing $ t\in{ \mathbb{N} }$, to simplify notation we let $ \operatorname{\mathbf{E}}'({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $ denote $ \operatorname{\mathbf{E}}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}|{ \mathscr{F} }(t)) $. We begin by proving . With $ Q_n(t) := q^{y_n(t)+n} $, a generic jump $ y_n(t) \mapsto y_{n}(t)+1 $ of particles decreases $ Q_n(t) $ by $ (1-q) Q_n(t) $. Consequently, $$\begin{aligned}
    Q_n(t+1) - Q_n(t)
    =  (q-1) Q_n(t) { K }_n(t)
    =  (q-1) Q_n(t) \operatorname{\mathbf{E}}'({ K }_n(t)) + Q_n(t) { W' }_n(t).\end{aligned}$$ With $ { K }_n(t) $ as in , we have $$\begin{aligned}
\label{eq:moveEx}
    \operatorname{\mathbf{E}}'({ K }_n(t))
    =
    \sum_{m:n\geq m}
    \frac{\nu+\alpha(t)}{1+\alpha(t)} q^{g_n(t)} \cdots \frac{\nu+\alpha(t)}{1+\alpha(t)} q^{g_{m+1}(t)}
    \frac{\alpha(t)}{1+\alpha(t)} (1-q^{g_m(t)}).\end{aligned}$$ Multiplying both sides by $ (q-1)Q_n(t) $, and then using the readily verify identity $$\begin{aligned}
\label{eq:Qshift}
    Q_n(t) q^{g_n(t)+g_{n-1}(t)+\ldots +g_{m'+1}(t)} = Q_{m'}(t),\end{aligned}$$ we obtain $$\begin{aligned}
\label{eq:moveQ}
    (q-1)Q_n(t) \operatorname{\mathbf{E}}'({ K }_n(t)) = [{ p' }(t+1,t)*Q(t)]_n - Q_n(t),\end{aligned}$$ whereby follows.

Turning to , without lost of generality we assume $ n_1\geq n_2 $. With $ { W' }_n(t) $ defined as in the proceeding, we have $
    \operatorname{\mathbf{E}}'({ W' }_{n_1}(t){ W' }_{n_2}(t)) = (q-1)^2 \operatorname{Cov}'({ K }_{n_1}(t),{ K }_{n_2}(t))
$, where $
    \operatorname{Cov}'({ K }_{n_1}(t),{ K }_{n_2}(t))
    := \operatorname{\mathbf{E}}'({ K }_{n_1}(t){ K }_{n_2}(t)) - \operatorname{\mathbf{E}}'({ K }_{n_1}(t)) \operatorname{\mathbf{E}}'({ K }_{n_2}(t))
$. Letting $ { \widetilde{I} }_{n_1,n_2}(t) := \prod_{n_1\geq k > n_2} ( { B' }_k(t) - { B }_k(t) ) $, with $ { K }_{n_1}(t) $ as in , we have $$\begin{aligned}
    { K }_{n_1}(t)
    =
    \sum_{n_1 \geq m > n_2} { I }_{n_1,m}(t) + { \widetilde{I} }_{n_1,n_2}(t) { K }_{n_2}(t),\end{aligned}$$ for all $ n_1 \geq n_2 $. Multiply both sides by $ { K }_{n_2}(t) $, using $ { K }_{n_2}(t)^2 = { K }_{n_2}(t) $, and then take the expectation $ \operatorname{\mathbf{E}}'({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $ on both sides. With $ \{{ B }_k(s),{ B' }_k(s)\}_k $ being independent, we obtain that $$\begin{aligned}
    \operatorname{\mathbf{E}}'({ K }_{n_1}(t) { K }_{n_2}(t))
    &=
    \Big( \sum_{n_1\geq m > n_2} \operatorname{\mathbf{E}}'({ I }_{n_1,m}(t)) + \operatorname{\mathbf{E}}'({ \widetilde{I} }_{n_1,n_2}(t)) \Big)
    \operatorname{\mathbf{E}}'({ K }_{n_2}(t)).\end{aligned}$$ Subtracting $ \operatorname{\mathbf{E}}'({ K }_{n_1}(t)) \operatorname{\mathbf{E}}'({ K }_{n_2}(t)) = [\sum_{n_1\geq m} \operatorname{\mathbf{E}}'({ I }_{n_1,m}(t))] \operatorname{\mathbf{E}}'({ K }_{n_2}(t)) $ from the last expression yields $$\begin{aligned}
    \notag
    \operatorname{Cov}'({ K }_{n_1}(s) { K }_{n_2}(t))
    &=
    \Big( -\sum_{m:n_2>m} \operatorname{\mathbf{E}}'( { I }_{n_1,m}(t) ) + \operatorname{\mathbf{E}}'( { \widetilde{I} }_{n_1,n_2}(t) ) \Big)
    \operatorname{\mathbf{E}}'({ K }_{n_2}(t))
\\
    \label{eq:moveCov}
    &=
    \operatorname{\mathbf{E}}'( { \widetilde{I} }_{n_1,n_2}(t) )
    \Big( -\operatorname{\mathbf{E}}'({ K }_{n_2}(t)) + 1 \Big) \operatorname{\mathbf{E}}'({ K }_{n_2}(t)).\end{aligned}$$ With $
    \operatorname{\mathbf{E}}'( { \widetilde{I} }_{n_1,n_2}(s) )
    = { {\left( \frac{\nu+\alpha(t)}{1+\alpha(t)} \right)} }^{n_2-n_1}
     q^{g_{n_1}(t)+\ldots+g_{n_2+1}(t)},
$ multiplying both sides of by $ (q-1)^{2}Q_{n_1}(t) Q_{n_2}(t) $, and then applying , we conclude .

We next introduce a centering to $ { R' }(t) $. Let $$\begin{aligned}
\label{eq:vart}
    { \sigma }(t) := (a_{{\operatorname{mod}_J}(t)})^2-(a_{{\operatorname{mod}_J}(t)+1})^2+(a_{{\operatorname{mod}_J}(t)}-a_{{\operatorname{mod}_J}(t)+1})(b+b').\end{aligned}$$

For any $ t\in{ \mathbb{N} }$, we have $
    \operatorname{\mathbf{E}}{ {\left(  { \lambda }(t)\rho^{{ R' }(t)}  \right)} } = 1,
$ so that $$\begin{aligned}
\label{eq:RW}
    \operatorname{\mathbf{P}}{ {\left(  { R }(t) + { \mu }(t) = n  \right)} } := { \lambda }(t) \rho^{n} \operatorname{\mathbf{P}}( { R' }(t)=n ),
    \quad
    n\in{ \mathbb{N} }\end{aligned}$$ defines an $({ \mathbb{N} }-{ \mu }(t))$-valued random variable. Further, $ \operatorname{\mathbf{E}}({ R }(t))=0 $ and $$\begin{aligned}
\label{eq:RWvar}
    \operatorname{\mathbf{E}}({ R }(t)^2) = { r_* }^2 { \sigma }(t).\end{aligned}$$

Fixing $ t\in{ \mathbb{N} }$, we consider the function $$\begin{aligned}
    \notag
    f(x) &:= \operatorname{\mathbf{E}}({ \lambda }(t) x^{{ R' }(t)})
\\
    \notag
    &=
    { \lambda }(t)
    { {\left[ 
        { {\left(  1 - \frac{(1-q)\alpha(t)}{1+\alpha(t)}  \right)} }
        +
        \sum_{i=1}^\infty x^i
        \frac{(1-q)\alpha(t)}{1+\alpha(t)}
        { {\left(  \frac{\nu+\alpha(t)}{1+\alpha(t)}  \right)} }^{i-1}
        \frac{1-\nu}{1+\alpha(t)}
         \right]} }
\\
    \label{eq:RWf}
    &=
    { \lambda }(t)
    \frac{1-\nu x +q\alpha(t) - q\alpha(t) x}{1-\nu x+\alpha(t)-\alpha(t) x}.\end{aligned}$$ With $ { \lambda }(t) $ defined as in , specializing at $ x={ \rho }$ we obtain $ f({ \rho })=1 $, thereby obtain $ \operatorname{\mathbf{E}}({ \lambda }(t){ \rho }^{{ R' }(t)})=1 $. Next, differentiating $ f(x) $ yields $$\begin{aligned}
    &
    \label{eq:RWfp}
    { {\left(  x \frac{df}{dx}  \right)} } ({ \rho }) = \operatorname{\mathbf{E}}{ {\left(  { \lambda }(t) { \rho }^{{ R' }(t)} { R' }(t)  \right)} } = \operatorname{\mathbf{E}}({ R }(t)+{ \mu }(t))
\\
    &
    \label{eq:RWfpp}
    { {\left(  x {\frac{d~}{dx}}{ {\left(  x {\frac{d~}{dx}}f \right)} }  \right)} } ({ \rho }) =  \operatorname{\mathbf{E}}{ {\left(  { \lambda }(t) { \rho }^{{ R' }(t)} { R' }(t)^2  \right)} }
     = \operatorname{\mathbf{E}}({ R }(t)+{ \mu }(t))^2.\end{aligned}$$ Plugging into the l.h.s. of – and specializing at $ x={ \rho }$, after some tedious but straightforward calculations, one obtains $ ( x \frac{df}{dx} ) ({ \rho }) = { \mu }(t) $ and $ ( x {\frac{d~}{dx}}( x {\frac{d~}{dx}}f) ) ({ \rho }) = { \mu }(t)^2 + { r_* }^2{ \sigma }(t) $, thereby concluding $ \operatorname{\mathbf{E}}({ R }(t))=0 $ and .

With $ [{ p' }(t+1,t)*Q(t)]_n = \operatorname{\mathbf{E}}(Q_{n-{ R' }(t)}(t)) $ and $ [{p}(t+1,t)*Z(t)](\xi) = \operatorname{\mathbf{E}}( Z(t,\xi-{ R }(t)) ) $, we have the readily verified identity $$\begin{aligned}
\label{eq:hkp2hk}
    { \widehat{\lambda} }(t+1) { \rho }^{\xi+{ \widehat{\mu} }(t+1)}
    [{ p' }(t+1,t) Q(t) ]_{\xi+{ \widehat{\mu}_\varepsilon }(t+1)}
    =
    [{p}(t+1,t)*Z(t)](\xi),\end{aligned}$$ for all $ \xi\in\Xi(t=1) $. In set $ n=\xi+{ \widehat{\mu} }(t+1) $, and multiply both sides by $ { \widehat{\lambda} }(t+1) { \rho }^{\xi+{ \widehat{\mu} }(t)} $. Using and , we obtain $$\begin{aligned}
\label{eq:dSHE:}
    Z(t+1,\xi)
    = \big[ {p}(t+1,t) * Z(t) \big](\xi)
     + Z(t,\xi+{ \mu }(t)) W(t,\xi+{ \mu_\varepsilon }(t)).\end{aligned}$$ Iterating this equation from $ t=t_2-1 $ until $ t=t_1 $, we thus conclude .

To derive , in , we set $ n_1=\xi_1+{ \widehat{\mu} }(t) $ and $ n_2=\xi_2+{ \widehat{\mu} }(t) $, and multiply both sides by $ { \widehat{\lambda} }(t+1)^2 { \rho }^{\xi_1+{ \widehat{\mu} }(t)} { \rho }^{\xi_2+{ \widehat{\mu} }(t)} $. Using and to express the resulting equation in terms of $ Z(t,{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $ and $ {p}(t+1,t,{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $, we thus conclude .

Moment Estimates: Proof of Propositions \[prop:tight\] and \[prop:stepEst\] {#sect:mom}
===========================================================================

Hereafter we let $ C(u_1,u_2,\ldots) $ denote a generic finite positive constant that depends only on the designated variables $ u_1,u_2,\ldots $ and possibly on $\alpha>0$, $\nu\in [0,1)$ and ${ \rho }\in (0,1)$, which are fixed throughout the paper.

\[lem:Taylor\] The function $ \phi(x,{ \varepsilon };t) := \operatorname{\mathbf{E}}(x^{{ R }_{ \varepsilon }(t)}) $ extends analytically in $ (x,{ \varepsilon }) $ to a neighbor of $ (1,0)\in{ \mathbb{C} }^2 $, with the Taylor expansion $$\begin{aligned}
\label{eq:RWTaylor}
    \phi(x,{ \varepsilon };t) = 1 + 2^{-1} \big( \partial_{xx{ \varepsilon }} \phi(1,0;t) \big) { \varepsilon }(x-1)^2 + O({ \varepsilon }|x-1|^3),\end{aligned}$$ and $ \partial_{xx{ \varepsilon }} \phi(1,0;t) \in (0,\infty) $.

Since $ { R }_{ \varepsilon }(t) $ is defined by $ { R' }_{ \varepsilon }(t) $ as in , we clearly have $$\begin{aligned}
\label{eq:phif}
    \phi(x,{ \varepsilon };t) = \operatorname{\mathbf{E}}( { \lambda }(t) { \rho }^{{ R' }_{ \varepsilon }(t)} x^{{ R' }_{ \varepsilon }(t)-{ \mu_\varepsilon }(t)} ).\end{aligned}$$ By , the function $ \operatorname{\mathbf{E}}( { \lambda }(t) x^{{ R' }_{ \varepsilon }(t)} ) $ function is analytic in $ (x,{ \varepsilon }) $ within a neighborhood of $ (\rho,0) $, whereby $ \phi(x,{ \varepsilon };t) $ is analytic within a neighborhood of $ (1,0) $. To obtain the Taylor expansion , we differentiate in $ x $, and then specialized at $ x=1 $, thereby obtaining $$\begin{aligned}
    &
    \label{eq:f1pf}
    \partial_x \phi(1,{ \varepsilon };t) = \operatorname{\mathbf{E}}{ {\left(  { \lambda_\varepsilon }(t) { \rho }^{{ R' }_{ \varepsilon }(t)} ({ R' }_{ \varepsilon }(t)-{ \mu_\varepsilon }(t))  \right)} }
    =
    \operatorname{\mathbf{E}}({ R }_{ \varepsilon }(t)) =0,
\\
    &
    \label{eq:f1ppf}
    \partial_{xx} \phi(x,{ \varepsilon };t) = \operatorname{\mathbf{E}}{ {\left(  { \lambda_\varepsilon }(t) { \rho }^{{ R' }_{ \varepsilon }(t)} ({ R' }_{ \varepsilon }(t)-{ \mu_\varepsilon }(t))^2  \right)} }
    =
    \operatorname{\mathbf{E}}({ R }_{ \varepsilon }(t))^2 = { r_* }^2 { \sigma_\varepsilon }(t).\end{aligned}$$ With $ { \sigma_\varepsilon }(t) $ defined as in , we have $$\begin{aligned}
\label{eq:vartApprox}
    { \sigma_\varepsilon }(t) = { \varepsilon }\alpha{ \gamma }(1+\alpha{ \gamma })^{-3} { {\left(  2\alpha{ \gamma }+ (b+b')(1+\alpha{ \gamma })  \right)} } + O({ \varepsilon }^2).\end{aligned}$$ From – we conclude that $ \partial^n_{x} \partial^m_{ \varepsilon }\phi(1,0;t)=0 $, unless $ n\geq 2 $ and $ m\geq 1 $, and that $ \partial_{xx{ \varepsilon }}\phi(1,0;t) >0 $, thereby obtaining .

Based on Lemma \[lem:Taylor\], we proceed to estimating of the heat kernel.

\[prop:hk\] Given any $ u,T> 0$ and $ v\in(0,1] $, there exists $ C=C(T,u) $ such that $$\begin{aligned}
    &
    \label{eq:hksum}
    \sum_{\zeta\in\Xi(t_2,t_1)}
    {p_\varepsilon}(t_2,t_1,\zeta) e^{ u{ \varepsilon }|\zeta| }    
    \leq C,
\\
    &
    \label{eq:hksumt}
    \sum_{\zeta\in\Xi(t_2,t_1)}
    |\zeta|^{v}{p_\varepsilon}(t_2,t_1,\zeta) e^{ u{ \varepsilon }|\zeta| } 
    \leq
    C ({ \varepsilon }|t_2-t_1|)^{v/2},
\\
    &
    \label{eq:hkbdd}
    {p_\varepsilon}(t_2,t_1,\xi)
    \leq
    C { \varepsilon }^{-1/2}(t_2-t_1+1)^{-1/2},
\\
    &
    \label{eq:hkx}
    { {\left|  {p_\varepsilon}(t_2,t_1,\xi) - {p_\varepsilon}(t_2,t_1,\xi')  \right|} }
    \leq
    C { \varepsilon }^{-(1+v)/2} |\xi-\xi'|^{v} (t_2-t_1+1)^{-(1+v)/2},\end{aligned}$$ for all $ t_1\leq t_2\in[0,{ \varepsilon }^{-3}T]\cap{ \mathbb{N} }$ and $ \xi,\xi'\in\Xi(t_2) $.

To prove , we consider $ F_1(u') := e^{u'({ R }_{ \varepsilon }(t_1)+\ldots+{ R }_{ \varepsilon }(t_2-1))} $. With $
    \operatorname{\mathbf{E}}(F_1(u')) = \sum_{\zeta\in\Xi(t_2,t_1)} {p_\varepsilon}(t_2,t_1,\zeta) e^{u'\zeta}
$ and $ e^{{ \varepsilon }u|\zeta|} \leq e^{{ \varepsilon }u \zeta} + e^{-{ \varepsilon }u \zeta} $, to show , it suffices to bound the expression $
    \operatorname{\mathbf{E}}(F_1(u'))
    = \prod_{s=t_1}^{t_2-1} \phi(e^{u'},{ \varepsilon };s),
$ for $ u'=\pm u{ \varepsilon }$. This, by , is bounded by $ [1+C{ \varepsilon }(e^{u'}-1)^2]^{t_2-t_1} $. With $ t_2-t_1 \leq { \varepsilon }^{-3}T $, the last expression is bounded by $ C=C(T,u) $, whereby we conclude .

Turning to showing , we let $ F_2 := { R }_{ \varepsilon }(s_1)+\ldots+{ R }_{ \varepsilon }(s_2-1) $. Similar to the preceding, it suffices to bound the expression $$\begin{aligned}
    \sum_{\zeta\in\Xi(s_1,s_2)} |\zeta|^{v} {p_\varepsilon}(s_1,s_2,\zeta) e^{u'\zeta} 
    =
    \operatorname{\mathbf{E}}{ {\left(  F_1(u') |F_2|^{v}   \right)} }
    \leq
    \big\Vert F_1(u') \big\Vert_{2/(2-v)}
    \big\Vert |F_2|^{v} \big\Vert_{2/v},\end{aligned}$$ for $ u'=\pm u{ \varepsilon }$, where we used Hölder’s inequality in the last inequality. With $ (F_1(u'))^{2/(2-v)} = F_1(2u'/(2-v)) $, applying for $ u = 2u/(2-v) $ we obtain $ { \Vert  F_1(\pm u{ \varepsilon })  \Vert }_{2/(2-v)} \leq C $. As for $ F_2 $, with $ \operatorname{\mathbf{E}}({ R }_{ \varepsilon }(s))=0 $ and , we have $$\begin{aligned}
    \big\Vert |F_2|^{v} \big\Vert_{2/v} = { {\left[  \operatorname{\mathbf{E}}(F_2)^2  \right]} }^{v/2}
    &=
    { {\left[  \operatorname{\mathbf{E}}({ R }_{ \varepsilon }(t_1)^2)+\ldots+\operatorname{\mathbf{E}}({ R }_{ \varepsilon }(t_2-1)^2)  \right]} }^{v/2}
\\
    &
    \leq C
    { {\left[  { \sigma }_{ \varepsilon }(t_1) + \ldots + { \sigma }_{ \varepsilon }(t_2-1)  \right]} }^{v/2}.\end{aligned}$$ Further using we thus obtain $
    \Vert |f_2|^{v} \Vert_{2/v} \leq C ({ \varepsilon }|t_2-t_1|)^{v/2},
$ thereby concluding .

Proceeding to showing –, first we apply the inversion formula of the characteristic function, $
    {p_\varepsilon}(t_2,t_1,\zeta)
    =
    \frac{1}{2\pi{\mathbf{i}}} \int_{-\pi}^{\pi}
    e^{-{\mathbf{i}}\zeta r} \prod_{s=t_1}^{t_2-1}\phi(e^{{\mathbf{i}}r},{ \varepsilon };s) dr
$ and the uniform $ v $-Hölder continuity of $ x \mapsto e^{{\mathbf{i}}x} $, $ x\in{ \mathbb{R} }$, to obtain $$\begin{aligned}
    &
    \label{eq:invshk}
    {p_\varepsilon}(t_2,t_1,\xi)
    \leq
    C \int_{-\pi}^{\pi} \prod_{s=t_1}^{t_2-1} { {\left|  \phi(e^{{\mathbf{i}}r},{ \varepsilon };s)  \right|} } dr,
\\
    &
    \label{eq:invshkx}
    { {\left|  {p_\varepsilon}(t_2,t_1,\xi) - {p_\varepsilon}(s_1,s_2,\xi')  \right|} }
    \leq
    C \int_{-\pi}^{\pi} (|\xi-\xi'|r)^{v} \prod_{s=t_1}^{t_2-1} { {\left|  \phi(e^{{\mathbf{i}}r},{ \varepsilon };s)  \right|} } dr.\end{aligned}$$ To further bond these expressions, we apply for $ x = e^{{\mathbf{i}}r} $ to obtain $$\begin{aligned}
\label{eq:chasmall}
    { {\left|  \phi(e^{{\mathbf{i}}r},{ \varepsilon };s)  \right|} }  \leq 1 - { \varepsilon }r^2/C, 
    \quad
    \forall s\in{ \mathbb{N} }, \ |r|\leq r_0,\end{aligned}$$ where $ r_0>0 $ is a constant. As for $ |r|>r_0 $, we let $ f(n,{ \varepsilon };s) := \operatorname{\mathbf{P}}({ R }_{{ \varepsilon }}(s)=n-{ \mu_\varepsilon }(s)) $, whereby $ \phi(e^{{\mathbf{i}}r},{ \varepsilon };s) = \sum_{n\in{ \mathbb{N} }} e^{{\mathbf{i}}r(n-{ \mu_\varepsilon }(s))}f(n,{ \varepsilon };s) $. Expressing the $ n=0 $ term as the sum of $ e^{-{\mathbf{i}}r{ \mu_\varepsilon }(s)}f(1,{ \varepsilon };s)$ and $e^{-{\mathbf{i}}r{ \mu_\varepsilon }(s)}(f(0,{ \varepsilon };s)-f(1,{ \varepsilon };s)) $, and then combining the former with the $ n=1 $ term, we obtain $$\begin{aligned}
    \phi(e^{{\mathbf{i}}r},{ \varepsilon };s)
    &=
    e^{-{\mathbf{i}}r{ \mu_\varepsilon }(s)} (1+e^{{\mathbf{i}}r}) f(1,{ \varepsilon };s)
    +  e^{-{\mathbf{i}}r{ \mu_\varepsilon }(s)}(f(0,{ \varepsilon };s)-f(1,{ \varepsilon };s))
\\
    &
    \quad
    + \sum_{n>1} e^{{\mathbf{i}}r(n-{ \mu_\varepsilon }(s))} f(n,{ \varepsilon };s).\end{aligned}$$ Taking the absolute value of this yields $$\begin{aligned}
\label{eq:cha}
    { {\left|  \phi(e^{{\mathbf{i}}r},{ \varepsilon };s)  \right|} }
    \leq
    |1+e^{{\mathbf{i}}r}| f(1,{ \varepsilon };s)
    +  |f(0,{ \varepsilon };s)-f(1,{ \varepsilon };s)|
    + \sum_{n>1} f(n,{ \varepsilon };s).\end{aligned}$$ By and , we find that $ f(0,{ \varepsilon };s) > f(1,{ \varepsilon };s) > { \varepsilon }/C $. Using this and $ \sum_{n=0}^\infty f(n,{ \varepsilon };s) =1 $ in , we then obtain the bound $$\begin{aligned}
\label{eq:chalarge}
    |\phi(e^{{\mathbf{i}}r},{ \varepsilon };s)|
    \leq
    f(1,{ \varepsilon };s) |1+e^{{\mathbf{i}}r}| + (1-2f(1,{ \varepsilon };s))
    \leq
    1- { \varepsilon }/C,
    \
    \forall s\in{ \mathbb{N} }, \ |r|> r_0.\end{aligned}$$ Now, combining and , we thus obtain $ |\phi(e^{{\mathbf{i}}r},{ \varepsilon };s)| \leq 1 - { \varepsilon }r^2/C $, for all $ r\in{ \mathbb{R} }$ and $ s\in{ \mathbb{N} }$. Plugging this in –, with $ t_2-t_1\leq { \varepsilon }^{-3} T $, we conclude $
    \prod_{s=t_1}^{t_2-1} { {\left|  \phi(e^{{\mathbf{i}}r},{ \varepsilon };s)  \right|} }
    \leq
     e^{-{ \varepsilon }r^2(t_2-t_1)/C},
$ for all $ { \varepsilon }\in(0,1] $ and $ r\in(-\pi,\pi) $. Using this, further integrating over $ r\in(-\pi,\pi) $, we conclude –.

Next, we derive bounds on moments of $ {Z_\text{mg}}(t_2,t_1,\xi) $ (as in ) and $ {Z_{\nabla,\text{mg}}}(t_2,t_1,\xi,\xi') := {Z_\text{mg}}(t_2,t_1,\xi) - {Z_\text{mg}}(t_2,t_1,\xi') $. Hereafter we adapt the shorthand notation $ \xi_{\pm t} := \xi\pm{ \mu }(t) $.

\[lem:Zmg\] For any $ k\geq 1 $ and $ v\in[0,1] $, there exists $ C=C(k) $ such that for all $ t_1\leq t_2\in{ \mathbb{N} }$ and $ \xi\in\Xi(t_1) $, $$\begin{aligned}
    \label{eq:Zmgbd}
    &
    { {\left\Vert  {Z_\text{mg}}(t_2,t_1,\xi)  \right\Vert} }^2_{2k}
    \leq
    C { \varepsilon }^{3/2} \sum_{s=t_{1}}^{t_2-1}
    \big[\overline{{p}}^{ \varepsilon }(s,\zeta) *{ \Vert Z(s) \Vert }^2_{2k}\big](\xi),
\\
    &
    \notag
    { {\left\Vert  {Z_{\nabla,\text{mg}}}(t_2,t_1,\xi,\xi')  \right\Vert} }^2_{2k}
    \leq
    C { \varepsilon }^{(3-v)/2} |\xi-\xi'|^{v}
    \sum_{s=t_{1}}^{t_2-1}
    \big(
        \big[\overline{{p}}^{ \varepsilon }_\nabla(s)*{ \Vert Z(s) \Vert }^2_{2k}\big](\xi)
\\
    \label{eq:Zmgxbd}
    &
    \hphantom{  { {\left\Vert  {Z_{\nabla,\text{mg}}}(t_2,t_1,\xi,\xi')  \right\Vert} }^2_{2k}
        \leq
        C { \varepsilon }^{(3-v)/2} |\xi-\xi'|^{v}
        \sum_{s=t_{1}}^{t_2-1}
        \big(
    }
        +\big[\overline{{p}}^{ \varepsilon }_\nabla(s)*{ \Vert Z(s) \Vert }^2_{2k}\big](\xi')
    \big).\end{aligned}$$ where $
    \overline{{p}}^{ \varepsilon }(s,\zeta)
    :=
    (|t_2-s|+1)^{-1/2}{p_\varepsilon}(t,s,\zeta)
$ and $
    \overline{{p}}^{ \varepsilon }_\nabla(s,\zeta)
    :=
    (|t_2-s|+1)^{-(1+v)/2}
    {p_\varepsilon}(t,s,\zeta).
$

Fix $ t_1\leq t_2\in{ \mathbb{N} }$ and $ \xi,\xi'\in\Xi(t_1) $. To prove , we estimate the corresponding quadratic variation. To this end, letting $ F(s,\zeta) := [ {p_\varepsilon}(t_2,s+1)* { {\left(  Z(s){ W }(s)  \right)} } ] (\zeta ) $, we consider the discrete time martingales $$\begin{aligned}
    &
    M(t)
    := \textstyle \sum_{s=t_{1}}^{t-1} F(s,\xi_{+s}),&
    &
    M_\nabla(t)
    := \textstyle\sum_{s=t_{1}}^{t-1} { {\left(  F(s,\xi_{+s}) - F(s,\xi_{+s}')  \right)} }\end{aligned}$$ with the respectively predictable compensator $ V_M(s) := \operatorname{\mathbf{E}}[F(s,\xi_{+s})^2|{ \mathscr{F} }(s)]  $ and $ V^\nabla_{M}(s) := \operatorname{\mathbf{E}}[(F(s,\xi_{+s}) - F(s,\xi_{+s}'))^2|{ \mathscr{F} }(s)] $. With $ M(t_2)={Z_\text{mg}}(t_2,t_1,\xi) $ and $ M_\nabla(t_2)= {Z_{\nabla,\text{mg}}}(t_2,t_1,\xi,\xi') $, applying Burkholder’s inequality we obtain $$\begin{aligned}
    \label{eq:Zmgbd:}
    { \Vert  {Z_\text{mg}}(t_2,t_1,\xi)  \Vert }_{2k}^2
    \leq
    C
    \sum_{s=t_{1}}^{t_2-1}
    { {\left\Vert  V_M(s)  \right\Vert} }_{2k},
    \   
    { \Vert  {Z_{\nabla,\text{mg}}}(t_2,t_1,\xi,\xi')  \Vert }_{2k}^2
    \leq
    C
    \sum_{s=t_{1}}^{t_2-1}
    { {\left\Vert  V^\nabla_{M}(s)  \right\Vert} }_{2k}.\end{aligned}$$

Having derived the inequality , we now proceed to estimating $ |V_M(s)| $ and $ |V^\nabla_{M}(s)| $. By , we have $$\begin{aligned}
    &
    \label{eq:qvbd:FM}
    V_M(s)
    =
    \sum_{\zeta_1,\zeta_2\in\Xi(s)}
    \widehat{{p}}^{ \varepsilon }(s,\zeta_1,\zeta_2)
    { {\left( \frac{(\nu+\alpha(s)){ \rho }}{1+\alpha(s)} \right)} }^{|\zeta_1-\zeta_2|}
    { \Theta }_1(s,\zeta_1\wedge\zeta_2) { \Theta }_2(s,\zeta_1\wedge\zeta_2),
\\
    &
    \label{eq:qvbd:FMp}
    V^\nabla_{M}(s)
    =
    \sum_{\zeta_1,\zeta_2\in\Xi(s)}
    \widehat{{p}}_\nabla^{ \varepsilon }(s,\zeta_1,\zeta_2)
    { {\left( \frac{(\nu+\alpha(s)){ \rho }}{1+\alpha(s)} \right)} }^{|\zeta_1-\zeta_2|}
    { \Theta }_1(s,\zeta_1\wedge\zeta_2) { \Theta }_2(s,\zeta_1\wedge\zeta_2),\end{aligned}$$ where $
    \widehat{{p}}^{ \varepsilon }(s,\zeta_1,\zeta_2) :=  {p_\varepsilon}(t_2,s,\xi_{+s}-\zeta_1)  {p_\varepsilon}(t_2,s,\xi_{+s}-\zeta_2),
$ and $$\begin{aligned}
    \widehat{{p}}^{ \varepsilon }_\nabla (s,\zeta_1,\zeta_2) :=
    \prod\nolimits_{k=1,2}
    \big[{p_\varepsilon}(t_2,s,\xi_{+s}-\zeta_k)-{p_\varepsilon}(t_2,s,\xi'_{+s}-\zeta_k)\big].\end{aligned}$$ To bound $ V_M(s) $ and $ V^\nabla_{M}(s) $, set $ (\zeta,\zeta') := (\zeta_1\wedge \zeta_2, \zeta_1-\zeta_2) $ in –, whereby $ \sum_{\zeta_1,\zeta_2\in\Xi(s)} = \sum_{\zeta'\in{ \mathbb{Z} }} \sum_{\zeta\in\Xi(s)} $; take absolute value on both sides of –; then, use in to bound $ |\widehat{{p}}^{ \varepsilon }(s,\zeta_1,\zeta_2)| $ by $
    C { \varepsilon }^{-1/2}\overline{{p}}^{ \varepsilon }(s+1,\xi_{+s}-\zeta);
$ and similarly use in to bound $ |\widehat{{p}}^{ \varepsilon }_\nabla(s,\zeta_1,\zeta_2)| $ by $
    C { \varepsilon }^{-(1+v)/2}|\xi-\xi'|^{v}
    [\overline{{p}}^{ \varepsilon }_\nabla(s+1,\xi_{-s}-\zeta) + \overline{{p}}^{ \varepsilon }_\nabla(s+1,\xi_{+s}'-\zeta) ].
$ Upon summing over $ \zeta' $, we obtain $$\begin{aligned}
    &
    \label{eq:qvbd:FMbd}
    { {\left|  V_M(s)  \right|} }    
    \leq
    C { \varepsilon }^{-1/2} \big[ \overline{{p}}^{ \varepsilon }(s+1) * { {\left(  { \Theta }_1(s) { \Theta }_2(s)  \right)} } \big] (\xi_{+s}),
\\
    &
    \notag
    { {\left|  V^\nabla_{M}(s)  \right|} }   
    \leq
    C { \varepsilon }^{-(1+v)/2} |\xi-\xi'|^{v}
    \big(
        [ \overline{{p}}^{ \varepsilon }_\nabla(s+1) * { {\left(  { \Theta }_1(s) { \Theta }_2(s)  \right)} } ] (\xi_{+s})
\\
    &
    \hphantom{{ {\left| f_{M}^\nabla(s) \right|} }\leq C{ \varepsilon }^{-(1+v)/2}|\xi-\xi'|^{v}}
    \label{eq:qvbd:FMpbd}
        +
        [ \overline{{p}}^{ \varepsilon }_\nabla(s+1) * { {\left(  { \Theta }_1(s) { \Theta }_2(s)  \right)} } ] (\xi'_{+s})
    \big).\end{aligned}$$

With –, we now turn to estimating $ | { \Theta }_1(s,\zeta) |$ and $| { \Theta }_2(s,\zeta) | $. To this end, we let $$\begin{aligned}
\label{eq:genps}
    {\widehat{\mathcal{L}}_\varepsilon}(s,\zeta) := {p_\varepsilon}(s+1,s,\zeta) - \operatorname{\mathbf{1}}_{ {\left\{ \zeta=-{ \mu_\varepsilon }(s) \right\}} },
    \quad
    \zeta \in ({ \mathbb{N} }-{ \mu_\varepsilon }(s))\end{aligned}$$ denote a pseudo generator (as the true generator is $ {\mathcal{L}_\varepsilon}(s,\zeta):={p_\varepsilon}(s+1,s,\zeta) - \operatorname{\mathbf{1}}_{ {\left\{ \zeta=0 \right\}} } $), and then rewrite $ { \Theta }_1(s,\zeta) $ and $ { \Theta }_2(s,\zeta) $ (as in –) as $$\begin{aligned}
    &
    \label{eq:qv1:gen}
    { \Theta }_1(s,\zeta)
    = ({ q_\varepsilon }{ \lambda_\varepsilon }(t) -1 ) Z(s,\zeta) - \big[ {\widehat{\mathcal{L}}_\varepsilon}(s) * Z(s) \big] (\zeta_{-s}),
\\
    &
    \label{eq:qv2:gen}
    { \Theta }_2(s,\zeta)
    = ({ \lambda_\varepsilon }(t) -1 ) Z(s,\zeta) + \big[ {\widehat{\mathcal{L}}_\varepsilon}(s) * Z(s) \big] (\zeta_{-s}).   \end{aligned}$$ By using and , with $ (1-{ q_\varepsilon }) \leq C { \varepsilon }$, it is not hard to show that $$\begin{aligned}
\label{eq:genBd}
    |{\widehat{\mathcal{L}}_\varepsilon}(s,\zeta)| \leq C { \varepsilon }{ {\left(  \frac{(\nu+{\alpha_\varepsilon}(s)){ \rho }}{1+{\alpha_\varepsilon}(s)}  \right)} }^{|\zeta|},\end{aligned}$$ for some $ C<\infty $. Further, with $ Z(s,\zeta') $ defined as in , using we have $$\begin{aligned}
    Z(s,\zeta')
    = { \rho }^{\zeta'-\zeta} { q_\varepsilon }^{\sum\text{gaps}} Z(s,\zeta)
    \leq { \rho }^{-|\zeta-\zeta'|} Z(s,\zeta),
    \quad
    \forall \zeta \geq \zeta'.\end{aligned}$$ Using this and in –, we thus obtain $ |{ \Theta }_1(s,\zeta)|, |{ \Theta }_2(s,\zeta)| \leq C{ \varepsilon }Z(s,\zeta) $. Plugging this in –, we now arrive at $$\begin{aligned}
    &
    \label{eq:qvbd:FM:}
    { {\left|  V_M(s)  \right|} }    
    \leq
    C { \varepsilon }^{3/2}
    \big[ \overline{{p}}^{ \varepsilon }(s+1) * Z(s)^2 \big] (\xi_{+s}),
\\
    &
    \notag
    { {\left|  V^\nabla_{M}(s)  \right|} }
    \leq
    C { \varepsilon }^{(3-v)/2}|\xi-\xi'|^{v}
    \big(
        [ \overline{{p}}^{ \varepsilon }_{\nabla}(s+1) * Z(s)^2 ] (\xi_{+s})
\\
    &
    \hphantom{  { {\left|  V^\nabla_{M}(s)  \right|} }
        \leq
        C { \varepsilon }^{(3-v)/2}|\xi-\xi'|^{v}
        \big(
    }
    \label{eq:qvbd:FMp:}
        + [ \overline{{p}}^{ \varepsilon }_{\nabla}(s+1) * Z(s)^2 ] (\xi'_{+s})
    \big).\end{aligned}$$ Further, by , we have $ {p_\varepsilon}(s+1,s,-{ \mu_\varepsilon }(s)) > 1 -C{ \varepsilon }$, whereby $ Z(s,\zeta)^2 \leq C [{p}(s+1,s)* Z(s)^2](\zeta_{-s}) $. Plugging this in –, using the semigroup property $ [{p_\varepsilon}(t,s+1)*{p_\varepsilon}(s+1,s)](\zeta) = {p_\varepsilon}(t,s,\zeta) $, we further obtain $$\begin{aligned}
    { {\left|  V_M(s)  \right|} }    
    &\leq
    C { \varepsilon }^{3/2}
    \big[ \overline{{p}}^{ \varepsilon }(s) * Z(s)^2 ](\xi),
\\
    { {\left|  V^\nabla_{M}(s)  \right|} }
    &\leq
    C { \varepsilon }^{(3-v)/2}|\xi-\xi'|^v
    \big(
        [ \overline{{p}}^{ \varepsilon }_\nabla(s) * Z(s)^2 ] (\xi)
        + [ \overline{{p}}^{ \varepsilon }_\nabla(s) * Z(s)^2 ] (\xi')
    \big).  \end{aligned}$$ Now, taking the $ L^k $-norm of both sides, and then combining the result with , we thus conclude –.

Based on , we establish a chaos-series-typed bounds on the moment of $ Z(t,\xi) $.

For any $ k\geq 1 $, $ t\in{ \mathbb{N} }$ and $ \xi\in\Xi(t) $, $$\begin{aligned}
\label{eq:Zchaos}
\begin{split}
    { {\left\Vert  Z(t,\xi)  \right\Vert} }^2_{2k}
    &\leq
    \big( [{p_\varepsilon}(t,0)*{ \Vert Z(0) \Vert }_{2k}](\xi) \big)
\\
    &
    \sum_{n=0}^{\infty}
    \sum_{{\vec{s}}\in\Delta_{n+1}(t) }
    h_{ \varepsilon }(s_1) \cdots h_{ \varepsilon }(s_{n}) \big( [{p_\varepsilon}(s_{n+1},0)*{ \Vert Z(0) \Vert }_{2k}](\xi) \big),
\end{split}\end{aligned}$$ where $ h_{ \varepsilon }(s) := C{ \varepsilon }^{3/2}(s+1)^{-1/2} $, $ C=C(k)<\infty $, and $ \Delta_n(t) := \{ (s_1,\ldots,s_n)\in ({ \mathbb{Z} }_{\geq 1})^n : s_1+\ldots + s_n =t \} $.

Note that $ \Delta_n(t) = \emptyset $ for all $ n>t $, so the sum over $ n $ in is in fact finite.

In , set $ (t_1,t_2)=(0,t) $, applying the elementary inequality $ |x+y|^2 \leq 2 |x|^2 + 2|y|^2 $ and then taking the $ L^{k} $-norm of both sides, we obtain $
    { {\left\Vert  Z(t,\xi)  \right\Vert} }^2_{2k}
    \leq
    2 { {\left\Vert {Z_\text{dr}}(t,0,\xi) \right\Vert} }^2_{2k}
    +
    2 { {\left\Vert  {Z_\text{mg}}(t,0,\xi)  \right\Vert} }^2_{2k}.
$ For the first term on the r.h.s., by the triangle inequality we clearly have $
    { \Vert  {Z_\text{dr}}(t,0,\xi)  \Vert }_{2k}
    \leq
    [{p_\varepsilon}(t,0)*{ \Vert Z(0) \Vert }_{2k}](\xi),
$ and for second term we apply , thereby obtaining $$\begin{aligned}
\label{eq:Zchaos:iter}
\begin{split}
    { {\left\Vert  Z(t,\xi)  \right\Vert} }^2_{2k}
    &\leq
    2 \big( [{p_\varepsilon}(t,0)*{ \Vert Z(0) \Vert }_{2k}](\xi) \big)^2
\\
    &
    \quad
    +
    C_1 \sum_{s=0}^{t-1} (t-s+1)^{-1/2} \big[ {p_\varepsilon}(t,s)*{ {\left\Vert  Z(s)  \right\Vert} }^2_{2k} \big] (\xi),
\end{split} \end{aligned}$$ for some $ C_1=C_1(k)<\infty $. Let $ h_{ \varepsilon }(s) := 2C_1 (s+1)^{-1/2} $. The bound now follows by iterating , using the semi-group property $ [{p_\varepsilon}(s_3,s_2)*{p}(s_2,s_1)](\zeta) = [{p_\varepsilon}(s_3,s_1)](\zeta) $.

Fix a collection of near equilibrium initial condition, with the corresponding $ u=u(k,v) $ (as in Definition \[def:neareq\]), and fix $ T<\infty $, $ k\geq 1 $ and $ v\in[0,1/2) $. We prove the following moment estimates: $$\begin{aligned}
    &
    \label{eq:mom1}
    { \Vert  Z(t,r)  \Vert }_{2k}
    \leq
    C e^{u{ \varepsilon }|r|},
\\
    &
    \label{eq:momx1}
    { \Vert  Z(\tau,r) - Z(\tau,r')  \Vert }_{2k}
    \leq
    C { {\left(  { \varepsilon }|r-r'|  \right)} }^{v} e^{u{ \varepsilon }(|r|+|r'|)} ,
\\
    &
    \label{eq:momt1}
    { \Vert  Z(\tau,r) - Z(\tau',r)  \Vert }_{2k}
    \leq
    C { {\left(  { \varepsilon }^3|\tau'-\tau|  \right)} }^{v/2} e^{2u{ \varepsilon }|r|},\end{aligned}$$ for some $ C=C(T,k,v)<\infty $ and for all $ \tau_1,\tau_2\in [0,{ \varepsilon }^{-3}T] $, $ r,r'\in{ \mathbb{R} }$ and $ { \varepsilon }>0 $ small enough. These estimates, by the Kolmogorov–Chentsov criterion of tightness [@kallenberg02 Corollary 14.9], immediately imply the tightness of $ \{{Z_\varepsilon}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}},{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}})\} $ in $ C({ \mathbb{R} }_+\times{ \mathbb{R} }) $.

By definition, $ Z(\tau,r) $ is defined on $ { \mathbb{R} }_+\times{ \mathbb{R} }$ by linear interpolation, so without lost of generality we assume $ \tau=t, \tau'=t'\in { \mathbb{N} }\cap[0,{ \varepsilon }^{-3}T] $ and $ r=\xi, r'=\xi'\in\Xi(t) $, and prove – as follows.

By , we have $
    [{p_\varepsilon}(t,0)*{ \Vert Z(0) \Vert }_{2k}](\xi)
    \leq
    \sum_{ \zeta\in\Xi(0) } {p_\varepsilon}(t,0,\xi-\zeta) ( C e^{u{ \varepsilon }|\zeta|} ).
$ In the last expression, using $ e^{u{ \varepsilon }|\zeta|} \leq e^{u{ \varepsilon }|\zeta-\xi|} e^{u{ \varepsilon }|\xi|} $ and then using to sum over $ \zeta $, we obtain $ [{p_\varepsilon}(t,0)*{ \Vert Z(0) \Vert }_{2k}](\xi) \leq C { \varepsilon }^{u{ \varepsilon }|\xi|} $. Now, plugging this in , we arrive at $$\begin{aligned}
\label{eq:mom:}
    { {\left\Vert  Z(t_2,\xi)  \right\Vert} }^2_{2k}
    \leq
    C e^{2u{ \varepsilon }|\xi|}
    \sum_{n=0}^{\infty}
    \sum_{{\vec{s}}\in\Delta_{n+1}(t) }
    h_{ \varepsilon }(s_1) \cdots h_{ \varepsilon }(s_{n}).\end{aligned}$$ Further, for each $ n $, applying the readily verified inequality $$\begin{aligned}
    \sum_{{\vec{s}}\in\Delta_{n+1}(t) } h_{ \varepsilon }(s_1) \cdots h_{ \varepsilon }(s_{n})
    \leq
    \int_{\tau_1+\ldots+\tau_{n} \leq t} \prod_{i=1}^{n} { {\left(  C{ \varepsilon }^{3/2} \tau_i^{-1/2} d\tau_i  \right)} }
    = \frac{[C\Gamma(1/2)({ \varepsilon }^{3}t)^{1/2}]^{n}}{n \Gamma(n/2)},\end{aligned}$$ and then summing over $ n $ in , we thus conclude .

Let $ {Z_{\nabla}}(t,\xi,\xi') := Z(t,\xi) - Z(t,\xi') $ and $ {Z_{\nabla,\text{dr}}}(t,\xi,\xi') := {Z_\text{dr}}(t,0,\xi) - {Z_\text{dr}}(t,0,\xi') $. By , we have $
    {Z_{\nabla}}(t,\xi,\xi') = {Z_{\nabla,\text{dr}}}(t,\xi,\xi') + {Z_{\nabla,\text{mg}}}(t,0,\xi,\xi'),
$ whereby $$\begin{aligned}
\label{eq:momx:tri}
    { \Vert  {Z_{\nabla}}(t,\xi,\xi')  \Vert }^2_{2k}
    \leq
    2 { \Vert {Z_{\nabla,\text{dr}}}(t,\xi,\xi') \Vert }^2_{2k}
    +
    2 { \Vert  {Z_{\nabla,\text{mg}}}(t,0,\xi,\xi')  \Vert }^2_{2k}.\end{aligned}$$ Letting $ 2 ({Z_{\nabla,\text{dr}}}^*)^2 $ and $ 2({Z_{\nabla,\text{mg}}}^*)^2 $ denote the respective terms on the r.h.s., we estimate these terms as follows.

For $ {Z_{\nabla,\text{dr}}}^* $, by the triangle inequality we clearly have $
    {Z_{\nabla,\text{dr}}}^*
    \leq
    \sum_{\zeta\in\Xi(t,0)} {p_\varepsilon}(t,0,\zeta) \Vert Z(0,\xi-\zeta) - Z(0,\xi'-\zeta) \Vert_{2k}.
$ Using in the last expression to replace $ \Vert Z(0,\xi-\zeta) - Z(0,\xi'-\zeta) \Vert_{2k} $ with $
    C ({ \varepsilon }|\xi-\xi'|)^{v} e^{u{ \varepsilon }(|\xi-\zeta|+|\xi'-\zeta|)}
    \leq
    C ({ \varepsilon }|\xi-\xi'|)^{v} e^{u{ \varepsilon }(|\xi|+|\xi'|)} e^{2u|\zeta|},
$ and then using to sum over $ \zeta $, we obtain the desired bound $
    {Z_{\nabla,\text{dr}}}^*
    \leq
     C ({ \varepsilon }|\xi-\xi'|)^{v} e^{u{ \varepsilon }(|\xi|+|\xi'|)}.
$ Turning to $ {Z_{\nabla,\text{mg}}}^* $, first we use to obtain $
    [{p_\varepsilon}(t,s)*{ \Vert Z(s) \Vert }^2_{2k}](\xi'')
    \leq
    \sum_{\zeta\in\Xi(s)} {p_\varepsilon}(t,s,\xi''-\zeta)(C e^{2u{ \varepsilon }|\zeta|} ).
$ Further replacing $ e^{2u{ \varepsilon }|\zeta|} $ with $ e^{2u{ \varepsilon }|\xi''-\zeta|} e^{2u{ \varepsilon }|\xi|} $, and then using to sum over $ \zeta $, we obtain $$\begin{aligned}
\label{eq:hkZZ}
    [{p_\varepsilon}(t,s)*{ \Vert Z(s) \Vert }^2_{2k}](\xi'') \leq C e^{2u{ \varepsilon }|\xi''|}.\end{aligned}$$ Now, specializing at $ (t_1,t_2)=(0,t) $, and combining the result with for $ \xi''=\xi $ and $ \xi''=\xi' $, we obtain $$\begin{aligned}
    \notag
    ({Z_{\nabla,\text{mg}}}^*)^2
    \leq&
    C ({ \varepsilon }|\xi-\xi'|)^{2v} { \varepsilon }^{3/2-3v} \sum_{s=0}^{t-1} (t-s+1)^{-(1+2v)/2}
\\
    &
    \label{eq:momx:}
    \quad
    \sum_{\zeta\in\Xi(s)} { {\left(  {p_\varepsilon}(t,s,\xi-\zeta) + {p_\varepsilon}(t,s,\xi'-\zeta)  \right)} } e^{2u{ \varepsilon }|\zeta|}.\end{aligned}$$ In , use $ e^{2u{ \varepsilon }|\zeta|} \leq (e^{u{ \varepsilon }|\xi-\zeta|}+e^{u{ \varepsilon }|\xi'-\zeta|}) e^{u{ \varepsilon }(|\xi|+|\xi'|)} $, and then use to sum over $ \zeta $. With $ t \leq { \varepsilon }^{-3} T $, further summing over $ s $ we obtain the desired bound $
    ({Z_{\nabla,\text{mg}}}^*)^2
    \leq
    C ({ \varepsilon }|\xi-\xi'|)^{2v} e^{u{ \varepsilon }(|\xi|+|\xi'|)}
$. Combining the preceding estimates of $ {Z_{\nabla,\text{dr}}}^* $ and $ {Z_{\nabla,\text{mg}}}^* $. with , we conclude .

Without lost of generality we assume $ t'\leq t $. By we have $$\begin{aligned}
    Z(t,\xi) - Z(t',\xi)
    =
    { {\left(  {Z_\text{dr}}(t,t',\xi) - Z(t',\xi)  \right)} }
    +
    {Z_\text{mg}}(t,t',\xi).\end{aligned}$$ Similar to the preceding, we bound separately $ {Z_\text{dr}}^{*} := { \Vert  {Z_\text{dr}}(t,t',\xi) - Z(t,\xi)  \Vert }_{2k} $ and $ {Z_\text{mg}}^{*} := { \Vert  {Z_\text{mg}}(t,t',\xi) \Vert }_{2k}  $.

For $ {Z_\text{dr}}^{*} $, with $ \sum_{\zeta\in\Xi(t')} {p_\varepsilon}(t,t',\xi-\zeta) =1 $, we have $$\begin{aligned}
    { {\left(  {Z_\text{dr}}(t,t',\xi) - Z(t',\xi)  \right)} }
    =
    \sum_{\zeta\in\Xi(t')} {p_\varepsilon}(t',t,\xi-\zeta)
    { {\left(  Z(t',\zeta) - Z(t',\xi)  \right)} }.\end{aligned}$$ Take the $ L^{2k} $-norm on both sides, and then use to replace $ \Vert Z(t',\zeta) - Z(t',\xi) \Vert_{2k} $ with $
    C({ \varepsilon }|\xi-\zeta|)^v e^{u{ \varepsilon }(|\xi|+|\zeta|)}
    \leq
    C({ \varepsilon }|\xi-\zeta|)^v e^{u{ \varepsilon }|\xi-\zeta|+2u{ \varepsilon }|\xi|}.
$ Using to sum over $ \zeta $, we then obtain the derided bound $
    {Z_\text{dr}}^*
    \leq
    C ({ \varepsilon }^3|t-t'|)^{v/2} e^{2u{ \varepsilon }|\xi|}.
$

As for $ {Z_\text{mg}}^{*} $, combining and , one obtains $
    ({Z_\text{mg}}^*)^2
    \leq
    C e^{2u{ \varepsilon }|\xi|} 
    \sum_{s=0}^{t-1} { \varepsilon }^{3/2} (t-s+1)^{-1/2}.
$ With $ t\leq { \varepsilon }^{-3}T $, summing over $ s $ we obtain the desired bound $
    ({Z_\text{mg}}^{*})^2
    \leq
    C e^{2u{ \varepsilon }|\xi|} ({ \varepsilon }^3|t-t'|)^{1/2}
    \leq
    C e^{4u{ \varepsilon }|\xi|} ({ \varepsilon }^3|t-t'|)^{v/2}.
$

This completes the proof of Proposition \[prop:tight\]. We now turn to the proof of Proposition \[prop:stepEst\].

As explained at the beginning of the proof of Proposition \[prop:tight\], without lost of generality we let $ \tau=t\in{ \mathbb{N} }\cap (0,{ \varepsilon }^{-3}T] $ and $ r=\xi, r'=\xi'\in\Xi(t) $.

To show , multiply both sides of by $ [{ r_* }{ \varepsilon }^{-1}(1-{ \rho })]^2 $ to obtain $$\begin{aligned}
    \notag
    { \Vert  {\widetilde{Z}}(t,\xi)  \Vert }^2_{2k}
    &\leq
    \big( [{p_\varepsilon}(t,0)*{\widetilde{Z}}(0)](\xi) \big)
\\
    &
    \label{eq:Zdchaos}
    \sum_{n=0}^{\infty}
    \sum_{{\vec{s}}\in\Delta_{n+1}(t) }
    h_{ \varepsilon }(s_1) \cdots h_{ \varepsilon }(s_{n}) \big( [{p_\varepsilon}(0,s_{n+1})*{\widetilde{Z}}(0)](\xi) \big).\end{aligned}$$ Note that here $ {\widetilde{Z}}(0,\xi) $ is deterministic. By and , we have $$\begin{aligned}
\label{eq:Zddrbd}
    [{p_\varepsilon}(t,0)*{\widetilde{Z}}(0)](\xi) \leq C [{ \varepsilon }^3(t+1)]^{-1/2}.\end{aligned}$$ Applying this to the last term in , we arrive at $$\begin{aligned}
    \notag
    { \Vert  {\widetilde{Z}}(t,\xi)  \Vert }^2_{2k}
\leq
    C \big( [{p_\varepsilon}(t,0)*{\widetilde{Z}}(0)](\xi) \big)
    \sum_{n=0}^{\infty}
    \sum_{{\vec{s}}\in\Delta_{n+1}(t) }
    h_{ \varepsilon }(s_1) \cdots h_{ \varepsilon }(s_{n}) [{ \varepsilon }^3(s_{n+1}+1)]^{-1/2}.\end{aligned}$$ Further, for each $ n $ applying the readily verified inequality $$\begin{aligned}
    &
    \sum_{{\vec{s}}\in\Delta_{n+1}(t) }
    h_{ \varepsilon }(s_1) \cdots  h_{ \varepsilon }(s_{n}) [{ \varepsilon }^3(s_{n+1}+1)]^{-1/2}
\\
    &
    \quad
    \leq
    \int_{\tau_1+\ldots+\tau_{n+1}=t} ({ \varepsilon }^{3}\tau_{n+1})^{-1/2} \prod_{i=1}^{n} { {\left(  C{ \varepsilon }^{3/2} \tau^{-1/2}_i d\tau_i  \right)} }
    = \frac{{ {\left( C\Gamma(1/2){ \varepsilon }^{3}t \right)} }^{n+1}}{\Gamma((n+1)/2)},\end{aligned}$$ and summing over $ n $, with $ t\leq { \varepsilon }^{-3}T $, we obtain $$\begin{aligned}
\label{eq:momd:}
    { \Vert  {\widetilde{Z}}(t,\xi)  \Vert }^2_{2k}
\leq
    C \big( [{p_\varepsilon}(t,0)*{\widetilde{Z}}(0)](\xi) \big) ({ \varepsilon }^{-3}t)^{-1/2}.\end{aligned}$$ Using again , we thus conclude .

Turning to showing , we let $ {\widetilde{Z}_{\nabla}}(t,\xi,\xi') $ denote $ {\widetilde{Z}}(t,\xi) - {\widetilde{Z}}(t,\xi') = { r_* }{ \varepsilon }^{-1}(1-{ \rho })(Z(t,\xi)-Z(t,\xi')) $, and similarly for $ {\widetilde{Z}_{\nabla,\text{dr}}}(t,\xi,\xi')  $ and $ {\widetilde{Z}_{\nabla,\text{mg}}}(t,\xi,\xi') $. Multiplying both sides of by $ ({ r_* }{ \varepsilon }^{-1}(1-{ \rho }))^2 $, we obtain $$\begin{aligned}
    &
    \label{eq:momxd:tri}
    { \Vert  {\widetilde{Z}_{\nabla}}(t,\xi,\xi')  \Vert }^2_{2k}
    \leq
    2{\widetilde{Z}_{\nabla,\text{dr}}}(t,\xi,\xi')^2 + 2{ \Vert  {\widetilde{Z}_{\nabla,\text{mg}}}(t,\xi,\xi')  \Vert }^2_{2k}.\end{aligned}$$ Let $ 2({\widetilde{Z}_{\nabla,\text{dr}}}^*)^2 $ and $ 2({\widetilde{Z}_{\nabla,\text{mg}}}^*)^2 $ denote the respective terms on the r.h.s.. We estimate them as follows.

For $ {\widetilde{Z}_{\nabla,\text{dr}}}^* $, combining for $ (s_1,s_2)=(0,t) $ and we obtain the desired bound $
    {\widetilde{Z}_{\nabla,\text{dr}}}^* \leq ({ \varepsilon }|\xi-\xi'|)^v ({ \varepsilon }^3 t)^{-(v+1)/2}.
$ As for $ {\widetilde{Z}_{\nabla,\text{mg}}}^* $, multiplying both sides of by $ ({ r_* }{ \varepsilon }^{-1}(1-{ \rho }))^2 $, we obtain $$\begin{aligned}
    &
    \notag
    ({\widetilde{Z}_{\nabla,\text{mg}}}^*)^2
    \leq
    C { \varepsilon }^{(3-6v)/2} ({ \varepsilon }|\xi-\xi'|)^{2v}
\\
    &
    \label{eq:momxd:mg}
    \quad
    \sum_{s=0}^{t-1} (t-s+1)^{-(2v+1)/2}
    { {\left( 
        \big[{p_\varepsilon}(t,s)*{ \Vert {\widetilde{Z}}(s) \Vert }^2_{2k}\big](\xi)
        +\big[{p_\varepsilon}(t,s)*{ \Vert {\widetilde{Z}}(s) \Vert }^2_{2k}\big](\xi')
         \right)} }.\end{aligned}$$ Plugging in , and summing over $ s $, with $ t\leq { \varepsilon }^{-3}T $, we obtain $$\begin{aligned}
    \notag
    ({\widetilde{Z}_{\nabla,\text{mg}}}^*)^2
    \leq
    C ({ \varepsilon }|\xi-\xi'|)^{2v}
    \big( [{p_\varepsilon}(t,0)*{\widetilde{Z}}(0)](\xi) + [{p_\varepsilon}(t,0)*{\widetilde{Z}}(0)](\xi') \big).\end{aligned}$$ In the last expression, further using , we obtain the desired bound $
    {\widetilde{Z}_{\nabla,\text{mg}}}^*
    \leq
    C ({ \varepsilon }|\xi-\xi'|)^{2v} ({ \varepsilon }^3 t)^{-1/2}
    \leq
    C ({ \varepsilon }|\xi-\xi'|)^{2v} ({ \varepsilon }^3 t)^{-(v+1)}.    
$ Combining the preceding bounds on $ {\widetilde{Z}_{\nabla,\text{dr}}}^* $ and $ {\widetilde{Z}_{\nabla,\text{dr}}}^* $ we conclude .

The Martingale Problem: Proof of Proposition \[prop:unique\] {#sect:mg}
============================================================

Hereafter we use $ { \mathcal{B}_{ \varepsilon }}(s,{\vec{\zeta}}) $ and $ { \mathcal{E}_{ \varepsilon }}(s,{\vec{\zeta}}) $, $ {\vec{\zeta}}=(\zeta_1,\ldots,\zeta_n) $, to denote respectively *generic* processes that are uniformly bounded and uniformly vanishing, i.e.$$\begin{aligned}
    &
    \sup{ {\left\{ 
        { \Vert { \mathcal{B}_{ \varepsilon }}(s,{\vec{\zeta}}) \Vert }_k:
        \zeta\in({ \varepsilon }^{-1}U)^n, s\leq T{ \varepsilon }^{-3}, { \varepsilon }\in(0,1]
         \right\}} }
     <\infty,
\\
    &
    \sup{ {\left\{ 
        { \Vert { \mathcal{E}_{ \varepsilon }}(s,{\vec{\zeta}}) \Vert }_k :
        {\vec{\zeta}}\in({ \varepsilon }^{-1}U)^n, s\leq T{ \varepsilon }^{-3}
         \right\}} }
    \longrightarrow 0,
    \quad
    \text{ as } { \varepsilon }\to 0,\end{aligned}$$ for any compact $ U\subset { \mathbb{R} }$, $ k\geq 1 $ and $ T>0 $.

We begin by deriving an approximate expression for the cross variance as in .

\[lem:qvApprox\] For near equilibrium initial conditions, we have $$\begin{aligned}
\label{eq:qvApprox}
\begin{split}
    &
    Z(s,\zeta_1) Z(s,\zeta_2) \operatorname{\mathbf{E}}{ {\left(  { W }(s,\zeta_1) { W }(s,\zeta_2) \middle| { \mathscr{F} }(s)  \right)} }
\\
    &
    \quad
    =
    { \varepsilon }^{2} \frac{\alpha{ \gamma }}{(1+\alpha{ \gamma })^2}
    { {\left(  \frac{(\nu+\alpha(s)){ \rho }}{1+\alpha(s)}  \right)} }^{|\zeta_1-\zeta_2|}
    { {\left(  Z(s,\zeta_1\wedge\zeta_2)^2 + { \mathcal{E}_{ \varepsilon }}(s,\zeta_1,\zeta_2) \right)} }.
\end{split}\end{aligned}$$

We prove by approximating the identities –, using the moment estimate . By we have $ ({ q_\varepsilon }{ \lambda_\varepsilon }(t)+1) = -{ \varepsilon }(1+\alpha{ \gamma })^{-1} + O({ \varepsilon }^2) $, so that $ ({ q_\varepsilon }{ \lambda_\varepsilon }(t)+1)Z(s,\zeta) = -{ \varepsilon }(1+\alpha{ \gamma })^{-1}Z(s,\zeta) + { \varepsilon }{ \mathcal{E}_{ \varepsilon }}(s,\zeta) $, and by , fixing arbitrary $ v\in(0,1/2) $, we have $ Z(s,\zeta') = Z(s,\zeta) + |\zeta'|^{v} { \mathcal{B}_{ \varepsilon }}(s,\zeta,\zeta') = Z(s,\zeta) + { \mathcal{E}_{ \varepsilon }}(s,\zeta,\zeta') $. In , using these approximations we obtain $$\begin{aligned}
    &
    \notag
    { \Theta }_1(s,\zeta)
    =
    -{ \varepsilon }(1+\alpha{ \gamma })^{-1} Z(s,\zeta) + { \varepsilon }{ \mathcal{E}_{ \varepsilon }}(t,\zeta)
\\
    &
    \label{eq:qv1Approx:}
    \quad
    - \Big( \sum_{\zeta'\in\Xi(s)} {\widehat{\mathcal{L}}_\varepsilon}(s,\zeta_{-s}-\zeta') \Big) Z(s,\zeta)
    + \sum_{\zeta\in\Xi(s)} {\widehat{\mathcal{L}}_\varepsilon}(s,\zeta_{-s}-\zeta') |\zeta-\zeta'|^{v} { \mathcal{E}_{ \varepsilon }}(s,\zeta,\zeta').\end{aligned}$$ With $ {\widehat{\mathcal{L}}_\varepsilon}(s,\zeta_{-s}-\zeta') $ as in , the first term in is zero since $ {\mathcal{L}_\varepsilon}(s,\zeta) $, and the second term is of the form $ { \varepsilon }{ \mathcal{E}_{ \varepsilon }}(s,\zeta) $ by . Consequently, $$\begin{aligned}
\label{eq:qv1Approx}
    { \Theta }_1(s,\zeta)
    =
    -{ \varepsilon }(1+\alpha{ \gamma })^{-1} Z(s,\zeta) + { \varepsilon }{ \mathcal{E}_{ \varepsilon }}(t,\zeta).\end{aligned}$$ Similarly, for $ { \Theta }_2(s,\xi_2) $ we have $$\begin{aligned}
\label{eq:qv2Approx}
    { \Theta }_2(s,\zeta)
    = -{ \varepsilon }\alpha{ \gamma }(1+\alpha{ \gamma })^{-1} Z(s,\zeta) + { \varepsilon }{ \mathcal{E}_{ \varepsilon }}(s,\zeta).\end{aligned}$$ Combining – with yields .

We proceed to proving Proposition \[prop:unique\]. Recall from [@bertini97] the following martingale problem of the .

\[defn:MGprob\] Let ${\mathcal{Z}}$ be a $C([0,\infty),C({ \mathbb{R} }))$-valued process such that given any $T>0$, there exists $u<\infty$ such that $$\label{eq:mgprob:mom}
    \sup_{\tau\in[0,T]}
    \sup_{r\in { \mathbb{R} }} e^{-u|r|} \operatorname{\mathbf{E}}{ {\left( {\mathcal{Z}}(\tau,r)^2 \right)} }
    <
    \infty.$$ For such $ {\mathcal{Z}}$ and for $ \psi\in C^\infty_c({ \mathbb{R} }) $, let $ { \langle {\mathcal{Z}}(\tau),\psi \rangle } := \int_{{ \mathbb{R} }} {\mathcal{Z}}(\tau,r) \psi(r) dr $. We say $ {\mathcal{Z}}$ solves the martingale problem with initial condition $ {\mathcal{Z}^{\text{ic}}}\in C({ \mathbb{R} }) $ if $ {\mathcal{Z}}(0,{{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}})={\mathcal{Z}^{\text{ic}}}({{\raisebox{-0.5ex}{\scalebox{1.8}{$\cdot$}}}}) $ in distribution, and $$\begin{aligned}
    &
    \tau
    \longmapsto
    N_\psi(\tau)
    :=
    { {\left <  {\mathcal{Z}}(\tau),\psi  \right >} }
    - { {\left <  {\mathcal{Z}}(0),\psi  \right >} }
    - \int_0^\tau 2^{-1} { {\left <  {\mathcal{Z}}(\tau'),\frac{d^2\psi}{dx^2}  \right >} } d\tau'.
\\
    &
    \tau
    \longmapsto 
    {\widehat{N}}_{\psi}(\tau)
    :=
     (N_\psi(\tau))^2
     - \int_0^\tau { {\left <  {\mathcal{Z}}(\tau')^2, \psi^2  \right >} } d\tau'\end{aligned}$$ are local martingale, for any $\psi\in C^\infty_c({ \mathbb{R} })$.

Recall from [@bertini97 Proposition 4.11] that for any initial condition $ {\mathcal{Z}^{\text{ic}}}$ satisfying $$\begin{aligned}
\label{eq:mgic}
    { {\left\Vert  {\mathcal{Z}^{\text{ic}}}(r)  \right\Vert} }_{2} \leq C e^{a|r|}, \quad\text{for some }a >0,\end{aligned}$$ the martingale problem of Definition \[defn:MGprob\] has a unique solution, which coincides with the law of the solution of the with initial condition $ {\mathcal{Z}^{\text{ic}}}$. By passing to the relative subsequence, we assume that $ {Z_\varepsilon}\Rightarrow {\mathcal{Z}}$, which, by , satisfies the moment condition . It hence suffices to show that $ {\mathcal{Z}}$ solves the martingale problem in Definition \[defn:MGprob\].

We begin by deriving the discrete analog of $ N_\psi(\tau) $ and $ {\widehat{N}}_\psi(\tau) $. To this end, fixing $ \psi\in C^\infty_c({ \mathbb{R} }) $, we consider the discrete approximation $$\begin{aligned}
\label{eq:Zpsie}
    { {\left <  Z(t), \psi  \right >} }_{ \varepsilon }:= { \varepsilon }{ r_* }^{-1} \sum_{\xi\in\Xi(t)} Z(t,\xi) \psi( { \varepsilon }{ r_* }^{-1}\xi )\end{aligned}$$ of $ { \langle  {Z_\varepsilon}( { \varepsilon }^{3}({ \tau^{\varepsilon}_* }J)^{-1}t ), \psi  \rangle } $, and similarly define $$\begin{aligned}
\label{eq:mgpsi}
    M_\psi(t) :=
    { \varepsilon }{ r_* }^{-1} \sum_{\xi\in\Xi(t+1)} Z(t,\xi_{+t})
    { W }(t,\xi_{+t}) \psi( { \varepsilon }{ r_* }^{-1}\xi ).\end{aligned}$$ In , multiply both sides by $ { \varepsilon }{ r_* }^{-1}\psi( { \varepsilon }{ r_* }^{-1}\xi ) $. Upon summing over $ \xi \in \Xi(t+1) $, we obtain $
    { \langle  Z(t+1), \psi  \rangle }_{ \varepsilon }=
    { \langle  Z(t), \psi_{{p_\varepsilon}(t+1,t)}  \rangle }_{ \varepsilon }+ M_\psi(t),
$ where $
    \psi_{{p_\varepsilon}(t+1,t)}(\zeta) := \sum_{\xi\in\Xi(t+1)} {p_\varepsilon}(t+1,t,\xi-\zeta) \psi(\xi).
$ Subtracting $ { \langle Z(t),\psi \rangle }_{ \varepsilon }$ from both sides, we further obtain $$\begin{aligned}
\label{eq:dmgprob}
    { {\left <  Z(t+1), \psi  \right >} }_{ \varepsilon }- { {\left <  Z(t), \psi  \right >} }_{ \varepsilon }=
    { \langle  Z(t), \psi_{{\mathcal{L}_\varepsilon}(t)}  \rangle }_{ \varepsilon }+ M_\psi(t,\xi),\end{aligned}$$ where $$\begin{aligned}
\label{eq:psigen}
    \psi_{{\mathcal{L}_\varepsilon}(t)}(\zeta) := \sum_{\xi\in\Xi(t+1)} {p_\varepsilon}(t+1,t,\xi-\zeta) \psi(\xi) - \psi(\zeta).\end{aligned}$$

Now, summing over $ s=0,\ldots,t-1 $, we arrive at $$\begin{aligned}
\label{eq:mgprob:d1}
    { {\left <  Z(t), \psi  \right >} }_{ \varepsilon }- { {\left <  Z(0), \psi  \right >} }_{ \varepsilon }-
    \sum_{s=0}^{t-1} { {\left <  Z(s), \psi_{{\mathcal{L}_\varepsilon}(t)}  \right >} }_{ \varepsilon }=
    \sum_{s=0}^{t-1} { M }_\psi(s)
    := {N^\varepsilon}_\psi(t).\end{aligned}$$ The process $ t\mapsto {N^\varepsilon}_\psi(t) $ is a discrete time martingale of quadratic variation\
$ \sum_{s=0}^{t-1} \operatorname{\mathbf{E}}[  { M }_\psi(s)^2 | { \mathscr{F} }(s) ] $, so $$\begin{aligned}
\label{eq:mgprob:d2}
    {\widehat{N}^\varepsilon}_\psi(t)
    :=
    { {\left(  {N^\varepsilon}_\psi(t)  \right)} }^2
    - \sum_{s=0}^{t-1} \operatorname{\mathbf{E}}{ {\left[   { M }_\psi(s)^2 \middle| { \mathscr{F} }(s)  \right]} }\end{aligned}$$ is also a discrete time martingale.

We proceed to showing that $ N_\psi(\tau) $ and $ {\widehat{N}}_\psi(\tau) $ are local martingales. Since, by , passing from discrete time to continuous time introduces only a negligible error, it suffices to show that terms in – converge in distribution to the corresponding terms. More precisely, recalling $ {t_\varepsilon}(\tau) :={ \varepsilon }^{-3}{ \tau^{\varepsilon}_* }J \tau $, our goal is to show $$\begin{aligned}
    { {\left < Z({t_\varepsilon}(\tau)),\psi \right >} }_{ \varepsilon }&\Longrightarrow
    { {\left <  {\mathcal{Z}}(\tau), \psi  \right >} },
\\
    \sum_{s<{t_\varepsilon}(\tau)} { {\left <  Z(s), \psi_{{\mathcal{L}_\varepsilon}}  \right >} }_{ \varepsilon }&\Longrightarrow
    \int_0^\tau 2^{-1} { {\left <  {\mathcal{Z}}(\tau'), \frac{d^2\psi}{dx^2}  \right >} } d\tau',
\\
    \sum_{s<{t_\varepsilon}(\tau)} \operatorname{\mathbf{E}}{ {\left(   { M }_\psi(s)^2 \middle| { \mathscr{F} }(s)  \right)} }
    &\Longrightarrow    
    \int_0^\tau { {\left <  {\mathcal{Z}}^2(\tau'), \psi(\tau')^2  \right >} } d\tau'.\end{aligned}$$ To this end, since $ {Z_\varepsilon}\Rightarrow {\mathcal{Z}}$, it clearly suffices to show $$\begin{aligned}
    \label{eq:cnvg:Zt}
    &
    \operatorname{\mathbf{E}}\Big|
        { {\left < Z({t_\varepsilon}(\tau)),\psi \right >} }_{ \varepsilon }- { {\left <  {Z_\varepsilon}(\tau), \psi  \right >} }
    \Big|
    \longrightarrow 0,
\\
    \label{eq:cnvg:drift}
    &
    \operatorname{\mathbf{E}}\Big|
        \sum_{s<{t_\varepsilon}(\tau)} { {\left <  Z(s), \psi_{{\mathcal{L}_\varepsilon}}  \right >} }_{ \varepsilon }-
        \int_0^\tau { {\left <  {Z_\varepsilon}(\tau'), \frac{d^2\psi}{dx^2}  \right >} } d\tau'
    \Big|
    \longrightarrow 0,
\\
    \label{eq:cnvg:mg}
    &
    \operatorname{\mathbf{E}}\Big|
        \sum_{s<{t_\varepsilon}(\tau)} \operatorname{\mathbf{E}}{ {\left(   { M }_\psi(s)^2 \middle| { \mathscr{F} }(s)  \right)} }
        -
        \int_0^\tau { {\left <  {Z_\varepsilon}^2(\tau'), \psi^2  \right >} } d\tau'
        \Big|
    \longrightarrow
    0.\end{aligned}$$ We now prove – as follows.

This amounts to show that $$\begin{aligned}
\label{eq:Zpsi}
    { {\left <  {Z_\varepsilon}(\tau), \psi  \right >} }
    =
    { \varepsilon }{ r_* }^{-1} \int_{{ \mathbb{R} }} Z({t_\varepsilon}(\tau),r) \psi({ \varepsilon }{ r_* }^{-1}r) dr\end{aligned}$$ and $ { {\left <  Z({t_\varepsilon}(\tau)), \psi  \right >} }_{ \varepsilon }$ are approximately equal. To this end, fixing arbitrary $ \zeta\in\Xi(t) $ and $ |r-\zeta|\leq 1 $, we use the smoothness of $ \psi $ and the moment estimates –, for arbitrary $ v\in(0,1/2) $, to obtain $
     Z(t,\zeta) \psi({ \varepsilon }{ r_* }^{-1}\zeta)
    - Z(t,r) \psi({ \varepsilon }{ r_* }^{-1} r)
    = { \varepsilon }^v{ \mathcal{B}_{ \varepsilon }}(\zeta,r)
    = { \mathcal{E}_{ \varepsilon }}(t,\zeta,r).
$ From this, with $ { \langle  {Z_\varepsilon}(\tau), \psi  \rangle } $ and $ { \langle  Z({t_\varepsilon}(\tau)), \psi  \rangle }_{ \varepsilon }$ as in and , we conclude $
    { \langle  {Z_\varepsilon}(\tau), \psi  \rangle }
    =
    { \langle  Z({t_\varepsilon}(\tau)), \psi  \rangle }_{ \varepsilon }+ { \mathcal{E}_{ \varepsilon }}(t),
$ whereby follows.

Taylor expanding $ \psi(\xi{ \varepsilon }{ r_* }^{-1}) $ around $ \xi=\zeta $ yields $$\begin{aligned}
    \psi({ \varepsilon }{ r_* }^{-1}\xi)
    =&
    \psi({ \varepsilon }{ r_* }^{-1}\zeta)
    + { {\left(  \frac{d\psi}{dx}({ \varepsilon }{ r_* }^{-1}\zeta) \right)} } { \varepsilon }{ r_* }^{-1}(\zeta-\xi)
\\
    &
    + { {\left(  2^{-1} { {\left(  \frac{d^2\psi}{dx^2}({ \varepsilon }{ r_* }^{-1}\zeta) \right)} } + { \mathcal{E}_{ \varepsilon }}(\xi,\zeta)  \right)} }
    { \varepsilon }^2{ r_* }^{-2} (\zeta-\xi)^2.\end{aligned}$$ Plug this in . With $
    \sum_{\xi\in\Xi(s+1)} {p_\varepsilon}(s+1,s,\xi-\zeta)(\xi-\zeta)^k = \operatorname{\mathbf{E}}({ R }_{ \varepsilon }(s))^k,
$ using $ \operatorname{\mathbf{E}}({ R }(s))=0 $, and $ { \sigma_\varepsilon }\leq C{ \varepsilon }$, we obtain $$\begin{aligned}
    \psi_{{\mathcal{L}_\varepsilon}(s)} (\zeta)
    =
    2^{-1} { {\left(  \frac{d^2\psi}{dx^2}({ \varepsilon }{ r_* }^{-1}\zeta) \right)} } { \varepsilon }^{2} { \sigma_\varepsilon }(s)
    +
    { \varepsilon }^3 { \mathcal{E}_{ \varepsilon }}(s,\zeta).  \end{aligned}$$ Now, plugging this expression of $ \psi_{{\mathcal{L}_\varepsilon}(s)} (\zeta) $ in the l.h.s. of , with $ {t_\varepsilon}(\tau) \leq { \varepsilon }^{-3} C $, we obtain $$\begin{aligned}
    \sum_{s<{t_\varepsilon}(\tau)} { {\left <  Z(s), \psi_{{\mathcal{L}_\varepsilon}(s)}  \right >} }_{ \varepsilon }=
    \sum_{s<{t_\varepsilon}(\tau)} { \varepsilon }^{2} { \sigma_\varepsilon }(s) { {\left <  Z(s), 2^{-1}\frac{d^2\psi}{dx^2}  \right >} }_{ \varepsilon }+ { \mathcal{E}_{ \varepsilon }}.\end{aligned}$$ Next, divide the sum on the r.h.s.into sums over the disjoint intervals $ T_t := { \mathbb{Z} }\cap[tJ,tJ+J) $. We neglect the boundary terms of $ T_{\tau{ \varepsilon }^{-3}/{ \tau^{\varepsilon}_* }} \cap [0,{t_\varepsilon}(\tau)) $, since, by , those terms contribute only $ { \varepsilon }^{2}{ \sigma_\varepsilon }(s) { \mathcal{B}_{ \varepsilon }}= { \mathcal{E}_{ \varepsilon }}$. Within each interval $ T_t $, use to replace $ { \langle  Z(s), \frac{d^2\psi}{dx^2}  \rangle }_{ \varepsilon }$ with $ { \langle  Z(t J), \frac{d^2\psi}{dx^2}  \rangle }_{ \varepsilon }+ { \mathcal{E}_{ \varepsilon }}(s) $. Further, with $ { \sigma_\varepsilon }(s) $ and $ { \tau^{\varepsilon}_* }$ as in and , we have $ \sum_{s\in T_t} { \sigma_\varepsilon }(s) = { \varepsilon }({ \tau^{\varepsilon}_* })^{-1} $. Consequently, $$\begin{aligned}
\label{eq:cnvgdr:}
    \sum_{s<{t_\varepsilon}(\tau)} { {\left <  Z(s), {\mathcal{L}_\varepsilon}\psi  \right >} }_{ \varepsilon }=
    \frac{{ \varepsilon }^{3}}{{ \tau^{\varepsilon}_* }} \sum_{t < { \varepsilon }^{-3}{ \tau^{\varepsilon}_* }\tau} { {\left <  Z(tJ), 2^{-1}\frac{d^2\psi}{dx^2}  \right >} }_{ \varepsilon }+ { \mathcal{E}_{ \varepsilon }}.\end{aligned}$$

The r.h.s. of represents a discrete approximation of $ \int_0^\tau { \langle  {Z_\varepsilon}(\tau'),2^{-1}\frac{d^2\psi}{dx^2}  \rangle } d \tau' $. In particular, by following the same procedure as in the proof of , one obtains\
$
    \frac{{ \varepsilon }^{3}}{{ \tau^{\varepsilon}_* }} \sum_{t < {t_\varepsilon}(\tau)}
    { \langle  Z(tJ), 2^{-1}\frac{d^2\psi}{dx^2}  \rangle }_{ \varepsilon }=
    \int_0^{\tau}
    { \langle Z(\tau'),2^{-1}\frac{d^2\psi}{dx^2} \rangle } d \tau'
    +
    { \mathcal{E}_{ \varepsilon }},
$ thereby concluding .

To calculate $  \operatorname{\mathbf{E}}[ M_\psi(s)^2 | { \mathscr{F} }(s) ] $, we plug in the expression , and then use the approximation for the cross variance, thereby obtaining $$\begin{aligned}
\label{eq:cnvgmg:id}
    &
    \operatorname{\mathbf{E}}{ {\left(  M_\psi(s)^2 \middle| { \mathscr{F} }(s)  \right)} }
    =
    \frac{{ \varepsilon }^4\alpha{ \gamma }}{(1+\alpha{ \gamma })^2{ r_* }^2}
    \sum_{\xi\in\Xi(s+1)} Z(s,\xi_{+s})^2 \psi({ \varepsilon }{ r_* }^{-1} \xi)
    F(s,\xi)
    +
    { \varepsilon }^{3} { \mathcal{E}_{ \varepsilon }}(s),\end{aligned}$$ where $$\begin{aligned}
\label{eq:cnvgmg:F}
    F(s,\xi) :=
    \sum_{n\in{ \mathbb{Z} }}
    \psi({ \varepsilon }{ r_* }^{-1}(\xi+|n|))
    { {\left(  \frac{(\nu+{\alpha_\varepsilon}(s)){ \rho }}{1+{\alpha_\varepsilon}(s)}  \right)} }^{|n|}.\end{aligned}$$ Let $ \eta_{ \varepsilon }:= \sum_{n\in{ \mathbb{Z} }} ( \frac{(\nu+{\alpha_\varepsilon}(s)){ \rho }}{1+{\alpha_\varepsilon}(s)} )^{|n|}  $. In , using the continuity of $ \psi $ at $ { \varepsilon }{ r_* }^{-1}\xi $, we further obtain $   
    F(s,\xi) = \eta_{ \varepsilon }\psi({ \varepsilon }{ r_* }^{-1}\xi) + { \mathcal{E}_{ \varepsilon }}(s,\xi).
$ Plugging this expression in , we arrive at $$\begin{aligned}
\label{eq:cnvgmg:s}
    &
    \operatorname{\mathbf{E}}{ {\left(  { M }_\psi(s)^2 \middle| { \mathscr{F} }(s)  \right)} }
    =
    { \varepsilon }^3 \alpha{ \gamma }\eta_{ \varepsilon }((1+\alpha{ \gamma })^2{ r_* })^{-1}  { {\left < Z^2(s),\psi^2 \right >} }_{ \varepsilon }+
    { \varepsilon }^{3} { \mathcal{E}_{ \varepsilon }}(s).\end{aligned}$$ Calculating $ \eta_{ \varepsilon }$ to the first order we have $ \eta_{ \varepsilon }= \frac{1+\alpha+\nu{ \rho }+\alpha{ \rho }}{1+\alpha-\nu{ \rho }-\alpha{ \rho }} + O({ \varepsilon }) $. Using this and , a tedious but straightforward calculation shows that $ \alpha{ \gamma }\eta_{ \varepsilon }((1+\alpha{ \gamma })^2{ r_* })^{-1} = (J{ \tau^{\varepsilon}_* })^{-1} + O({ \varepsilon }) $. Consequently, summing over $ s<{t_\varepsilon}(\tau) $ yields $$\begin{aligned}
    &
    \sum_{s<{t_\varepsilon}(\tau)} \operatorname{\mathbf{E}}{ {\left(  { M }_\psi(s)^2 \middle| { \mathscr{F} }(s)  \right)} }
    =
    \frac{{ \varepsilon }^{3}}{{ \tau^{\varepsilon}_* }J} \sum_{s< {t_\varepsilon}(\tau)} { {\left < Z^2(s),\psi^2 \right >} }_{ \varepsilon }+
    { \mathcal{E}_{ \varepsilon }}(s).\end{aligned}$$ The r.h.s. represents a discrete approximation of $ \int_0^\tau { \langle  {Z_\varepsilon}(\sigma)^2,\psi^2  \rangle } d \sigma $, so following the same procedure as in the proof of , one concludes .
