Introduction

Background

In this paper, we extend the main results from [4] to reaction–diffusion systems evolving on infinite sets. As in [4], the class of stochastic differential equations we consider here is:

$$\begin{aligned} {\left\{ \begin{array}{ll}d\zeta _t(x) = \left( \Delta _p \zeta _t(x) - b \cdot (\zeta _t(x))^\kappa \right) dt + \sqrt{a \cdot (\zeta _t(x))^\ell }\;dB^x_t,\quad x \in {\mathbb V},\\ [.2cm] \zeta _0 = \bar{\zeta } \in [0,\infty )^{\mathbb V}, \end{array}\right. } \end{aligned}$$
(1.1)

where \(a,b,\kappa ,\ell \) are positive real numbers with \(\kappa ,\ell \ge 1\)\({\mathbb V}\) is a discrete set, \(\Delta _p\) is the Laplacian induced by a probability kernel p on \({\mathbb V}\), that is,

$$\begin{aligned} \Delta _p\zeta (x):= \sum _{y \in {\mathbb V}}\big (p(y,x)\cdot \zeta (y) - p(x,y)\cdot \zeta (x)\big ), \end{aligned}$$
(1.2)

and \(\{B^x_\cdot \}_{x \in {\mathbb V}}\) is a family of independent standard Brownian motions on \(\mathbb {R}\).

This system of equations can be used to model a reaction–diffusion system associated for instance with chemical reactions or population dynamics. In the setting of chemical reactions, space is divided into cells, corresponding to points of \({\mathbb V}\), and each cell contains a certain density of particles. Within each cell, particles are subject to a reaction that can lead to a change in their density. As an image, consider the evolution of the density of ozone subject to the reaction ozone \(\leftrightharpoons \) oxygen in a confined region. This modeling framework is inspired by auto-catalytic models as presented in Nicolis and Prigogine [12, Chap. 7] and resembles the modeling adopted by Blount [3], the main difference being that we keep the size of reaction cells constant. Note that the system in (1.1) has \(\zeta \equiv 0\) as a stable point, and the interaction term \(-b(\zeta _t(x))^\kappa \)) can then be interpreted as a restoring force, driving the system back to equilibrium. Hence, a solution to (1.1) represents how these processes converge to equilibrium in a path-wise sense.

The focus of [4] was on the finite-dimensional setting, that is, when \({\mathbb V}\) is finite. Therein a sequence of interacting particle systems \(\{\eta ^n_\cdot \}_{n \ge 1}\) on \(\{0,1,\ldots \}^{\mathbb V}\) is shown to converge after being properly rescaled to a solution of (1.1). This solution was moreover proved to be unique. For each n, the dynamics of \(\eta ^n_\cdot \) can be encoded by the following formal generator expression, for \(\eta \in \{0,1\ldots \}^{\mathbb V}\) and a local function \(f:\{0,1\ldots \}^{\mathbb V}\rightarrow \mathbb {R}\):

$$\begin{aligned} \begin{aligned} L^nf(\eta )&= \sum _{x,y \in {\mathbb V}} \eta (x)\cdot p(x,y)\cdot (f(\eta + \delta _y - \delta _x) - f(\eta )) \\&\quad + \sum _{x \in {\mathbb V}} \left[ F^{n,+}(\eta (x))\cdot (f(\eta + \delta _x) - f(\eta )) + F^{n,-}(\eta (x))\cdot (f(\eta - \delta _x) - f(\eta )) \right] . \end{aligned} \end{aligned}$$
(1.3)

In words, a pile of \(\eta _t^n(x)\) particles occupies site x at time t; each particle moves with rate one according to the kernel p, and in addition, particles are born and die at x with rates \(F^{n,+}(\eta ^n_t(x))\) and \(F^{n,-}(\eta ^n_t(x))\), respectively. The motion of distinct particles and births and deaths at distinct sites are independent.

The functions \(F^{n,+}\) and \(F^{n,-}\) are defined, for every \(u \ge 0\), as

$$\begin{aligned}&F^{n,-}(u):= \frac{an^2}{2}\cdot \left( \frac{u}{n}\right) ^\ell + \min \left\{ \frac{an^2}{2}\cdot \left( \frac{u}{n}\right) ^\ell ;\;\frac{bn}{2}\cdot \left( \frac{u}{n}\right) ^\kappa \right\} , \end{aligned}$$
(1.4)
$$\begin{aligned}&F^{n,+}(u):= \frac{an^2}{2}\cdot \left( \frac{u}{n}\right) ^\ell - \min \left\{ \frac{an^2}{2}\cdot \left( \frac{u}{n}\right) ^\ell ;\;\frac{bn}{2}\cdot \left( \frac{u}{n}\right) ^\kappa \right\} . \end{aligned}$$
(1.5)

These rates are chosen so that, for every \(z \ge 0\), we have

$$\begin{aligned}&\lim _{n \rightarrow \infty } \frac{1}{n}(F^{n,+}(nz) - F^{n,-}(nz)) = -bz^\kappa \quad \text {and}\\&\quad \lim _{n \rightarrow \infty } \frac{1}{n^2}(F^{n,+}(nz) + F^{n,-}(nz)) = az^\ell . \end{aligned}$$

The idea is to make it so that the interacting particle system resembles, with increasing precision as \(n \rightarrow \infty \), a solution to (1.1). In addition, these functions satisfy other important properties. First, \(F^{n,-}(0) = F^{n,+}(0)= 0\), so there is no birth (and evidently no death) of particles at empty sites. Second, \(F^{n,-}(u) \ge F^{n,+}(u) \ge 0\) for all u; this guarantees that the number of particles in the system is stochastically decreasing, so that the dynamics has no finite-time explosion.

This leads to the result that, given a sequence \(\{\eta ^n_0\}_{n \ge 1}\) with \(\frac{1}{n}\eta ^n_0 \rightarrow \bar{\zeta } \in [0,\infty )^{\mathbb V}\), and letting \(\eta ^n_\cdot \) denote the process started from \(\eta ^n_0\) and with dynamics governed by \(L^n\), we have that the sequence of processes \(\{\tfrac{1}{n}\eta ^n_\cdot \}_{n \ge 1}\) converges to a weak solution of (1.1) [4, Theorem 1]. We will review the meaning of a weak solution of the SDE (1.1) in Section 2.4. A limit obtained through this sort of scaling procedure, where there is no scaling of space, but the “mass” of individual particles is taken to zero, is often referred to as a fluid limit.

Results

Here we are interested in obtaining the fluid limit described above in the case where \({\mathbb V}\) is a countably infinite set. Through this extension, one can hope to achieve a better understanding of stability properties of the solution with respect to the underlying space. Apart from this, the extension has theoretical interest, as it brings forward some important challenges.

Our approach requires an assumption on the transition kernel \(p(\cdot ,\cdot )\), as well as a restriction on the set of allowed initial conditions for the SDE. As in [11], we assume that there exists a function \(\alpha :{\mathbb V}\rightarrow (0,\infty )\) such that

$$\begin{aligned} \sup _{x \in {\mathbb V}}\alpha (x)< \infty \qquad \text {and}\qquad \mathsf {C}:= \sup _{x\in {\mathbb V}}\sum _{y \in V}p(x,y)\frac{\alpha (y)}{\alpha (x)} < \infty . \end{aligned}$$
(1.6)

For instance, if \({\mathbb V}= \mathbb {Z}^d\) and \(p(x,y) = \frac{1}{2d}\cdot \mathbb {1}{\{x \sim y\}}\) (nearest-neighbors diffusion), then these conditions are satisfied by \(\alpha (x) := \exp \{-|x|\}\), where \(\Vert \cdot \Vert \) denotes any norm in \({\mathbb R}^d\). Next, we define

$$\begin{aligned} \Vert \zeta \Vert := \sum _{x \in {\mathbb V}}\alpha (x)\cdot |\zeta (x)| \in [0,\infty ], \qquad \zeta \in {\mathbb R}^{{\mathbb V}} \end{aligned}$$
(1.7)

and the set of configurations

$$\begin{aligned} \mathcal {E} := \left\{ \zeta \in [0,\infty )^{{\mathbb V}}:\;\sum _{x \in {\mathbb V}}\alpha (x)\cdot \zeta (x)<\infty \right\} . \end{aligned}$$

We will only consider initial conditions of (1.1) belonging to \(\mathcal {E}\). Assumptions and restrictions of this type are common in the treatment of systems involving diffusions on infinite environments; see for instance [1, 11]. The key point here is to avoid explosion from diffusion, that is, situations where infinite amounts of mass can enter a finite set instantaneously due to excessive growth of the initial configuration.

The next step in establishing a fluid limit result is to construct processes \(\eta ^n_\cdot \) on \({\mathbb N}_0^{\mathbb V}\) whose limit should be a solution to the stochastic differential equation. As the dynamics linked to the generator (1.1) has unbounded jump rates, and the space \({\mathbb N}_0^{\mathbb V}\) is not compact (or locally compact), such a construction does not fall into the most standard framework of the theory, by means of the Hille–Yosida theorem, as presented in Chapter I of [9]. While there are still ways to construct the process under our assumptions (see for instance Chapter IX of [9], or the aforementioned references [1, 11]), here we avoid this construction issue by only constructing particle systems with finite mass. That is, we define the (countable) set

$$\begin{aligned} E:= \left\{ \eta \in {\mathbb N}_0^{\mathbb V}:\; \sum _{x \in {\mathbb V}}\eta (x) < \infty \right\} \end{aligned}$$
(1.8)

and only consider particle systems \(\eta ^n_\cdot \) with initial configuration in E. This way, since the dynamics of (1.1) causes the number of particles to decrease stochastically, it ends up producing a non-explosive continuous-time Markov chain on the countable state space E.

We are now ready to state our main result.

Theorem 1.1

  1. (a)

    Let \(\zeta _0 \in \mathcal {E}\) and let \(\{\eta ^n_0\}\) be a sequence in E with \(\Vert \tfrac{1}{n}\eta ^n_0 - \zeta _0\Vert \xrightarrow {n \rightarrow \infty } 0\). For each n, let \((\eta ^n_t)_{t \ge 0}\) denote the Markov chain on E with transitions encoded by (1.3) started from \(\eta ^n_0\). Then, as \(n \rightarrow \infty \), the processes \(\frac{1}{n}\eta ^n_\cdot \) converge in distribution (with respect to the Skorokhod topology) to an \(\mathcal {E}\)-valued process \(\zeta _\cdot \) with continuous trajectories which is a weak solution to (1.1) with initial condition \(\bar{\zeta } = \zeta _0\). The law of this process does not depend on the choice of sequence \(\{\eta ^n_0\}\) with \(\Vert \tfrac{1}{n}\eta ^n_0 -\zeta _0\Vert \xrightarrow {n \rightarrow \infty } 0\).

  2. (b)

    In case \(\sum _{x \in {\mathbb V}}\zeta _0(x) < \infty \), the process obtained through this limit is the unique weak solution to (1.1) with initial condition \(\bar{\zeta } = \zeta _0\) in the sense that it has the same distribution as any other solution of the same equation.

  3. (c)

    The mapping \(\mathcal {E}\ni \zeta _0 \mapsto (\zeta _t)_{t \ge 0}\) of initial conditions to corresponding solutions obtained through the limit of part (a) is continuous when \(\mathcal {E}\) is endowed with the norm \(\Vert \cdot \Vert \) and the set of processes on \(C([0,\infty ),\mathcal {E})\) is endowed with the topology of weak convergence of probability measures.

Outline of Methods and Organization of the Paper

Let us give an outline of our methods. Using generator estimates, we prove that a collection of processes \(\{\eta ^n_\cdot \}\) as in the statement of Theorem 1.1(a), with \(\Vert \tfrac{1}{n}\eta ^n_0 - \zeta _0\Vert \xrightarrow {n \rightarrow \infty } 0\) for some \(\zeta _0 \in \mathcal {E}\), is tight. This allows us to extract convergent subsequences, say \(\{\eta ^{n_k}_\cdot \}\) converging to a process \(\zeta _\cdot \). We then prove that the Dynkin martingales associated to \(\eta ^{n_k}_\cdot \), as defined in Lemma 2.1, converge to processes of the form

$$\begin{aligned} f(\zeta _t)-f(\zeta _0)-\int _0^t \mathcal {L}^*f(\zeta _s)ds,\qquad t \ge 0, \end{aligned}$$
(1.9)

where \(\mathcal {L}^*\), the generator associated to (1.1), is given by

$$\begin{aligned} (\mathcal {L}^*f)(\zeta ) := \sum _{x \in {\mathbb V}} (\Delta _p\zeta (x) - b\cdot (\zeta (x))^\kappa )\cdot \partial _xf(\zeta ) + \frac{1}{2}\sum _{x \in {\mathbb V}} a \cdot (\zeta (x))^\ell \cdot \partial ^2_x f(\zeta ),\quad \zeta \in \mathcal {E}, \end{aligned}$$

for a suitable collection of functions f. This convergence allows us to obtain that (1.9) is a local martingale. Using classical results from the theory of stochastic differential equations, we then conclude that the subsequential limit \(\zeta _\cdot \) is a solution of (1.1). An adaptation of the argument in [13] gives us Theorem 1.1(b), that is, that (1.1) has at most one solution in case \(\zeta _0\) has finite mass. Combining these ideas, we get that if \(\zeta _0\) has finite mass, then \(\{\eta ^n_\cdot \}\) has a single accumulation point, so the whole sequence converges. From this, we finish the proof of Theorem 1.1(a), that is, we prove convergence for any \(\zeta _0 \in \mathcal {E}\) with infinite mass by approximation : any \(\zeta _0 \in \mathcal {E}\) is arbitrarily close to configurations with finite mass.

A key tool that we rely on for this approximation and for several other arguments is a coupling inequality, Lemma 3.4, allowing us to compare pairs of processes with same generator but different initial configurations.

The rest of the paper is organized as follows. In Sect. 2, we review several technical concepts and results, including notions of convergence of probability measures, local martingales defined from Markov chains, and classical results about stochastic differential equations. In Sect. 3, we study particle systems with finite mass on \({\mathbb N}_0^{\mathbb V}\), and obtain the key coupling inequality in Lemma 3.4. In Sect. 4.1, we state our tightness result and use it to follow the rest of the outline given above, proving our main results. In Sect. 5, we prove the tightness result. Section A. is an appendix where we include some proofs to ease the flow of the exposition in the paper.

Technical Preliminaries

In this section, we collect remarks, definitions, and properties that will be useful in the study of convergence of a family of stochastic processes as mentioned in the previous Section.

Configuration Spaces

We let \({\mathbb V}\) be a countable set and \(p: {\mathbb V}\times {\mathbb V}\rightarrow [0,1]\) be a probability transition function (that is, \(\sum _y p(x,y) = 1\) for all x) and assume that there exists a function \(\alpha : {\mathbb V}\rightarrow [0,\infty )\) for which (1.6) holds. We define \(\Vert \cdot \Vert \), \(\mathcal {E}\) and E as in (1.7)–(1.8). Note that E is countable, that \(\Vert \cdot \Vert \) is a norm on the linear subspace of \({\mathbb R}^{\mathbb V}\) where it is finite, and that the metric induced by \(\Vert \cdot \Vert \) turns \(\mathcal {E}\) into a complete and separable metric space. For the sake of clarity, we mostly denote the (integer-valued) elements of E by the letter \(\eta \) rather than \(\zeta \) and processes taking values on E by \(\eta _\cdot \) rather than \(\zeta _\cdot \).

It will be useful to observe that the assumptions (1.6) yield:

$$\begin{aligned} p(x,y) \le \frac{1}{\alpha (y)}\sum _{z} p(x,z)\cdot \alpha (z)\le \frac{\mathsf {C}\alpha (x)}{\alpha (y)}. \end{aligned}$$
(2.1)

This implies that, for \(\zeta \in \mathcal {E}\),

$$\begin{aligned} \sum _{x \in {\mathbb V}} \zeta (x) \cdot p(x,y) \le \mathsf {C}\cdot \sum _{x \in {\mathbb V}} \frac{\zeta (x)\cdot \alpha (x)}{\alpha (y)}= \mathsf {C}\frac{\Vert \zeta \Vert }{\alpha (y)}, \qquad y \in {\mathbb V}. \end{aligned}$$

In particular, \(\Delta _p \zeta (x)\) in (1.2) is well defined for all \(\zeta \in \mathcal {E}\) and all \(x \in {\mathbb V}\).

Convergence of Probability Measures on Trajectory Spaces

To study convergence of probability measures on trajectory spaces, we first define a metric on the space of trajectories, then we consider a family of \(\sigma \)-algebras associated to this metric and finally we define a distance between probability measures on such \(\sigma \)-algebras.

Metric. Let \(\mathcal {X} = (\mathcal {X},\mathrm {d}_\mathcal {X})\) be a complete, separable metric space. In most cases, this will be either \(({\mathbb R},|\cdot |)\) or \(\mathcal {E}\) or E with the metric induced by \(\Vert \cdot \Vert \). We denote by \(D_{\mathcal {X}} = D([0,\infty ), \mathcal {X})\) the space of càdlàg functions \(\gamma : [0,\infty ) \rightarrow \mathcal {X}\), and by \(C_{\mathcal {X}}\) the set of functions in \(D_{\mathcal {X}}\) which are continuous. The Skorokhod metric on \(D_{\mathcal {X}}\) is defined by

$$\begin{aligned} \mathrm {d}_{\mathrm {S}}(\gamma , \gamma '):= \int _0^\infty e^{-t}\cdot \mathrm {d}^{(t)}_{\mathrm {S}}(\gamma ,\gamma ')dt, \end{aligned}$$

where

$$\begin{aligned} \mathrm {d}^{(t)}_{\mathrm {S}}(\gamma ,\gamma '):= 1 \wedge \inf _\varphi \left( \sup _{s \in [0,t]}\mathrm {d}_{\mathcal {X}}( \gamma _{\varphi (s)}, \gamma '_{s}) \vee \sup _{r,s \in [0,t]} \log \frac{|\varphi (r)-\varphi (s)|}{|r-s|} \right) , \end{aligned}$$

where the infimum is taken over all increasing bijections \(\varphi :[0,t] \rightarrow [0,t]\). This turns \((D_{\mathcal {X}},\mathrm {d}_{\mathrm {S}})\) into a complete and separable metric space, and we denote by \({\mathcal D}_{\mathcal {X}}\) its Borel \(\sigma \)-algebra. We refer the reader to [2, Chapter 3] and [5, Chapter 3] for expositions on this metric. Here let us only make one further observation, see [2, Sect. 12, p. 124]:

$$\begin{aligned}&\text {if } \{\gamma ^n\}_{n \ge 1} \subset D_{\mathcal {X}},\; \gamma \in C_{\mathcal {X}} \text { and } \mathrm {d}_{\mathrm {S}}(\gamma ^n,\gamma ) \xrightarrow {n \rightarrow \infty }0, \nonumber \\&\text {then } \sup _{0 \le s \le t} \mathrm {d}_{\mathcal {X}}(\gamma ^n_s,\gamma _s) \xrightarrow {n \rightarrow \infty } 0\; \text {for all } t\ge 0, \end{aligned}$$
(2.2)

that is, convergence in the Skorokhod topology to a continuous function implies uniform convergence on compact intervals.

Sigma-algebras. Given a stochastic process \(X_\cdot \) on \({\mathcal X}\) with càdlàg trajectories, we denote by \({\mathcal F}_t = {\mathcal F}^X_t\) the \(\sigma \)-algebra generated by \((X_s)_{0\le s \le t}\) and by \({\mathcal N} := \{A \in {\mathcal D}_{\mathcal {X}} :{\mathbb P}(X_\cdot \in A) = 0\}\). We refer to \(({\mathcal F}_t)_{t \ge 0}\) as the natural filtration of \((X_t)_{t \ge 0}\).

Convergence. Finally, we recall the definition of the Lèvy-Prohorov distance in the case of two probability measures \(\mu \) and \(\nu \) defined on \(D_{\mathcal X} = ({D}_{\mathcal {X}},{\mathcal D}_{\mathcal {X}})\):

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}}(\mu , \nu ) := \inf \left\{ \varepsilon > 0 : \; \mu (A) \le \nu (A^\epsilon ) + \epsilon \text { and } \nu (A) \le \mu (A^\epsilon ) + \epsilon \text { for all } A \in \mathcal {D}_{\mathcal {X}}\right\} , \end{aligned}$$

where \(A^\epsilon := \{ y\in D_{{\mathcal X}} :\mathrm {d}_{\mathrm {S}}(x,y) < \epsilon \text { for some }x \in A\}\). Convergence in this metric is equivalent to weak convergence of probability measures, see [5, Theorem 3.3.1, p. 108], that is, \(\mathrm {d}_{\mathrm {LP}}(\mu _n,\mu )\xrightarrow {n \rightarrow \infty } 0\) is equivalent to having \(\int f\,\mathrm {d}\mu _n \xrightarrow {n \rightarrow \infty } \int f\, \mathrm {d}\mu \) for all continuous and bounded functions \(f:D_{\mathcal {X}} \rightarrow \mathbb {R}\). Denote by \({\mathcal C}(\mu ,\nu )\) the set of all measures \(\hat{\lambda }\) on \(D_{\mathcal {X}} \times D_{\mathcal {X}}\) that couple \(\mu ,\nu \). By [5, Theorem 3.1.2, p. 98], we remark that

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}}(\mu , \nu ) = \inf _{\lambda \in {\mathcal C}(\mu ,\nu )} \quad \inf \big \{\varepsilon >0 :\lambda \{(\gamma ,\gamma ') \in D_{\mathcal {X}} \times D_{\mathcal {X}}: \mathrm {d}_{\mathrm {S}}(\gamma , \gamma ')\ge \varepsilon \} \le \varepsilon \big \}. \end{aligned}$$
(2.3)

Continuous-Time Markov Chains and Martingales

Given a stochastic process X on \(D_{{\mathcal X}}\), there is a sequence of stopping times \(\tau ^X_0 := 0 \) and \(\tau ^X_n := \inf \{t > \tau ^X_{n-1} :X_t \ne X_{t-}\}\) that exhaust the jumps of X, see Proposition 2.26 in [7, Chapter 1, p. 10]. Let \(\tau ^X_\infty := \lim _n \tau ^X_n\). Following [10, Remark 2.27], we say that a process on \(D_{{\mathcal X}}\) is non-explosive if

$$\begin{aligned} {\mathbb P}(\tau ^X_\infty < t ) = 0\quad \text { for all } t \ge 0. \end{aligned}$$

For the following two results, let S be a countable set and \((X_t)_{t \ge 0}\) be a non-explosive continuous-time Markov chain on S. For distinct \(x,y \in S\), let \(q(x,y) \ge 0\) be the jump rate from x to y, and let \(q(x,x) = -\sum _{y \ne x}q(x,y)\). For a function \(f: S \rightarrow \mathbb {R}\) satisfying

$$\begin{aligned} \sum _{y \in S}q(x,y)\cdot |f(y)-f(x)| < \infty \qquad \text {for all }x \in S, \end{aligned}$$
(2.4)

we define \(Lf(x):= \sum _{y\in S} q(x,y)\cdot (f(y)-f(x))\). If f also satisfies

$$\begin{aligned} \sum _{y \in S}q(x,y)\cdot (f(y)-f(x))^2 < \infty \qquad \text {for all }x \in S, \end{aligned}$$
(2.5)

we define \(Qf(x):= \sum _{y\in S} q(x,y)\cdot (f(y)-f(x))^2\).

Lemma 2.1

Let \(f: S \rightarrow \mathbb {R}\) be a function satisfying (2.4). Then, the process

$$\begin{aligned} M^f_t:= f(X_t) - f(X_0) - \int _0^t Lf(X_s)ds,\qquad t\ge 0 \end{aligned}$$

is a local martingale with respect to the natural filtration of \((X_t)_{t \ge 0}\). If f also satisfies (2.5), then the process

$$\begin{aligned} N^f_t:= M_t^2 - \int _0^t Qf(X_s)ds,\qquad t \ge 0 \end{aligned}$$

is also a local martingale with respect to the natural filtration of \((X_t)_{t \ge 0}\).

Proof

Fix an arbitrary initial state \(x_0 \in S\), and let \((\Lambda ^j)_{j \ge 1}\) be an increasing sequence of finite subsets of S with \(x_0 \in \Lambda ^1\) and \(\cup _j\Lambda ^j = S\). Define \(\tau ^j := \inf \{t \ge 0:\;X_t \notin \Lambda ^j\}\). We then have that \(\tau ^j \le \tau ^{j+1}\) for each j and, because \((X_t)_{t\ge 0}\) is non-explosive, \(\tau ^j \xrightarrow {j \rightarrow \infty } \infty \) almost surely. Then, under the assumptions (2.4) and (2.5), classical arguments establish that \(M^f_\cdot \) and \(N^f_\cdot \) are local martingales with respect to the natural filtration of \((X_t)_{t \ge 0}\), see for instance [8, Appendix 1, Lemma 5.1]. \(\square \)

Remark 2.2

The martingales \(M^f_\cdot \) for functions f that satisfy (2.4) are commonly referred to as Dynkin Martingales. Unless the generator L is not clear from the context, we will omit the dependence (as we have done here) to alleviate notation.

To obtain stochastic bounds, we prove a supermartingale inequality associated to the martingales \(M^f_\cdot \).

Lemma 2.3

Let \(f: S \rightarrow \mathbb {R}\) be a non-negative function satisfying (2.4). Assume that there exists \(C > 0\) such that

$$\begin{aligned} \sum _{y \in S}q(x,y)\cdot (f(y)-f(x)) \le Cf(x) \qquad \text { for all }x \in S. \end{aligned}$$

Then, the process \((e^{-Ct}\cdot f(X_t))_{t \ge 0}\) is a supermartingale with respect to the natural filtration of \((X_t)_{t \ge 0}\).

Proof

Fix an arbitrary initial state \(x_0 \in S\). By the Markov property, it is sufficient to prove that, for any \(t \ge 0\),

$$\begin{aligned} \mathbb {E}[e^{-Ct}\cdot f(X_t)] \le f(x_0). \end{aligned}$$

Let \((\Lambda ^j)_{j \ge 1}\) and \(\tau ^j\) be as in the proof of Lemma 2.1. Define the processes

$$\begin{aligned} X^j_t := {\left\{ \begin{array}{ll} X_t&{}\text {if }t \le \tau ^j;\\ \star &{}\text {otherwise,} \end{array}\right. } \qquad t\ge 0 \end{aligned}$$

where \(\star \) denotes a cemetery state. Then, \(X^j_\cdot \) is a Markov chain on the finite state space \(S^j := \Lambda ^j \cup \{\star \}\), with jump rates, for distinct \(x,y \in S_j\), given by

$$\begin{aligned} q^j(x,y) = {\left\{ \begin{array}{ll} 0&{}\text {if }x = \star ;\\ q(x,y)&{} \text {if } x \ne \star ,\;y \ne \star ;\\ \sum _{y \in (\Lambda ^j)^c}q(x,y)&{}\text {if } x \ne \star ,\;y = \star . \end{array}\right. } \end{aligned}$$

Let \(f^j: S^j \rightarrow \mathbb {R}\) be defined by \(f^j(x) = f(x)\) for \(x \ne \star \) and \(f^j(\star ) = 0\). We have, for any \(x \in S^j\backslash \{\star \}\),

$$\begin{aligned} \sum _{y \in S^j} q^j(x,y)\cdot (f^j(y)-f^j(x)) \le \sum _{y \in S} q(x,y)\cdot (f(y)-f(x)) \le Cf(x), \end{aligned}$$

since \(0 = f^j(\star ) \le f(y)\) for \(y \in S \backslash \Lambda ^j\). It then follows from the elementary theory of Markov chains that \((e^{-Ct}\cdot X^j_t)_{t \ge 0}\) is a supermartingale. Since \(f(X^j_t)\xrightarrow {j \rightarrow \infty } f(X_t)\) almost surely, it follows from Fatou’s Lemma that

$$\begin{aligned} \mathbb {E}[e^{-Ct}\cdot f(X_t)] \le \liminf _{j \rightarrow \infty } \mathbb {E}[e^{-Ct}\cdot f(X_t^j)] \le f(x_0). \end{aligned}$$

\(\square \)

Some Properties of Solutions of the SDE (1.1)

Let \((\zeta _t)_{t \ge 0}\) be a stochastic process (defined on some space \((\Omega ,\mathcal {F},{\mathbb P})\)) with values in \(\mathcal {E}\) and continuous trajectories. We say that \(\zeta _\cdot \) is a weak solution to the SDE (1.1) if there exists a space \((\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb P})\) in which we have defined

  • A process \(X_\cdot \) with values in \(\mathcal {E}\), continuous trajectories, and same distribution as \(\zeta _\cdot \), and

  • A family \((B^x_\cdot )_{x \in {\mathbb V}}\) of independent, standard one-dimensional Brownian motions,

and moreover, \(\tilde{\mathbb P}\) almost surely we have

$$\begin{aligned} X_t(x)= & {} X_0(x) + \int _0^t (\Delta _pX_s(x) - b(X_s(x))^\kappa )ds\\&+\int _0^t \left( a(X_s(x))^\ell \right) ^{\frac{1}{2}}dB^x_s, \quad \text {for} \,t \ge 0\, \mathrm{and}\, x \in {\mathbb V}. \end{aligned}$$

We will need the following result, which gives uniqueness for solutions of (1.1) with finite mass. The essence of its proof is taken from [13]. We present in the appendix (Sect. A.1) the proof with slight modifications to adjust to our setting.

Proposition 2.4

Let \(\bar{\zeta } \in \mathcal {E}\) satisfy \(\Vert \zeta \Vert _1 = \sum _{x \in {\mathbb V}}\zeta (x) < \infty \). Let \(\zeta ^1_\cdot \) and \(\zeta ^2_\cdot \) be two weak solutions of the SDE (1.1) with initial condition \(\bar{\zeta }\). Then, \(\zeta ^1_\cdot \) and \(\zeta ^2_\cdot \) have the same distribution.

Let \(f:\mathcal {E} \rightarrow \mathbb {R}\) be a function for which there exists a finite set \(\{x_1,\ldots , x_k\} \subset {\mathbb V}\) so that f only depends on \((\zeta (x_1),\ldots , \zeta (x_k))\), and moreover f is a twice continuously differentiable function of this vector. Define

$$\begin{aligned} (\mathcal {L}^*f)(\zeta ) := \sum _{x \in {\mathbb V}} (\Delta _p\zeta (x) - b\cdot (\zeta (x))^\kappa )\cdot \partial _xf(\zeta ) + \frac{1}{2}\sum _{x \in {\mathbb V}} a \cdot (\zeta (x))^\ell \cdot \partial ^2_x f(\zeta ),\quad \zeta \in \mathcal {E}. \end{aligned}$$

In particular, we denote by \(f_x(\zeta ) := \zeta (x)\) the coordinate projection on \(x \in {\mathbb V}\), and by \(f_{xy}(\zeta ) := \zeta (x)\zeta (y)\). Note that all those functions are indeed differentiable and depend only on a finite number of sites.

Proposition 2.5

Let \(\zeta _\cdot \) be a stochastic process with values in \(\mathcal {E}\) and continuous trajectories. Assume that for every \(f \in \{f_x,f_{xy}:x,y \in {\mathbb V}\}\), the process

$$\begin{aligned} f(\zeta _t) - f(\zeta _0) - \int _0^t \mathcal {L}^*f(\zeta _s)ds,\qquad t \ge 0. \end{aligned}$$

is a local martingale. Then, \(\zeta _\cdot \) is a weak solution of (1.1).

This proposition is the same as Proposition 4.6, page 315 in [7], except that here we deal with infinite-dimensional processes, whereas the setup of the proposition in [7] is finite-dimensional. However, largely due to the fact that the cross-variation of our SDE is trivial (that is, the expression for \(d\zeta _t(x)\) in (1.1) does not involve \(dB^y_t\) for \(y \ne x\)), the proof in [7] carries through to our setting. In order to highlight the differences and the main steps, we sketch the proof in the appendix (Section A.2).

Construction of Diffusive Birth-and-Death Particle Systems

Recall that \(E = \{\eta \in {\mathbb N}_0^{\mathbb V}:\sum _x \eta (x) < \infty \}\). In this section, we will construct continuous-time Markov chains on E that will later be used in the fluid limit for the proof of our main result. Rather than having an index n and functions \(F^{n,+}\) and \(F^{n,-}\) as in (1.4) and (1.5), for now we will have no index, and functions \(F^+\) and \(F^-\) satisfying certain properties (see (3.1) below).

We define the set of marks

$$\begin{aligned} \mathcal {M} = \{(x,+),(x,-),(x,y):x, y\in {\mathbb V}\}. \end{aligned}$$

Marks will serve as instructions for the dynamics. Marks of the form \((x,+)\) and \((x,-)\) represent the birth and the death of a particle at site x, respectively, and a mark of the form (xy) represents that a particle from site x jumps to site y. Marks are thus associated with the transition operators

$$\begin{aligned} \Gamma ^{x,+}(\eta ) :=\eta + \delta _x, \qquad \Gamma ^{x,-}(\eta ) := \eta - \delta _x, \qquad \text {and} \qquad \Gamma ^{(x,y)}_x(\eta ) :=\eta - \delta _x+ \delta _{y}, \end{aligned}$$

where for \(x\in {\mathbb V}\), \(\delta _x \in E\) is the configuration with only one particle at \(x\). For a configuration \(\eta \in E\) and \(x,y\in {\mathbb V}\), we define transition rates by

$$\begin{aligned} R^{x,+}(\eta ) := F^+(\eta (x)), \qquad R^{x,-}(\eta ) := F^-(\eta (x)), \qquad R^{(x,y)}(\eta ) := p(x,y) \cdot \eta (x), \end{aligned}$$

where we assume that the reaction functions \(F^+\) and \(F^-\) satisfy

$$\begin{aligned}&F^+(0) = F^-(0) = 0,\qquad 0 \le F^+ \le F^-,\nonumber \\&\qquad {\mathbb N}_0 \ni z \mapsto F^+(z) - F^-(z) \text { is decreasing.} \end{aligned}$$
(3.1)

We now define a continuous-time Markov chain on E with the prescription that

$$\begin{aligned} \text {for each }\mathsf {a}\in \mathcal {M},\; \eta \,\, \mathrm{jumps}\, \mathrm{to}\, \Gamma ^\mathsf {a}(\eta )\, \mathrm{with}\,\mathrm{rate}\, R^\mathsf {a}(\eta ). \end{aligned}$$

Noting that \(\sum _{\mathsf {a}\in \mathcal {M}}R^\mathsf {a}(\eta ) < \infty \) for any \(\eta \in E\), this indeed describes jump rates of a continuous-time Markov chain. The assumption \(F^+ \le F^-\) combined with a simple stochastic comparison argument (see [4, Lemma 2]) implies that the chain is non-explosive. It will be important to leave the initial condition explicit, so we will denote the chain started from \(\eta \in E\) by \((\Phi _t(\eta ))_{t \ge 0}\).

For any function \(f:E \rightarrow \mathbb {R}\) satisfying

$$\begin{aligned} \sum _{\mathsf {a}\in \mathcal {M}} R^\mathsf {a}(\eta )\cdot \max \left( |f(\Gamma ^\mathsf {a}(\eta ))-f(\eta )|,\;|f(\Gamma ^\mathsf {a}(\eta ))-f(\eta )|^2\right) < \infty \quad \text {for all }\eta \in E, \end{aligned}$$
(3.2)

we define

$$\begin{aligned}&Lf(\eta ) := \sum _{\mathsf {a}\in \mathcal {M}} R^\mathsf {a}(\eta )\cdot (f(\Gamma ^\mathsf {a}(\eta ))-f(\eta )),\\&\qquad Qf(\eta ):= \sum _{\mathsf {a}\in \mathcal {M}} R^\mathsf {a}(\eta )\cdot (f(\Gamma ^\mathsf {a}(\eta ))-f(\eta ))^2. \end{aligned}$$

Recall that a function defined on a subset of \(\mathbb {R}^{\mathbb V}\) is called local if there exists a finite set \({\mathbb V}' \subset {\mathbb V}\) such that the function depends on \(\eta \in \mathbb {R}^{\mathbb V}\) only through \((\eta (x):x \in {\mathbb V}')\). We then have

Lemma 3.1

Any local function \(f: E \rightarrow \mathbb {R}\) satisfies (3.2), and the processes

$$\begin{aligned} M^f_t&:= f(\Phi _t(\eta )) - f(\eta ) - \int _0^t Lf(\Phi _s(\eta ))ds, \quad t \ge 0,\text { and }\\ N^f_t&:= \left( M^f_t \right) ^2 - \int _0^t Qf(\Phi _s(\eta ))ds, \quad t \ge 0 \end{aligned}$$

are local martingales with respect to the natural filtration of \((\Phi _t(\eta ))_{t \ge 0}\).

Proof

The first statement is straightforward to check: since f is local and its argument is an element of E, the sum in (3.2) only has finitely many non-zero terms. The second statement then follows from Lemma 2.1. \(\square \)

Lemma 3.2

(1-norm bound in E) For any \(\eta \in E\), the process \(\Vert \Phi _\cdot (\eta )\Vert _1\) is a supermartingale. In particular, for any \(T \ge 0\) and \(A > 0\),

$$\begin{aligned} {\mathbb P}\left( \sup _{0\le t \le T} \Vert \Phi _t(\eta )\Vert _1 > A \right) \le \frac{\Vert \eta \Vert _1}{A}. \end{aligned}$$
(3.3)

Proof

Let \(f:E \rightarrow \mathbb {R}\) be given by \(f(\eta ) := \Vert \eta \Vert _1\). It is readily seen that f satisfies (3.2), and then it follows from Lemma 2.3 that \(\Vert \Phi _\cdot (\eta )\Vert _1\) is a supermartingale. Equation (3.3) then follows from the optional stopping theorem. \(\square \)

Now, given a pair \(\eta ,\eta ' \in E\) we will construct a coupled process

$$\begin{aligned} \hat{\Phi }_t(\eta ,\eta ') = (\hat{\Phi }_{t,1}(\eta ,\eta '),\hat{\Phi }_{t,2}(\eta ,\eta ')),\qquad t \ge 0 \end{aligned}$$

on \(E \times E\) so that the coordinate processes are distributed as \(\Phi _\cdot (\eta )\) and \(\Phi _\cdot (\eta ')\), respectively. The coupling is given as the Markov chain on \(E \times E\) with transition rates described by

$$\begin{aligned} \text {for each } \mathsf {a}\in \mathcal {M},\; (\eta ,\eta ') \text { jumps to } {\left\{ \begin{array}{ll} (\Gamma ^\mathsf {a}(\eta ),\Gamma ^\mathsf {a}(\eta '))&{}\text {with rate } \min (R^\mathsf {a}(\eta ),R^\mathsf {a}(\eta '));\\ (\Gamma ^\mathsf {a}(\eta ),\eta ')&{}\text {with rate } \max (R^\mathsf {a}(\eta )-R^\mathsf {a}(\eta '),0);\\ (\eta ,\Gamma ^\mathsf {a}(\eta '))&{}\text {with rate } \max (R^\mathsf {a}(\eta ')-R^\mathsf {a}(\eta ),0). \end{array}\right. } \end{aligned}$$

We then define

$$\begin{aligned} \begin{aligned} \hat{L}g(\eta ,\eta '):= \sum _{\mathsf {a}\in \mathcal {M}}&[\min (R^\mathsf {a}(\eta ),R^\mathsf {a}(\eta '))\cdot (g(\Gamma ^\mathsf {a}(\eta ),\Gamma ^\mathsf {a}(\eta ')) - g(\eta ,\eta ') ) \\&\;\;+ \max (R^\mathsf {a}(\eta )-R^\mathsf {a}(\eta '),0)\cdot (g(\Gamma ^\mathsf {a}(\eta ),\eta ') - g(\eta ,\eta '))\\&\;\;+ \max (R^\mathsf {a}(\eta ')-R^\mathsf {a}(\eta ),0)\cdot (g(\eta ,\Gamma ^\mathsf {a}(\eta ')) - g(\eta ,\eta ')) ] \end{aligned} \end{aligned}$$
(3.4)

for functions \(g:E\times E \rightarrow \mathbb {R}\) for which the sum on the right-hand side is absolutely convergent for all \((\eta ,\eta ')\).

Lemma 3.3

For \(g(\eta ,\eta '):= \Vert \eta - \eta '\Vert \), we have that \(\hat{L}g\) is well defined and satisfies

$$\begin{aligned} \hat{L}g(\eta ,\eta ') \le \mathsf {C}\cdot g(\eta ,\eta ')\quad \text {for all }\eta ,\eta ' \in E, \end{aligned}$$

where \(\mathsf {C}\) is the constant from (1.6).

Proof

For this choice of g, the first line in the right-hand side of (3.4) vanishes, so we can write \(\hat{L}g(\eta ,\eta ') = \sum _{\mathsf {a}\in \mathcal {M}} (\Xi ^\mathsf {a}_1 + \Xi ^\mathsf {a}_2)(\eta ,\eta ')\), where

$$\begin{aligned} \Xi ^\mathsf {a}_{1}(\eta ,\eta ')&= \left( R^\mathsf {a}(\eta ) - \min \{R^\mathsf {a}(\eta ),R^\mathsf {a}(\eta ')\}\right) \cdot \left( \Vert \Gamma ^\mathsf {a}(\eta ) - \eta '\Vert - \Vert \eta - \eta '\Vert \right) ,\\ \Xi ^\mathsf {a}_{2}(\eta ,\eta ')&= \left( R^\mathsf {a}(\eta ') - \min \{R^\mathsf {a}(\eta ),R^\mathsf {a}(\eta ')\}\right) \cdot \left( \Vert \eta - \Gamma ^\mathsf {a}(\eta ')\Vert - \Vert \eta - \eta '\Vert \right) . \end{aligned}$$

We first deal with the reaction terms, that is, the terms corresponding to marks of the form \((x,+)\) and \((x,-)\). Due to the nature of the rates we have chosen, the only cases we have to look at are those for which \(\eta (x) \ne \eta '(x)\). If \(\eta (x) > \eta '(x)\), the contribution from \((x,+)\) marks is:

$$\begin{aligned} \big ( \Xi _1^{(x,+)} + \Xi _2^{(x,+)} \big )(\eta , \eta ')&= [F^+(\eta (x)) - F^+(\eta '(x))]_+ \cdot \alpha (x)\\&\quad + [F^+(\eta '(x)) - F^+(\eta (x))]_+ \cdot (-\alpha (x)) \\&= [F^+(\eta (x)) - F^+(\eta '(x))] \cdot \alpha (x). \end{aligned}$$

Doing similarly for marks \((x,-)\), we get:

$$\begin{aligned}&\sum _{\sigma \in \{+,-\}} (\Xi _1^{(x,\sigma )} + \Xi _2^{(x,\sigma )})(\eta , \eta ')\nonumber \\&\quad \quad \quad = \alpha (x)\cdot \left[ (F^+-F^-)(\eta (x)) - (F^+ - F^-)(\eta '(x)) \right] \le 0, \end{aligned}$$
(3.5)

where the last inequality follows from our hypothesis (3.1) ensuring that \(F = F^+ - F^-\) is decreasing. Observe that we don’t need to assume (although it would be natural to) that each \(F^+\) and \(F^-\) are increasing.

The same argument shows (3.5) for the case \(\eta (x) < \eta '(x)\). We thus conclude that

$$\begin{aligned} \sum _{x \in {\mathbb V}} \sum _{\sigma \in \{+,-\}} (\Xi ^{(x,\sigma )}_1 + \Xi ^{(x,\sigma )}_2)(\eta ,\eta ') \le 0. \end{aligned}$$

We now turn to the diffusion terms. Fix \(x,y \in {\mathbb V}\), and first assume that \(\eta (x) > \eta '(x)\). Again, note that reaction rates are equal when \(\eta (x) = \eta '(x)\), and the contribution to \(\hat{L}g\) is zero in those cases. We then have \(\Xi ^{(x,y)}_2(\eta ,\eta ') = 0\) and

$$\begin{aligned} \Xi ^{(x,y)}_1(\eta ,\eta ')&= (\eta (x)-\eta '(x)) \cdot p(x,y) \cdot \left( \Vert \eta -\delta _x+\delta _y - \eta ' \Vert - \Vert \eta - \eta ' \Vert \right) \\&\le (\eta (x)-\eta '(x)) \cdot p(x,y) \cdot (- \alpha (x) + \alpha (y)). \end{aligned}$$

Treating the case \(\eta (x) < \eta '(x)\) analogously, we obtain

$$\begin{aligned} \sum _{x,y \in {\mathbb V}} (\Xi ^{(x,y)}_1 + \Xi ^{(x,y)}_2)(\eta ,\eta ')&\le \sum _{x:\eta (x) > \eta '(x)} \sum _y \Xi _1^{(x,y)}(\eta ,\eta ') \quad + \sum _{x:\eta (x) < \eta '(x)} \sum _y \Xi _2^{(x,y)}(\eta ,\eta ') \\&\le \sum _{x \in {\mathbb V}} |\eta (x) - \eta '(x)| \sum _{y \in {\mathbb V}} p(x,y) ( - \alpha (x) + \alpha (y)) \\&{\mathop {\le }\limits ^{(1.2)}} (\mathsf {C}- 1)\sum _{x \in {\mathbb V}} |\eta (x)-\eta '(x)|\alpha (x) \le \mathsf {C}\Vert \eta -\eta '\Vert . \end{aligned}$$

\(\square \)

Lemma 3.4

For any \(\eta ,\eta ' \in E\), the process \((e^{-\mathsf {C}t} \cdot \Vert \hat{\Phi }_{t,1}(\eta ,\eta ') - \hat{\Phi }_{t,2}(\eta ,\eta ')\Vert )_{t \ge 0}\) is a supermartingale with respect to its natural filtration. In particular, for any \(T > 0\) and \(A > 0\), we have

$$\begin{aligned} {\mathbb P}\left( \sup _{0 \le t \le T}\Vert \hat{\Phi }_{t,1}(\eta ,\eta ')-\hat{\Phi }_{t,2}(\eta ,\eta ')\Vert > A\right) \le \frac{e^{\mathsf {C}T}\cdot \Vert \eta -\eta '\Vert }{A}. \end{aligned}$$
(3.6)

Proof

The first statement is a consequence of Lemmas 2.3 and 3.3. To prove the second statement, abbreviate

$$\begin{aligned}Y_t:= \Vert \hat{\Phi }_{t,1}(\eta ,\eta ') - \hat{\Phi }_{t,2}(\eta ,\eta ')\Vert ,\qquad X_t := e^{-\mathsf {C}t}\cdot Y_t. \end{aligned}$$

For \(a > 0\), define \(\tau _a := \inf \{t \ge 0:\;X_t > a\}\). We have \(\Vert \eta -\eta '\Vert \ge {\mathbb E}[X_{\tau _a \wedge T}] \ge a \cdot {\mathbb P}(\tau _a \le T)\) for any \(T > 0\) and \(a > 0\), so

$$\begin{aligned} {\mathbb P}\left( \sup _{0 \le t \le T}X_t > a\right) \le \frac{\Vert \eta - \eta '\Vert }{a}. \end{aligned}$$

We then obtain

$$\begin{aligned} {\mathbb P}\left( \sup _{0 \le t \le T}Y_t> A \right) \le {\mathbb P}\left( \sup _{0\le t \le T}X_t > Ae^{-\mathsf {C}T}\right) \le \frac{e^{\mathsf {C}T}\cdot \Vert \eta -\eta '\Vert }{A}. \end{aligned}$$

\(\square \)

Convergence to Solutions of Reaction–Diffusion Equations

Sequence of Particle Systems: Definition and First Estimates

In this section, following the program outlined in the Introduction, we consider a sequence of processes of the type constructed in the previous section and prove that this sequence converges to solutions of the system of reaction–diffusion Eq. (1.1).

We recall that we define, for \(u \ge 0\),

$$\begin{aligned}&F^{n,-}(u):= \frac{an^2}{2}\cdot \left( \frac{u}{n}\right) ^\ell + \min \left\{ \frac{an^2}{2}\cdot \left( \frac{u}{n}\right) ^\ell ;\;\frac{bn}{2}\cdot \left( \frac{u}{n}\right) ^\kappa \right\} ,\\&F^{n,+}(u):= \frac{an^2}{2}\cdot \left( \frac{u}{n}\right) ^\ell - \min \left\{ \frac{an^2}{2}\cdot \left( \frac{u}{n}\right) ^\ell ;\;\frac{bn}{2}\cdot \left( \frac{u}{n}\right) ^\kappa \right\} . \end{aligned}$$

We denote by \((\Phi ^n_t(\eta ))_{t \ge 0}\) the Markov chains, as constructed in Sect. 3, corresponding to the rate functions \(F^+= F^{n,+}\) and \(F^-=F^{n,-}\) (we leave the diffusion part of the dynamics constant for all n, that is, particles jump with rate one regardless of n). Conditions in (3.1) are satisfied with this choice of rate functions. We also denote by \(\hat{\Phi }^n_\cdot (\eta ,\eta ')\) the corresponding coupling, as described in the previous section. Finally, we write

$$\begin{aligned} \varphi ^n_t(\zeta ):= & {} \tfrac{1}{n}\Phi ^n(n\zeta ),\quad \hat{\varphi }^n_t(\zeta ,\zeta ')\\= & {} \Big (\varphi _{t,1}(\zeta ,\zeta '),\hat{\varphi }_{t,2}(\zeta ,\zeta ')\Big ) := \tfrac{1}{n}\hat{\Phi }^n_t(n\zeta ,n\zeta '),\quad \zeta ,\zeta ' \in \tfrac{1}{n}E. \end{aligned}$$

Hence, \(\varphi ^n_\cdot (\zeta )\) and \(\hat{\varphi }^n_{\cdot ,i}(\zeta ,\zeta ')\), for \(i \in \{1,2\}\), are processes on \(\tfrac{1}{n}E\).

From (3.6), we obtain that, for any n and any \(\zeta ,\zeta ' \in \tfrac{1}{n}E\),

$$\begin{aligned} {\mathbb P}\left( \sup _{0 \le t \le T}\Vert \hat{\varphi }_{t,1}^n(\zeta ,\zeta ')-\hat{\varphi }_{t,2}^n(\zeta ,\zeta ')\Vert > A\right) \le \frac{e^{\mathsf {C}T}\cdot \Vert \zeta -\zeta '\Vert }{A}. \end{aligned}$$
(4.1)

We have the following consequence of Lemma 3.4.

Corollary 4.1

For every \(\epsilon > 0\) and \(T > 0\), there exists \(\delta > 0\) such that, for all \(n \in {\mathbb N}\) and all \(\zeta ,\zeta ' \in \tfrac{1}{n}E\) with \(\Vert \zeta -\zeta '\Vert < \delta \) we have

$$\begin{aligned} {\mathbb P}\left( \sup _{0 \le t \le T}\left\| \hat{\varphi }^n_{t,1}(\zeta ,\zeta ') - \hat{\varphi }^n_{t,2}(\zeta ,\zeta ')\right\| > \epsilon \right) < \epsilon . \end{aligned}$$

We may now state a distance bound with respect to the Lèvy-Prohorov metric.

Lemma 4.2

For every \(\epsilon > 0\), there exists \(\delta > 0\) such that, for all \(n \in {\mathbb N}\) and all \(\zeta ,\zeta ' \in \tfrac{1}{n}E\) with \(\Vert \zeta -\zeta '\Vert < \delta \) we have

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}}\left( \varphi ^n_\cdot (\zeta ),\;\varphi ^n_\cdot (\zeta ')\right) < \epsilon . \end{aligned}$$
(4.2)

Proof

Fix \(\epsilon > 0\). It follows from the definition of the Skorokhod metric that

$$\begin{aligned} \gamma ,\gamma ' \in D([0,\infty ),\mathcal {E}),\; \sup _{0 \le t \le \log (2/\epsilon )}\Vert \gamma _t - \gamma '_t\Vert \le \frac{\epsilon }{2} \quad \Longrightarrow \quad \mathrm {d}_\mathrm {S}(\gamma ,\gamma ')< \epsilon . \end{aligned}$$

Together with Lemma 4.1 (with \(T = \log (2/\epsilon )\)), this implies that there exists \(\delta > 0\) such that for any n and any \(\zeta ,\zeta '\) with \(\Vert \zeta - \zeta '\Vert < \delta \) we have

$$\begin{aligned} {\mathbb P}\left( \mathrm {d}_\mathrm {S}(\hat{\varphi }^n_{1,\cdot }(\zeta ,\zeta '), \hat{\varphi }^n_{2,\cdot }(\zeta ,\zeta ')) > \epsilon \right) < \epsilon . \end{aligned}$$

The desired result now follows from (2.3) and the fact that \(\hat{\varphi }^n_{1,\cdot }(\eta ,\eta ')\) and \(\hat{\varphi }^n_{2,\cdot }(\eta ,\eta ')\) have the same distribution of \(\varphi ^n_\cdot (\eta )\), \(\varphi ^n_\cdot (\eta ')\), respectively. \(\square \)

For a local function \(f:\frac{1}{n}E \rightarrow \mathbb {R}\), define

$$\begin{aligned}&\mathcal {L}^nf(\zeta ):= \sum _{\mathsf {a}\in \mathcal {M}} R^\mathsf {a}(n\zeta )\cdot \left( f\left( \tfrac{1}{n}\Gamma ^\mathsf {a}(n\zeta )\right) - f(\zeta )\right) ,\\&\mathcal {Q}^nf(\zeta ):= \sum _{\mathsf {a}\in \mathcal {M}} R^\mathsf {a}(n\zeta )\cdot \left( f\left( \tfrac{1}{n}\Gamma ^\mathsf {a}(n\zeta )\right) - f(\zeta )\right) ^2,\quad \zeta \in \tfrac{1}{n}E. \end{aligned}$$

It follows from Lemma 3.1 that the processes

$$\begin{aligned} M^f_t&:= f(\varphi ^n_t(\zeta )) - f(\zeta ) - \int _0^t \mathcal {L}^n f(\varphi ^n_s(\zeta ))ds, \quad \text {and}\\ N^f_t&:= \left( M^f_t \right) ^2 - \int _0^t \mathcal {Q}^nf(\varphi ^n_s(\zeta ))ds \end{aligned}$$

defined for \(t \ge 0\) are local martingales.

In the following lemma, we give explicit expressions for \(\mathcal {L}^n\) and \(\mathcal {Q}^n\) applied to functions of the form \(f_x(\zeta ) = \zeta (x)\) and \(f_{xy}(\zeta ) = \zeta (x)\cdot \zeta (y)\). We postpone the calculations to Appendix A.3.

Lemma 4.3

We have, for any \(\zeta \in \frac{1}{n}E\) and \(x \in {\mathbb V}\),

$$\begin{aligned}&(\mathcal {L}^nf_x)(\zeta ) = (\Delta _p \zeta )(x) - \min \left\{ an\cdot (\zeta (x))^\ell ; \;b\cdot (\zeta (x))^\kappa \right\} \end{aligned}$$
(4.3)
$$\begin{aligned}&(\mathcal {Q}^nf_x)(\zeta ) = a\cdot (\zeta (x))^\ell + \frac{1}{n}\sum _{y \ne x} (p(y,x)\zeta (y)+p(x,y)\zeta (x)), \end{aligned}$$
(4.4)
$$\begin{aligned} (&\mathcal {L}^nf_{xx})(\zeta ) = (2f_x \cdot \mathcal {L}^nf_x)(\zeta ) + a\cdot (\zeta (x))^\ell + \frac{1}{n}\sum _{y \ne x} (\zeta (y)p(y,x)+\zeta (x)p(x,y)). \end{aligned}$$
(4.5)

Moreover, for distinct \(x,y\in {\mathbb V}\),

$$\begin{aligned} (\mathcal {L}^nf_{x,y})(\zeta ) = (f_x \cdot \mathcal {L}^n f_y + f_y \cdot \mathcal {L}^n f_x)(\zeta ) - \frac{1}{n} (\zeta (y) p(y,x)+ \zeta (x)p(x,y)). \end{aligned}$$
(4.6)

Finally, we state our tightness result, which will allow us to extract convergent subsequences of a sequence of processes of the form \(\{\varphi ^n_\cdot (\zeta ^n)\}_{n \ge 1}\).

Proposition 4.4

Let \(\zeta \in \mathcal {E}\) and \(\{\zeta ^n\}_{n \ge 1}\) be a sequence with \(\zeta ^n \in \tfrac{1}{n}E\) for each n and \(\Vert \zeta ^n - \zeta \Vert \xrightarrow {n \rightarrow \infty } 0\). Then, the family of processes \(\{\varphi ^n_\cdot (\zeta ^n)\}_{n \ge 1}\) is tight in \(D( [0,\infty ),\mathcal {E} )\).

The proof of this proposition will be carried out in Sect. 5.

Limit Points are Solutions

Recall from Sect. 2.4 that we defined

$$\begin{aligned} (\mathcal {L}^*f)(\zeta ) := \sum _{x \in {\mathbb V}} (\Delta _p\zeta (x) - b\cdot (\zeta (x))^\kappa )\cdot \partial _xf(\zeta ) + \frac{1}{2}\sum _{x \in {\mathbb V}} a \cdot (\zeta (x))^\ell \cdot \partial ^2_x f(\zeta ),\quad \zeta \in \mathcal {E} \end{aligned}$$

for any local and twice continuously differentiable function f. Substituting \(f_x\), \(f_{xx}\) and \(f_{xy}\) gives

$$\begin{aligned}&(\mathcal {L}^*f_x)(\zeta ) = \Delta _p\zeta (x) - b\cdot (\zeta (x))^\kappa , \end{aligned}$$
(4.7)
$$\begin{aligned}&(\mathcal {L}^*f_{xx})(\zeta )= (2f_x\cdot \mathcal {L}^*f_x)(\zeta ) + a \cdot (\zeta (x))^\ell , \end{aligned}$$
(4.8)
$$\begin{aligned}&(\mathcal {L}^*f_{xy})(\zeta ) = (f_x \cdot \mathcal {L}^*f_y + f_y \cdot \mathcal {L}^*f_x)(\zeta )\quad \text {for }x \ne y. \end{aligned}$$
(4.9)

Lemma 4.5

For any \(f \in \{f_x,f_{xy}:x,y \in {\mathbb V}\}\) and \(A > 0\), we have

$$\begin{aligned} \sup \left\{ |(\mathcal {L}^*f)(\zeta ) - (\mathcal {L}^nf)(\zeta )| :\; \zeta \in \tfrac{1}{n}E,\; \Vert \zeta \Vert \le A \right\} \xrightarrow {n \rightarrow \infty }0. \end{aligned}$$
(4.10)

Proof

Let us fix \(x \in {\mathbb V}\) and consider the case \(f = f_x\). Comparing (4.3) and (4.7), and noting that \(\zeta (x) \le \Vert \zeta \Vert /\alpha (x)\), the supremum on the left-hand side of (4.10) is at most

$$\begin{aligned}&\sup _{z \in [0,A/\alpha (x)]} \left( b z^\kappa - an z^\ell \right) \cdot \mathbb {1}\left\{ b z^k > an z^\ell \right\} \xrightarrow {n \rightarrow \infty }0; \end{aligned}$$

the convergence can be checked by separately considering the cases \(\kappa > \ell \) and \(\kappa \le \ell \).

Next, for \(f = f_{xx}\), comparing (4.5) and (4.8) and using the case of \(f_x\) that we just treated, it suffices to note that, by (5.8), we have

$$\begin{aligned} \frac{1}{n}\cdot \sup \left\{ \sum _{y \ne x}(\zeta (y)p(y,x) + \zeta (x)p(x,y)): \;\zeta \in \tfrac{1}{n}E,\;\Vert \zeta \Vert \le A \right\} \xrightarrow {n \rightarrow \infty }0. \end{aligned}$$

Finally, the case \(f=f_{xy}\) with given \(x\ne y\) is easier: comparing (4.6) and (4.9) and using the convergence for \(f_x\) and \(f_y\), it suffices to note that

$$\begin{aligned} \frac{1}{n}\cdot \sup \left\{ \zeta (y)p(y,x) + \zeta (x)p(x,y):\; \zeta \in \tfrac{1}{n}E,\;\Vert \zeta \Vert \le A \right\} \xrightarrow {n \rightarrow \infty } 0. \end{aligned}$$

\(\square \)

Proposition 4.6

Let \(\zeta \in \mathcal {E}\) and \(\{\zeta ^n\}_{n \ge 1}\) be a sequence with \(\zeta ^n \in \tfrac{1}{n}E\) for each n and \(\Vert \zeta ^n - \zeta \Vert \xrightarrow {n \rightarrow \infty }0\). Assume that the sequence of processes \(\{\varphi ^{n}_\cdot (\zeta ^n)\}_{n \ge 1}\) has a subsequence that converges in distribution in \(D([0,\infty ),\mathcal {E})\) to a process \(\zeta _\cdot ^*\). Then, the distribution of \(\zeta _\cdot ^*\) is supported on \(C([0,\infty ),\mathcal {E})\), and \(\zeta _\cdot ^*\) is a solution of the SDE (1.1).

Proof of Proposition 4.6

Denote the convergent subsequence by \(\{\varphi ^{n_k}_\cdot (\zeta ^{n_k})\}_{k \ge 1}\). By Skorokhod’s representation theorem [2, p. 70], we may consider a probability space \((\widetilde{\Omega },\widetilde{\mathcal {F}},\widetilde{P})\) with processes \(\{Z^k_\cdot \}_{k \ge 1}\) and \(Z^*_\cdot \) so that

  • For each k\(Z^k_\cdot \) has trajectories in \(D([0,\infty ),\frac{1}{n}E)\) and same distribution as \(\varphi ^{n_k}_\cdot (\zeta ^{n_k})\);

  • \(Z^*_\cdot \) has trajectories in \(D([0,\infty ),\mathcal {E})\) and same distribution as \(\zeta ^*_\cdot \);

  • For each \(\omega \in \widetilde{\Omega }\), we have \(\mathrm {d}_\mathrm {S}(Z^k_\cdot (\omega ),Z^*_\cdot (\omega ))\xrightarrow {n \rightarrow \infty }0.\)

For \(\gamma \in D([0,\infty ),\mathcal {E})\) and \(t \ge 0\), define \(J_t(\gamma ):= \sup _{s \le t}\Vert \gamma _s - \gamma _{s-}\Vert \), the largest jump size of \(\gamma \) until time t. We have that

$$\begin{aligned} J_t\left( Z^k_\cdot (\omega )\right) \le \frac{2}{n_k}\cdot \max _{x \in {\mathbb V}}\alpha (x) \xrightarrow {k \rightarrow \infty } 0\quad \text { for all }\omega \in \widetilde{\Omega }, \end{aligned}$$

so, by continuity of \(J_t\) in the Skorokhod topology [2,  p. 125], we obtain \(J_t\left( Z^*_t(\omega )\right) = 0\) for all \(\omega \) and t. This implies that the trajectories of \(Z^*_\cdot \) are continuous, and we proved the first part.

Then, using (2.2), we obtain

$$\begin{aligned} \sup _{0\le s \le t} \Vert Z^k_s(\omega )-Z^*_s(\omega )\Vert \xrightarrow {k \rightarrow \infty } 0\quad \text {for all }t \ge 0,\;\omega \in \widetilde{\Omega }. \end{aligned}$$
(4.11)

We now define, for \(f \in \{f_x,f_{xy}:x,y \in {\mathbb V}\}\),

$$\begin{aligned}&M^{k,f}_t := f(Z^k_t) - f(Z^k_0) - \int _0^t \mathcal {L}^{n_k}f(Z^k_s)ds,\\&M^{*,f}_t := f(Z^*_t) - f(Z^*_0) - \int _0^t \mathcal {L}^*f(Z^*_s)ds,\quad t \ge 0. \end{aligned}$$

As observed in Sect. 4.1, we have that \(M^{k,f}_\cdot \) is a local martingale for each k. We now claim that

$$\begin{aligned} \sup _{0\le s \le t} \left| M^{k,f}_s(\omega ) - M^{*,f}_s(\omega )\right| \xrightarrow {k \rightarrow \infty } 0 \quad \text {for all } t \ge 0,\; \omega \in \widetilde{\Omega }. \end{aligned}$$
(4.12)

Let us first show how this convergence will allow us to conclude. Since the trajectories of \(M^{*,f}_\cdot \) are continuous, (2.2) and (4.12) imply that

$$\begin{aligned} \mathrm {d}_\mathrm {s}(M_\cdot ^{k,f}(\omega ),M_\cdot ^{*,f}(\omega ))\xrightarrow {k \rightarrow \infty } 0 \quad \text {for all}~\omega \in \widetilde{\Omega }. \end{aligned}$$

Since almost sure convergence implies convergence in distribution, this gives

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}}(M^{k,f}_\cdot ,M^{*,f}_\cdot )\xrightarrow {k \rightarrow \infty } 0. \end{aligned}$$

Now, Corollary 1.19, page 527 in [6] states that if a sequence of càdlàg local martingales converges in distribution (with respect to the Skorokhod topology), then the limiting process is also a local martingale. We then obtain that \(M^{*,f}_\cdot \) is a local martingale. By Proposition 2.5, this implies that \(Z^*_\cdot \) is a solution to the SDE (1.1).

It remains to prove (4.12). To do so, fix \(f \in \{f_x,f_{xy}:x,y \in \mathbb {V}\}\), \(\omega \in \widetilde{\Omega }\) and \(t \ge 0\). Using (4.11), we obtain that

$$\begin{aligned} \sup _{0 \le s \le t} \Big | f(Z^k_s(\omega )) - f(Z^*_s(\omega )) \Big | \xrightarrow {n \rightarrow \infty }0, \end{aligned}$$

and also that \(A:= \sup \{\Vert Z^k_s(\omega )\Vert : s \le t,\;k \ge 1\}\) is finite. By this latter point and Lemma 4.5, we then have that

$$\begin{aligned} \sup _{0\le s \le t}|\mathcal {L}^{n_k}f(Z^k_s(\omega )) - \mathcal {L}^*f(Z^k_s(\omega ))|\xrightarrow {k \rightarrow \infty }0. \end{aligned}$$

Next, from the generator expressions in (4.7), (4.8) and (4.9), it follows that \(\mathcal {L}^*f\) is uniformly continuous on \(\{\zeta \in \mathcal {E}: \Vert \zeta \Vert \le A\}\); this and (4.11) imply that

$$\begin{aligned} \sup _{0 \le s \le t} \Big | \mathcal {L}^*f(Z^k_s(\omega )) - \mathcal {L}^*f(Z^*_s(\omega )) \Big | \xrightarrow {k \rightarrow \infty } 0. \end{aligned}$$

The desired convergence (4.12) follows. \(\square \)

Convergence to Solutions: Proof of Theorem 1.1

Proof of Theorem 1.1

We split the proof in two parts: first with finite initial condition, then the general case.

Finite case Fix \(\mathring{\zeta }\in \mathcal {E}\) with \(\Vert \mathring{\zeta }\Vert _1 = \sum _{x\in {\mathbb V}} \mathring{\zeta }(x) < \infty \). Also, fix a sequence \(\{\mathring{\zeta }^n\}_{n \ge 1}\) with \(\mathring{\zeta }^n \in \tfrac{1}{n}E\) for each n and \(\Vert \mathring{\zeta }^n - \mathring{\zeta }\Vert \xrightarrow {n \rightarrow \infty } 0\). By Proposition 4.4, there exists a subsequence \(\{\mathring{\zeta }^{n_k}\}_{k \ge 1}\) such that the sequence of processes \(\{\varphi ^{n_k}_\cdot (\mathring{\zeta }^{n_k})\}_{k \ge 1}\) converges in distribution to a process on \(D([0,\infty ),\mathcal {E})\). Let us denote this limiting process by \(\psi ^*_\cdot (\mathring{\zeta })\). By Proposition 4.6, \(\psi ^*_\cdot (\mathring{\zeta })\) has trajectories in \(C([0,\infty ),\mathcal {E})\) and is a solution to (1.1) with initial configuration \(\bar{\zeta } = \mathring{\zeta }\). Then, by Proposition 2.4, any other subsequence of \(\{\varphi ^n(\mathring{\zeta }^n)\}_{n \ge 1}\) which converges in distribution must have the same limit \(\psi ^*_\cdot (\mathring{\zeta })\). This implies that the entire sequence \(\{\varphi ^n(\mathring{\eta }^n)\}_{n \ge 1}\) converges in distribution to \(\psi ^*_\cdot (\mathring{\zeta })\), that is,

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}}\left( \varphi ^n_\cdot (\mathring{\zeta }^n), \;\psi ^*_\cdot (\mathring{\zeta })\right) \xrightarrow {n \rightarrow \infty } 0. \end{aligned}$$

General case Now, fix \({\zeta } \in \mathcal {E}\) and a sequence \(\{\zeta ^n\}_{n \ge 1}\) with \(\zeta ^n \in \tfrac{1}{n}E\) for each n and \(\Vert {\zeta }^n - {\zeta }\Vert \xrightarrow {n \rightarrow \infty } 0\). We claim that the sequence of processes \(\{\varphi ^n_\cdot (\zeta ^n)\}_{n \ge 1}\) is Cauchy with respect to \(\mathrm {d}_\mathrm {LP}\). To see this, fix \(\epsilon > 0\). Next, choose \(\delta > 0\) such that Lemma 4.1 ensures that the left-hand side in (4.2) is smaller than \(\frac{\epsilon }{3}\). Next, because \(\Vert \zeta ^n - \zeta \Vert \rightarrow 0\), we may choose \(R > 0\) such that

$$\begin{aligned} \Vert \zeta ^n \cdot \mathbb {1}_{B_0(R)} - \zeta ^n\Vert < \delta \quad \text { for all}~n. \end{aligned}$$

This implies that

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}} \left( \varphi ^n_\cdot (\zeta ^n \cdot \mathbb {1}_{B_0(R)}),\; \varphi ^{n}_\cdot (\zeta ^{n})\right) < \frac{\epsilon }{3} \quad \text {for all}\, n \in {\mathbb N}. \end{aligned}$$

Finally, using the finite case, with \(\{\zeta ^n \cdot \mathbb {1}_{B_0(R)}\}_{n}\) as approximating sequence to \(\zeta \cdot \mathbb {1}_{B_0(R)}\), which is finite, we may choose \(n_0 \in {\mathbb N}\) such that

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}}\left( \varphi ^n_\cdot (\zeta ^n \cdot \mathbb {1}_{B_0(R)}),\; \varphi ^{n'}_\cdot (\zeta ^{n'} \cdot \mathbb {1}_{B_0(R)})\right) < \frac{\epsilon }{3}\quad \text {for all }n,n' \ge n_0. \end{aligned}$$

By the triangle inequality, it follows that \(\{\varphi ^n_\cdot (\zeta ^n)\}_{n \ge 1}\) is Cauchy.

Now, from the tightness of this sequence, given by Proposition 4.4 (or alternatively, the completeness of the metric space \((D([0,\infty ),\mathcal {E}),\mathrm {d}_{\mathrm {LP}}) \)), we obtain that \(\{\varphi ^n_\cdot (\eta ^n)\}_{n \ge 1}\) converges in distribution to a process on \(D([0,\infty ),\mathcal {E})\), which we denote by \(\psi ^*_\cdot (\zeta )\). To see that this process does not depend on the sequence \(\{\zeta ^n\}_{n \ge 1}\) that we fixed, take an alternative sequence \(\{\tilde{\zeta }^n\}_{n \ge 1}\) with \(\tilde{\zeta }^n \in \frac{1}{n}E\) for each n and \(\Vert \tilde{\zeta }^n - \zeta \Vert \xrightarrow {n \rightarrow \infty }0\), and note that Lemma 4.1 gives

$$\begin{aligned} \lim _{n \rightarrow \infty } \mathrm {d}_{\mathrm {LP}}\left( \varphi ^n_\cdot (\zeta ^n),\; \varphi ^n_\cdot (\tilde{\zeta }^n) \right) \xrightarrow {n \rightarrow \infty }0. \end{aligned}$$

Finally, by Proposition 4.6, \(\psi ^*_\cdot ({\zeta })\) has trajectories in \(C([0,\infty ),\mathcal {E})\), and is a solution to (1.1) with initial configuration \(\bar{\zeta } = {\zeta }\).

It remains to prove part (c) of the statement of the theorem. Fix \(\epsilon > 0\), and choose \(\delta > 0\) corresponding to \(\epsilon \) in Lemma 4.2. Fix \(\zeta ,\zeta ' \in \mathcal {E}\) with \(\Vert \zeta - \zeta '\Vert < \delta /2\). Take sequences \(\{\zeta ^n\}_{n\ge 1}\), \(\{\zeta '^n\}_{n \ge 1}\) with \(\zeta ^n,\zeta '^n \in \tfrac{1}{n}E\) for each n and

$$\begin{aligned} \Vert \zeta ^n - \zeta \Vert \xrightarrow {n \rightarrow \infty } 0, \qquad \Vert \zeta '^n - \zeta '\Vert \xrightarrow {n \rightarrow \infty } 0. \end{aligned}$$

In particular, for n large enough we have \(\Vert \zeta ^n - \zeta '^n\Vert < \delta \), so \(\mathrm {d}_{\mathrm {LP}}(\varphi ^n_\cdot (\zeta ),\varphi ^n_\cdot (\zeta ')) < \epsilon \). By the previous results, we have

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}}(\varphi ^n_\cdot (\zeta ^n),\psi ^*_\cdot (\zeta )) \xrightarrow {n \rightarrow \infty } 0, \qquad \mathrm {d}_{\mathrm {LP}}(\varphi ^n_\cdot (\zeta '^n),\psi ^*_\cdot (\zeta ')) \xrightarrow {n \rightarrow \infty } 0, \end{aligned}$$

so

$$\begin{aligned} \mathrm {d}_{\mathrm {LP}}(\psi ^*_\cdot (\zeta ),\psi ^*_\cdot (\zeta ')) = \lim _{n \rightarrow \infty } \mathrm {d}_{\mathrm {LP}}(\varphi ^n_\cdot (\zeta ^n),\varphi ^n_\cdot (\zeta '^n)) \le \epsilon . \end{aligned}$$

\(\square \)

Tightness: Proof of Proposition 4.4

Aldous’ Criterion

Throughout this section, we fix \(\zeta \) and \(\{\zeta ^n\}_{n \ge 1}\) as in the statement of Proposition 4.4. We will abbreviate

$$\begin{aligned} \zeta ^n_t:= \varphi ^n_t(\zeta ^n),\quad n \ge 1,\; t \ge 0. \end{aligned}$$

We will prove Proposition 4.4 using Aldous’ criterion [2, p. 51]. We need to verify that

$$\begin{aligned}&\forall t \ge 0, \;\forall \epsilon > 0,\; \exists K \subset \mathcal {E} \text { compact}: \;\sup _{n \in {\mathbb N}}\; {\mathbb P}(\zeta ^n_t \notin K) < \epsilon , \text { and} \end{aligned}$$
(5.1)
$$\begin{aligned}&\forall T> 0,\;\forall \epsilon> 0,\quad \lim _{\delta \rightarrow 0}\;\sup _{n \in {\mathbb N}}\;\sup _{\tau \in \mathcal {T}^n_T}\; {\mathbb P}\left( \left\| \zeta ^n_{(\tau +\delta )\wedge T} - \zeta ^n_\tau \right\| > \epsilon \right) = 0, \end{aligned}$$
(5.2)

where \(\mathcal {T}^n_T\) is the set of stopping times (with respect to the natural filtration of \(\zeta ^n_\cdot \)) that are bounded by T.

To verify the first criterion, we will rely on a definition and a lemma. For any \(r > 0\), we define

$$\begin{aligned} \Lambda (r):= \{x \in {\mathbb V}: \;\alpha (x) > 1/r\}. \end{aligned}$$
(5.3)

Lemma 5.1

(Negligible norm near infinity) For any \(T > 0\) and \(\epsilon > 0\), there exists \(R > 0\) such that

$$\begin{aligned} {\mathbb P}\left( \sup _{0 \le t \le T} \Vert \zeta ^n_t \cdot \mathbb {1}_{\Lambda (R)^c}\Vert > \epsilon \right) < \epsilon \quad \text {for all }n \ge 1. \end{aligned}$$

We postpone the proof of this lemma to Sect. 5.2.

Proof of Proposition 4.4, condition (5.1)

Fix \(t \ge 0\) and \(\epsilon > 0\). Using (4.1) with \(\zeta ' \equiv 0\), we obtain that there exists \(A > 0\) such that

$$\begin{aligned} \sup _n {\mathbb P}\left( \Vert \zeta ^n_t\Vert > A\right) \le \frac{\epsilon }{2}. \end{aligned}$$

Furthermore, by Lemma 5.1, for any \(k \in {\mathbb N}\) there exists \(R_k\) such that

$$\begin{aligned} \sup _n {\mathbb P}\left( \Vert \zeta ^n_t \cdot \mathbb {1}_{\Lambda (R_k)^c} \Vert > \frac{1}{k} \right) < \frac{\epsilon }{2^{k+1}}. \end{aligned}$$

Now, defining

$$\begin{aligned} K:= \left\{ \zeta \in \mathcal {E}:\;\Vert \zeta \Vert \le A,\; \Vert \zeta \cdot \mathbb {1}_{\Lambda (R_k)^c}\Vert \le 1/k \text { for all }k \right\} , \end{aligned}$$

we have that \({\mathbb P}(\zeta _t^n \in K) > 1-\epsilon \) for all n.

We claim that K is compact. To verify this, fix a sequence \(\{\zeta ^j\}_{j \ge 1}\) of elements of K. For every \(x \in {\mathbb V}\) we have that \(\zeta ^j(x) \le A/\alpha (x)\) for every j, so, using a diagonal argument, we can obtain a subsequence \(\{\zeta ^{j'}\}_{j' \ge 1}\) so that \(\zeta ^{j'}(x)\) is convergent for each x. Let \(\bar{\zeta }\) be defined by \(\bar{\zeta }(x):= \lim _{j'} \zeta ^{j'}(x)\) for each x. Next, Fatou’s Lemma gives \(\Vert \bar{\zeta }\cdot \mathbb {1}_{\Lambda (R_k)^c}\Vert \le 1/k\) for all k, so \(\bar{\zeta }\in \mathcal {E}\) (since \(\Lambda (R_k)\) is finite) and

$$\begin{aligned} \limsup _{j' \rightarrow \infty } \Vert \bar{\zeta }- \zeta ^{j'}\Vert \le \limsup _{j' \rightarrow \infty } \sum _{x \in \Lambda (R_k)}\alpha (x)\cdot |\bar{\zeta }(x) - \zeta ^{j'}(x)| + \frac{2}{k} = \frac{2}{k}, \end{aligned}$$

which can be made as small as desired by taking k large. This shows that K is compact and completes the proof of (5.1). \(\square \)

For the proof of (5.2), again we will need a preliminary result.

Lemma 5.2

(Oscillation of coordinates) For any \(T > 0\)\(\epsilon > 0\) and \(x \in {\mathbb V}\), we have

$$\begin{aligned} \lim _{\delta \rightarrow 0}\;\sup _{n \in {\mathbb N}}\;\sup _{\tau \in \mathcal {T}_T^n} \; {\mathbb P}\left( \left| \zeta ^n_{(\tau +\delta )\wedge T}(x) - \zeta ^n_\tau (x) \right| > \epsilon \right) = 0. \end{aligned}$$

We postpone the proof of this lemma to Sect. 5.3.

Proof of Proposition 4.4, condition (5.2)

Fix \(T > 0\) and \(\epsilon > 0\). By Lemma 5.1, we may choose \(R> 0\) such that, for any n,

$$\begin{aligned} {\mathbb P}\left( \sup _{0\le t \le T} \Vert \zeta ^n_t \cdot \mathbb {1}_{\Lambda (R)^c}\Vert > \frac{\epsilon }{3}\right) <\frac{\epsilon }{3}. \end{aligned}$$

Next, by Lemma 5.2 and a union bound for \(x \in \Lambda (R)\), we can choose \(\delta _0 > 0\) such that, for any \(\delta \le \delta _0\)\(n \in {\mathbb N}\) and \(\tau \in \mathcal {T}_T^n\), we have

$$\begin{aligned} \mathbb {P}\left( \Vert (\zeta ^n_{(\tau +\delta )\wedge T} - \zeta ^n_t)\cdot \mathbb {1}_{\Lambda (R)} \Vert > \frac{\epsilon }{3} \right) <\frac{\epsilon }{3}. \end{aligned}$$

Therefore, by the triangle inequality,

$$\begin{aligned} {\mathbb P}\left( \Vert \zeta ^n_{(\tau + \delta )\wedge T} - \zeta ^n_t \Vert> \epsilon \right) \le&{\mathbb P}\left( \Vert (\zeta ^n_{(\tau +\delta )\wedge T}- \zeta ^n_\tau ) \cdot \mathbb {1}_{\Lambda (R)}\Vert> \frac{\epsilon }{3} \right) \\&+{\mathbb P}\left( \Vert \zeta ^n_{(\tau +\delta )\wedge T} \cdot \mathbb {1}_{\Lambda (R)^c} \Vert> \frac{\epsilon }{3}\right) \\&+ {\mathbb P}\left( \Vert \zeta ^n_\tau \cdot \mathbb {1}_{\Lambda (R)^c}\Vert > \frac{\epsilon }{3} \right) < \epsilon . \end{aligned}$$

Since \(\epsilon \) is arbitrary, the proof is complete. \(\square \)

Norm Near Infinity: Proof of Lemma 5.1

Let us define

$$\begin{aligned} \zeta ^{n,r}_t:= \varphi ^n_t(\zeta ^n\cdot \mathbb {1}_{\Lambda (r)}),\quad n \ge 1,\;t \ge 0. \end{aligned}$$

We observe that, by (4.1), for any \(\epsilon > 0\)\(T > 0\) and \(n \ge 1\) we have

$$\begin{aligned} {\mathbb P}\left( \sup _{0 \le t \le T} \Vert \zeta ^n_t - \zeta ^{n,r}_t\Vert > \epsilon \right) \le \frac{e^{\mathsf {C}T}\cdot \Vert \zeta _0^n \cdot \mathbb {1}_{\Lambda (r)^c}\Vert }{\epsilon }. \end{aligned}$$
(5.4)

Proof of Lemma 5.1

Fix \(T > 0\) and \(\epsilon > 0\). By (5.4), we can choose r large enough that

$$\begin{aligned} \sup _n {\mathbb P}\left( \sup _{0\le t \le T} \Vert \zeta ^n_t - \zeta ^{n,r}_t\Vert > \frac{\epsilon }{2}\right) \le \frac{\epsilon }{2}. \end{aligned}$$
(5.5)

Next, note that for any nR and t we have

$$\begin{aligned} \Vert \zeta ^{n,r}_t \cdot \mathbb {1}_{\Lambda (R)^c}\Vert = \sum _{x \notin \Lambda (R)} \alpha (x)\cdot |\zeta ^{n,r}_t(x)| {\mathop {\le }\limits ^{(5.3)}} \frac{\Vert \zeta ^{n,r}_t\Vert _1}{R}, \end{aligned}$$

so, for any n and R,

$$\begin{aligned} {\mathbb P}\left( \sup _{0 \le t \le T} \Vert \zeta ^{n,r}_t \cdot \mathbb {1}_{\Lambda (R)^c} \Vert> \frac{\epsilon }{2} \right) \le {\mathbb P}\left( \sup _{0\le t \le T} \Vert \zeta ^{n,r}_t\Vert _1 > \frac{R\epsilon }{2} \right) {\mathop {\le }\limits ^{(3.3)}}\frac{2\Vert \zeta ^{n,r}_0\Vert _1}{R\epsilon }. \end{aligned}$$

Now, the assumption that \(\Vert \zeta _0^n -\zeta ^*\Vert \xrightarrow {n \rightarrow \infty }0\) implies that \(\sup _n \Vert \zeta _0^{n,r}\Vert _1 < \infty \). We thus have

$$\begin{aligned} \sup _n {\mathbb P}\left( \sup _{0 \le t \le T}\Vert \zeta ^{n,r}_t \cdot \mathbb {1}_{\Lambda (R)^c}\Vert > \frac{\epsilon }{2} \right) \le \frac{\epsilon }{2} \end{aligned}$$
(5.6)

if R is large enough. Combining (5.5) and (5.6) with the bound

$$\begin{aligned}&\Vert \zeta ^n_t \cdot \mathbb {1}_{\Lambda (R)^c} \Vert \le \Vert (\zeta ^n_t - \zeta ^{n,r}_t)\cdot \mathbb {1}_{\Lambda (R)^c} \Vert \\&+ \Vert \zeta ^{n,r}_t\cdot \mathbb {1}_{\Lambda (R)^c} \Vert \le \Vert \zeta ^n_t - \zeta ^{n,r}_t\Vert + \Vert \zeta ^{n,r}_t\cdot \mathbb {1}_{\Lambda (R)^c} \Vert \end{aligned}$$

gives the desired bound. \(\square \)

Oscillation of Coordinates: Proof of Lemma 5.2

In the proof of Lemma 5.2, it will be useful to note that, for any \(n \in {\mathbb N}\)\(x \in {\mathbb V}\) and \(A > 0\) we have

$$\begin{aligned} \sup \left\{ |\mathcal {L}^nf_x(\zeta )|\vee |\mathcal {Q}^nf_x(\zeta )|:\; \zeta \in \tfrac{1}{n}E,\; \Vert \zeta \Vert \le A\right\} < \infty . \end{aligned}$$
(5.7)

This follows from the expressions in (4.3), (4.4), the fact that \(\Vert \zeta \Vert \le A\) implies \(|\zeta (x)| \le A / \alpha (x)\), and the bound

$$\begin{aligned} \sum _{y \ne x}(\zeta (y)p(y,x) + \zeta (x)p(x,y)) {\mathop {\le }\limits ^{(2.1)}} \sum _{y \in {\mathbb V}} \zeta (y) \cdot \frac{\mathsf {C}\alpha (y)}{\alpha (x)} + \zeta (x) \le \frac{\mathsf {C}+1}{\alpha (x)}\cdot \Vert \zeta \Vert . \end{aligned}$$
(5.8)

Proof of Lemma 5.2

As noted in Sect. 4.1, writing

$$\begin{aligned} {\mathcal M}^{n,x}_t := \zeta ^n_t(x) - \zeta ^n_0(x) - \int _0^{t}{\mathcal L}^nf_{x}(\zeta ^n_s)\, ds, \qquad \text {for}\, \,t \ge 0, \end{aligned}$$

we have that \({\mathcal M}^{n}_\cdot \) is a local martingale. Its quadratic variation is given by

$$\begin{aligned} \langle {\mathcal M}^{n,x}\rangle _t= & {} \int _0^{t}{\mathcal Q}^nf_{x}(\zeta ^n_s)\, ds \\= & {} \int _0^t a \cdot (\zeta ^n_s(x))^\ell + \sum _{y} \frac{p(y,x)\zeta ^n_s(y) + p(x,y)\zeta ^n_s(x)}{n} \, ds, \quad \text {for}\, t \ge 0. \end{aligned}$$

Now, fix \(T > 0\), \(\epsilon > 0\) and \(\tau \in \mathcal {T}^n_T\). Given \(A > 0\), define \(\tau ^n_A := T \wedge \inf \{\,t \in [0,T]: \left\| \zeta ^n_t \right\| > A\,\}\). We have that, for any \(\delta > 0\),

$$\begin{aligned} \begin{aligned}&{\mathbb P}\left( \left| \zeta ^n_{(\tau +\delta )\wedge T}(x) - \zeta ^n_\tau (x) \right| \ge \varepsilon \right) \\&\le {\mathbb P}\left( \left| {\mathcal M}^n_{(\tau +\delta )\wedge \tau ^n_A} - {\mathcal M}^n_{\tau \wedge \tau ^n_A} \right| \ge \varepsilon /2 \right) \\&\quad + {\mathbb P}\left( \left| \int _{\tau \wedge \tau ^n_A}^{(\tau + \delta ) \wedge \tau ^n_A}{\mathcal L}^nf_{x}(\zeta ^n_s)\,ds \right| \ge \varepsilon /2 \right) + {\mathbb P}(\tau ^n_A<T)\\&\le \frac{4}{\varepsilon ^2} {\mathbb E}\left[ \int _{\tau \wedge \tau ^n_A}^{(\tau + \quad \delta )\wedge \tau ^n_A}{\mathcal Q}^nf_x(\zeta ^n_s)\, ds \right] \\&\quad + {\mathbb P}\left( \left| \int _{\tau \wedge \tau ^n_A}^{(\tau + \delta ) \wedge \tau ^n_A}{\mathcal L}^nf_x(\zeta ^n_s)\,ds \right| \ge \varepsilon /2 \right) + {\mathbb P}(\tau ^n_A <T). \end{aligned} \end{aligned}$$

Now, using (5.7), we can choose \(\delta > 0\) small enough that

$$\begin{aligned}&\frac{4}{\varepsilon ^2} {\mathbb E}\left[ \int _{\tau \wedge \tau ^n_A}^{(\tau + \delta )\wedge \tau ^n_A}{\mathcal Q}^nf_x(\zeta ^n_s)\, ds \right] < \frac{\epsilon }{2} \qquad \text {and}\qquad \\&{\mathbb P}\left( \left| \int _{\tau \wedge \tau ^n_A}^{(\tau + \delta ) \wedge \tau ^n_A}{\mathcal L}^nf_x(\zeta ^n_s)\,ds \right| \ge \varepsilon /2 \right) = 0. \end{aligned}$$

To conclude the proof, we choose \(A > 0\) large enough that \({\mathbb P}(\tau ^n_A< T) < \tfrac{\epsilon }{2}\) . \(\square \)