# Stochastic dynamics of particle systems on unbounded degree graphs

Georgy Chargaziya  Alexei Daletskii

###### Abstract

We consider an infinite system of coupled stochastic differential equations
(SDE) describing dynamics of the following infinite particle system. Each
partricle is characterised by its position $x\in\mathbb{R}^{d}$ and internal
parameter (spin) $\sigma_{x}\in\mathbb{R}$. While the positions of particles
form a fixed (”quenched”) locally-finite set (configuration) $\gamma\subset$
$\mathbb{R}^{d}$, the spins $\sigma_{x}$ and $\sigma_{y}$ interact via a pair
potential whenever $\left|x-y\right|<\rho$, where $\rho>0$ is a fixed
interaction radius. The number $n_{x}$ of particles interacting with a
particle in positionn $x$ is finite but unbounded in $x$. The growth of
$n_{x}$ as $x\rightarrow\infty$ creates a major technical problem for solving
our SDE system. To overcome this problem, we use a finite volume approximation
combined with a version of the Ovsjannikov method, and prove the existence and
uniqueness of the solution in a scale of Banach spaces of weighted sequences.
As an application example, we construct stochastic dynamics associated with
Gibbs states of our particle system.

Keywords: interacting particle systems, infinite systems of stochastic
equations, scale of Banach spaces, Ovsyannikov’s method, dissipativity

2010 Mathematics Subject Classification: 82C20; 82C31; 60H10; 46E99

###### Contents

  1. 1 Introduction
  2. 2 The setup and main results
  3. 3 Existence, uniqueness and properties of the solution.
    1. 3.1 Truncated System
    2. 3.2 One Dimensional Special Case
    3. 3.3 Proof of Existence and Uniqueness
    4. 3.4 Markov semigroup
  4. 4 Stochastic dynamics associated with Gibbs measures
    1. 4.1 Construction of Gibbs measures
    2. 4.2 Construction of the stochastic dynamics
  5. 5 Appendix
    1. 5.1 Linear operators in the spaces of sequences
    2. 5.2 Comparison theorem and Gronwall-type inequality
    3. 5.3 Estimates of the solutions

##  1 Introduction

In recent decades, there has been an increasing interest in studying countable
systems of particles randomly distributed in the Euclidean space
$\mathbb{R}^{d}$. In such systems, each particle is characterized by its
position $x\in X:=\mathbb{R}^{d}$ and an internal parameter (spin)
$\sigma_{x}\in S:=\mathbb{R}^{n}$, see for example [27], [25, Section 11], [6]
and [12, 13], pertaining to modelling of non-crystalline (amorphous)
substances, e.g. ferrofluids and amorphous magnets. Throughout the paper we
suppose, mostly for simplicity, that $n=1$.

Let us denote by $\Gamma(X)$ the space of all locally finite subsets
(configurations) of $X$ and consider a particle system with positions forming
a given fixed (“quenched”) configuration $\gamma\in\Gamma(X)$. Two spins
$\sigma_{x}$ and $\sigma_{y}$, $x,y\in\gamma$, are allowed to interact via a
pair potential if the distance between $x$ and $y$ is no more than a fixed
interaction radius $\rho>0$, that is, they are neighbors in the geometric
graph defined by $\gamma$ and $\rho$. The evolution of spins is described then
by a system of coupled stochastic differential equations.

Namely, we consider, for a fixed $\gamma\in\Gamma(X)$, a system of stochastic
differential equations in $S=\mathbb{R}^{1}$ of the following form:

$d\xi_{x,t}=\Phi_{x}(\Xi_{t})dt+\Psi_{x}(\Xi_{t})dW_{x,t},\ \quad x\in\gamma,\
t\geq 0,$ (1.1)

where $\Xi_{t}=\left(\xi_{x,t}\right)_{x\in\gamma}$ and
$\left(W_{x,t}\right)_{x\in\gamma}$ are, respectively, families of real-valued
stochastic processes and independent Wiener processes on a suitable
probability space. Here the drift and diffusion coefficients $\Phi_{x}$ and
$\Psi_{x}$ are real-valued functions, defined on the Cartesian power
$S^{\gamma}:=\left\\{\bar{\sigma}=(\sigma_{x})_{x\in\gamma}\left|\sigma_{x}\in
S,x\in\gamma\right.\right\\}$. Both $\Phi_{x}$ and $\Psi_{x}$ are constructed
using pair interaction between the particles and their self-interaction
potentials, see Section 2, and are independent of $\sigma_{y}$ if
$\left|y-x\right|>\rho$.

The aim of including the diffusion term in (1.1) is two-fold. On the one hand,
it allows to consider the influence of random forces on our particle system
and, on the other hand, to construct and study stochastic dynamics associated
with the equilibrium (Gibbs) states of the system. The Gibbs states of spin
systems on unbounded degree graphs have been studied in [22, 12, 13], see also
references given there.

The case where vertex degrees of the graph are globally bounded (in
particular, if $\gamma$ has a regular structure, e.g. $\gamma=\mathbb{Z}^{d}$)
has been well-studied (in both deterministic and stochastic cases), see e.g.
[24, 15, 4, 5, 2, 3, 19], and references therein. However, aforementioned
applications to non-crystalline substances require to deal with unbounded
vertex degree graphs. An important example of such graphs is served by
configurations $\gamma$ distributed according to a Poisson or, more generally,
Gibbs measure on $\Gamma(X)$ with a superstable low regular interaction
energy, in which case the typical number of “neighbors” of a particle located
at $x\in X$ is proportional to $\sqrt{1+\log|x|}$, see e.g. [28] and [21,
p.1047].

There are two main technical difficulties in the study of system (1.1). The
first one is related to the fact that the number of particles interacting with
a tagged particle $x$ is finite but unbounded in $x\in\gamma$. As a
consequence, the system cannot be considered as an equation in a fixed Banach
space and studied by standard methods of e.g. [11], [14]. The way around it
has been proposed in [10], where a system of (deterministic) equations was
considered in an expanding scale of embedded Banach spaces and solved using a
version of the Ovsjannikov method. Stochastic equations in a scale of Banach
spaces were considered in [8] and [7].

The second difficulty, which is related to the following. The results of the
latter two works are applicable to system (1.1) only in the case where the
drift coefficients $\Psi_{x}$, $x\in\gamma$, are globally Lipschitz. However,
in order to construct the dynamics associated with a Gibbs state of
interacting particle system, one has to consider the drift coefficients that
are only locally Lipschitz. The existence of such dynamics, under certain
dissipativity condition of the drift, is well-known in the situation of a
regular lattice, see [4, 5] (observe that these works deal with the more
complicated quantum systems but are applicable to classical systems, too,
albeit only for the additive noise). For deterministic systems on unbounded
degree graphs, the dissipative case was considered in [10]. In the present
work, we combine aforementioned approaches and prove the existence and
uniqueness result for the system (1.1) in the case of dissipative single-
particle potentials.

The structure of the paper is as follows. In Section 2 we introduce the
framework and formulate our main results. Section 3 is devoted to the proof of
the existence and uniqueness result for (1.1). In a short Section 3.4, we
discuss Markov semigroup generated by the solution of (1.1). In Section 4, we
study stochastic dynamics associated with Gibbs states of our system.

Finally, the Appendix contains auxiliary results on linear operators in the
scales of Banach spaces, estimates of the solutions of system (1.1) and,
notably, a crucial for our techniques generalization of the classical
comparison theorem and a Gronwall-type inequality, suitable for our framework.

Acknowledgment. We are very gratefull to Zdzislaw Brzezniak, Dmitry
Finkelshtein, Jiang-Lun Wu and Yuri Kondratiev for their interest in this work
and stimulating discussions.

##  2 The setup and main results

Our aim is to find a strong solution of SDE system (1.1), that is, a solution
of a system of integral equations

$\displaystyle\xi_{x,t}$
$\displaystyle=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\Xi_{s})ds+\int_{0}^{t}\Psi_{x}(\Xi_{s})dW_{x,s},\
x\in\gamma,$ (2.1) $\displaystyle\Xi_{t}$
$\displaystyle=\left(\xi_{x,t}\right)_{x\in\gamma},\ t\in\mathcal{T}:=[0,T],\
T>0,\ \zeta_{x}\in S,$

in $S=\mathbb{R}$, where $\gamma\in\Gamma(X),$ that is, $\gamma$ is a locally
finite (countable) subset of $X=\mathbb{R}^{d}$, and
$\left(W_{x,t}\right)_{x\in\gamma}$ is a family of independent Wiener
processes on a suitable filtered and complete probability space
$\mathbf{P}:=(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$. The coefficients
$\Phi_{x}$ and $\Psi_{x}$ are defined explicitly in Assumption 1 below.

First we need to introduce some notations. We fix $\rho>0$ and denote by
$n_{x}$, $x\in\gamma$, the number of elements in the set

$\bar{\gamma}_{x}:=\left\\{y\in\gamma:\left|x-y\right|\leq\rho\right\\}.$

Observe that $n_{x}\geq 1$ for all $x\in\gamma$, because
$x\in\bar{\gamma}_{x}$. We will also use the notation
$\gamma_{x}:=\bar{\gamma}_{x}\setminus\left\\{x\right\\}\equiv\left\\{y\in\gamma:\left|x-y\right|\leq\rho,y\neq
x\right\\}$.

For a fixed $\gamma\in\Gamma(X)$, we will consider the Cartesian product
$S^{\gamma}$ of identical copies $S_{x}$, $x\in\gamma$, of $S$, and denote its
elements by $\bar{z}:=\left(z_{x}\right)_{x\in\gamma}$, etc. When dealing with
multiple configurations $\eta\in\Gamma(X)$, we will sometimes write
$\bar{z}_{\eta}:=\left(z_{x}\right)_{x\in\eta}$, to emphasize the dependence
on $\eta$.

We will work under the following assumption.

###### Assumption 1

  1. (A)

There exists a constant $C>0$ such that

$n_{x}\leq C(1+\log(1+|x|))\ \text{for all }x\in\gamma;$ (2.2)

  2. (B)

the drift coefficients $\Phi_{x},\ x\in\gamma$, have the form

$\Phi_{x}(\bar{z}):=\phi(z_{x})+\sum_{y\in\gamma_{x}}a(x-y)z_{y},\text{for all
}x\in\gamma,$ (2.3)

where $\phi:S\rightarrow\mathbb{R}$ and $a:X\rightarrow\mathbb{R}$ are
measurable functions; moreover, $\left|a\right|$ is bounded by a constant
$\bar{a}>0$;

  3. (C)

there exist constants $c,R>0$ such that

$|\phi(\sigma)|\leq c(1+|\sigma|^{R}),\ \sigma\in S;$ (2.4)

  4. (D)

there exists $b>0$ such that

$(\sigma_{1}-\sigma_{2})(\phi(\sigma_{1})-\phi(\sigma_{2}))\leq
b(\sigma_{1}-\sigma_{2})_{.}^{2},\ \sigma_{1},\sigma_{2}\in S;$ (2.5)

  5. (E)

the diffusion coefficients $\Psi_{x},\ x\in\gamma$, have the form

$\Psi_{x}((z_{y})_{y\in\gamma}):=\sum_{y\in\bar{\gamma}_{x}}\psi_{xy}(z_{x},z_{y})\
\text{for all }x\in\gamma,$ (2.6)

where $\psi_{xy}:S^{2}\rightarrow\mathbb{R}^{1}$ are measurable functions
satisfying uniform Lipschitz condition

$\left|\psi_{xy}(\sigma_{1},s_{1})-\psi_{xy}(\sigma_{2},s_{2})\right|\leq
M\left(\left|\sigma_{1}-\sigma_{2}\right|+\left|s_{1}-s_{2}\right|\right),$
(2.7)

for some constant $M>0$ and all $x,y\in\gamma,\
\sigma_{1},\sigma_{2},s_{1},s_{2}\in\mathbb{R}.$

The specific form of the coefficients requires development of a special
framework. Indeed, we will be looking for a solution of (2.1) in a scale of
expanding Banach spaces of weighted sequences, which we introduce below.

We start with a general definition and consider a family $\mathfrak{B}$ of
Banach spaces $B_{\alpha}$ indexed by
$\alpha\in\mathcal{A}:=[\alpha_{\ast},\alpha^{\ast})$ with fixed
$0\leq\alpha_{\ast},\alpha^{\ast}<\infty$, and denote by
$\left\|\cdot\right\|_{B_{\alpha}}$ the corresponding norms. When speaking of
these spaces and related objects, we will always assume that the range of
indices is $[\alpha_{\ast},\alpha^{\ast})$, unless stated otherwise. The
interval $\mathcal{A}$ remains fixed for the rest of this work. We will also
use the corresponding closed interval
$\mathcal{\bar{A}}:=[\alpha_{\ast},\alpha^{\ast}].$

###### Definition 2

The family $\mathfrak{B}$ is called a scale if

$B_{\alpha}\subset B_{\beta}\ {\text{and
}}\left\|u\right\|_{B_{\beta}}\leq\left\|u\right\|_{B_{\alpha}}{\text{ for any
}}\alpha<\beta,\ u\in B_{\alpha},\ \alpha,\beta\in\mathcal{\bar{A}},$

where the embedding means that $B_{\alpha}$ is a dense vector subspace of
$B_{\beta}$.

For any $\ \alpha,\beta\in\mathcal{A}$, we will use the notation

$B_{\alpha+}:={\bigcap}_{\beta>\alpha}B_{\beta}.$

The two main scales we will be working with are given by the spaces
$l_{\alpha}^{p}$ of weighted sequences and $l_{\alpha}^{p}$-valued random
processes, respectively, defined as follows.

  * (1)

For all $p\geq 1$ and $\alpha\in\mathcal{\bar{A}}$ let

$\displaystyle l_{\mathfrak{\alpha}}^{p}$ $\displaystyle:=\left\\{\bar{z}\in
S^{\gamma}\ \left|\
\|\bar{z}\|_{l_{\alpha}^{p}}:=\left(\sum_{x\in\gamma}e^{-a|x|}|z_{x}|^{p}\right)^{\frac{1}{p}}<\infty\right.\right\\}$
(2.8) $\displaystyle\text{and }\mathcal{L}^{p}$
$\displaystyle:=\\{l_{\alpha}^{p}\\}_{\mathfrak{\alpha}\in\mathcal{A}}$

be, respectively, a Banach space of weighted real sequences and the scale of
such spaces.

  * (2)

For all $p\geq 1$ and $\alpha\in\mathcal{\bar{A}}$ let
$\mathcal{R}_{\alpha}^{p}$ denote the Banach space of $l_{\alpha}^{p}$-valued
adapted random processes $\bar{\xi}_{t},\ t\in\mathcal{T}$, on probability
space $\mathbf{P}$, with finite norm

$\|\bar{\xi}\|_{\mathcal{R}_{\alpha}^{p}}:=\left(\sup\left\\{\mathbb{E}\|\xi_{t}\|_{l_{\alpha}^{p}}^{p}\
\left|\ t\in\mathcal{T}\right.\right\\}\right)^{\frac{1}{p}}<\infty,$

and let
$\mathcal{R}^{p}:=\\{\mathcal{R}_{\alpha}^{p}\\}_{\alpha\in\mathcal{A}}$ be
the scale of such spaces.

Our main result is the following theorem.

###### Theorem 3

Suppose that Assumption 1 holds. Then, for all $p\geq\max\left\\{R,2\right\\}$
and any initial condition $\bar{\zeta}:=(\zeta_{x})_{x\in\gamma}\in
l_{\alpha}^{p}$, stochastic system (2.1) admits a unique solution
$\Xi\in\mathcal{R}_{\alpha+}^{p}$. Moreover, the map

$l_{\alpha}^{p}\ni\bar{\zeta}\mapsto\Xi\in\mathcal{R}_{\beta}^{p}$

is continuous for any $\beta>\alpha$.

###### Remark 4

Assumption $p\geq R$ ensures that given $\bar{\xi}\in\mathcal{R}_{\beta}^{p}$
the random variable $\phi(\bar{\xi}_{t})$ is integrable for any $t\geq 0$.

The proof of Theorem 3 will be given in Section 3.

Our second main result is about the construction of non-equlibrium stochastic
dynamics associated with Gibbs states of our system. We consider a Gibbs
measure $\nu$ on $S^{\gamma}$ defined by the pair interaction
$W_{x,y}(\sigma_{x},\sigma_{y})=a(x-y)\sigma_{x}\sigma_{y}$,
$\sigma_{x},\sigma_{y}\in S$, $x,y\in\gamma$, where $a$ is as in (2.3), and a
single particle potential $V:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the
lower bound

$V(\sigma)\geq a_{V}\left|\sigma\right|^{2+\varepsilon}-b_{V},\ \sigma\in S,\
\text{for some }a_{V},b_{V}>0\text{ and }\varepsilon>R-2,$

which is supported on $l_{\alpha}^{p}$ for some $\alpha\in\mathcal{A}$ and
$p\in\left[\max\left\\{R,2\right\\},2+\varepsilon\right]$, see Section 4.1 for
details. Suppose now that $\phi$ in (2.3) has a gradient form, that is,
$\phi=-\nabla V$, and $\left\\{\begin{array}[]{c}\psi_{xy}=0,\ x\neq y\\\
\psi_{xx}=1\end{array}\right.$ for all $x,y\in\gamma$, so that our noise is
additive, cf. (2.6). Let $T_{t}$ be the Markov semigroup defined by the
process $\Xi_{t}$ in a standard way. This semigroup acts in the space
$C_{b}(l_{\alpha+}^{p})$ of bounded continuous functions on space
$l_{\alpha+}^{p}=\cap_{\beta>\alpha}l_{\beta}^{p}$ equipped with the
projective limit topology, see Section 3.4 below for details.

###### Theorem 5

Gibbs measure $\nu$ is a symmetrizing measure for the solution of (2.1), that
is,

$\int T_{t}f(\bar{\zeta})\nu(d\bar{\zeta})=\int
f(\bar{\zeta})\nu(d\bar{\zeta})$

for any $\alpha\in\mathcal{A}$ and $f\in C_{b}(l_{\alpha+}^{p})$.

The proof of this result will be given in Section 4.2.

From now on, the constant $p\geq\max\left\\{R,2\right\\}$ will be fixed.

##  3 Existence, uniqueness and properties of the solution.

In this section we give the proof of Theorem 3. It will go along the following
lines.

  1. (1)

Consider a sequence of processes
$\left\\{\Xi_{t}^{n}\right\\}_{n\in\mathbb{N}}$, $t\in\mathcal{T}$, that solve
finite cutoffs of system (2.1), and prove their uniform bound in
$\mathcal{R}_{\beta}^{p}$ for any $\beta>\alpha$. For this, we use our version
of the comparison theorem and Gronwall-type inequality in the scale of spaces,
which is in turn based on the Ovsjannikov method, see Appendix 5.3.

  2. (2)

The uniform bound above implies the convergence of sequence $\Xi^{n}$,
$n\rightarrow\infty$, to a process
$\Xi=(\xi_{x})_{x\in\gamma}\in\mathcal{R}_{\beta}^{p}$, $\beta>\alpha$. Our
next goal is to prove that the process $\Xi$ solves system (2.1). The
multiplicative noise term does not allow to achieve this by a direct limit
transition. Therefore, we construct an $\mathbb{R}$-valued process $\eta_{t}$
that solves an equation describing the dynamics of a tagged particle $x$,
while processes $\xi_{y,t}$, $y\in\gamma,y\neq x$, are fixed, and prove that
$\eta_{t}=\xi_{x,t}$.

  3. (3)

The uniqueness and continuous dependence on the initial data is proved by
using our version of a Gronwall-type inequality, as above in part (1).

Finally, in Subsection 3.4, we introduce Markov semigroup defined by the
solution of (2.1).

###  3.1 Truncated System

Let us fix an expanding sequence $\\{\Lambda_{n}\\}_{n\in\mathbb{N}}$ of
finite subsets of $\gamma$ such that $\Lambda_{n}\uparrow\gamma$ as
$n\rightarrow\infty$ and consider the following system of equations:

$\displaystyle\xi_{x,t}^{n}$ $\displaystyle=$
$\displaystyle\zeta_{x}+\int_{0}^{t}\Phi_{x}(\Xi_{s}^{n})ds+\int_{0}^{t}\Psi_{x}(\Xi_{s}^{n})dW_{x,s},\
x\in\Lambda_{n},$ (3.1) $\displaystyle\xi_{x,t}^{n}$ $\displaystyle=$
$\displaystyle\zeta_{x},\quad x\not\in\Lambda_{n},\ \ t\in\mathcal{T}.$

Observe that for each $n\in\mathbb{N}$ system (3.1) is a truncated version of
our original stochastic system (2.1).

Assume that $\zeta=(\zeta_{x})_{x\in\gamma}\in l_{\alpha}^{p}$ for some
$\alpha\in\mathcal{A}$.

###### Theorem 6

For any $n\in\mathbb{N}$ system (3.1) admits a (classical strong) solution
$\Xi^{n}\in\mathcal{R}_{\alpha}^{p}$.

Proof. The existence of strong solutions of the non-trivial finite part of
system (3.1) is well-known, see [1, Theorem 3.1], where the case of $p=2$ is
considered, and also references given there. The proof for $p>2$ is similar.
The inclusion $\Xi^{n}\in\mathcal{R}_{\alpha}^{p}$ follows then from the fact
that $\xi_{x,t}^{n}=\zeta_{x},\ t\in\mathcal{T}$, for $x\not\in\Lambda_{n}$.

Our next goal is to show that the sequence $\\{\Xi^{n}\\}_{n\in\mathbb{N}}$
converges in $\mathcal{R}_{\beta}^{p}$ for any $\beta>\alpha$. We start with
the following uniform estimate, which is rather similar to the one from [22],
adapted to the framework of the scale of Banach spaces using our version of
the Gronwall inequality.

###### Theorem 7

Let $\Xi^{n}=(\xi_{x}^{n})_{x\in\gamma}$, $n\in\mathbb{N}$, be the sequence of
process defined by Theorem 6. Then for all $\beta>\alpha$ we have

$\sum_{x\in\gamma}e^{-\beta|x|}\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}<\infty.$
(3.2)

Proof. It follows from the first part of Lemma 22 in Appendix (with
$\xi^{1}\equiv\xi^{n}$) that for all $x\in\Lambda_{n}$ and $t\in\mathcal{T}$
we have

$\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}\leq|\zeta_{x}|^{p}+C_{1}n_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}\mathbb{E}\bigg{[}|\xi_{y,s}^{n}|^{p}\bigg{]}ds+C_{2}.$
(3.3)

We remark that inequality above trivially holds for $x\notin\Lambda_{n}$,
because in this case $\xi_{x,t}^{n}=\zeta_{x}$ and all terms in the right-hand
side of the inequality are non-negative.

We now define a measurable map $\varphi^{n}:\mathcal{T}\rightarrow
l_{\alpha_{\ast}}^{1}$ via the following formula

$\varphi_{x}^{n}(t):=\max_{m\leq n}\
\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]},\quad\forall(t\in\mathcal{T}).$

It is immediate that its components satisfy inequality similar to (3.3), that
is,

$\varphi_{x}^{n}(t)\leq|\zeta_{x}|^{p}+C_{1}n_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}\varphi_{y}^{n}(s)ds+C_{2}.$

Set $\bar{\gamma}_{x}=|\zeta_{x}|^{p}+C_{2}$ and observe that
$(\bar{\gamma}_{x})_{x\in\gamma}\in l_{\alpha}^{1}$. Then the map
$\varphi^{n}$ fulfills the conditions of Lemma 23 in Appendix, which implies
that for all $n\in\mathbb{N}$ and $\beta>\alpha$ we have

$\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}\varphi_{x}^{n}(t)\leq
K(\alpha,\beta)\sum_{x\in\gamma}e^{-\alpha|x|}b_{x}<\infty.$

Observe that the left-hand side forms an increasing sequence, which implies
that it converges and

$\lim_{n\rightarrow\infty}\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}\varphi_{x}^{n}(t)\leq
K(\alpha,\beta)\sum_{x\in\gamma}e^{-\alpha|x|}b_{x}<\infty.$

Then, for any finite set $\eta\subset\gamma$, we have

$\bigg{\\{}\sum_{x\in\eta}e^{-\beta|x|}\lim_{n\rightarrow\infty}\sup_{t\in\mathcal{T}}\varphi_{x}^{n}(t)\bigg{\\}}=\lim_{n\rightarrow\infty}\bigg{\\{}\sum_{x\in\eta}e^{-\beta|x|}\sup_{t\in\mathcal{T}}\varphi_{x}^{n}(t)\bigg{\\}}\\\
\leq K(\alpha,\beta)\sum_{x\in\gamma}e^{-\alpha|x|}b_{x}.$

On the other hand, it is clear that

$\lim_{n\rightarrow\infty}\varphi_{x}^{n}(t)=\sup_{n\in\mathbb{N}}\max_{m\leq
n}\sup_{t\in\mathcal{T}}\
\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}=\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}$

for any $x\in\gamma.$ Thus

$\sum_{x\in\eta}e^{-\beta|x|}\sup_{t\in\mathcal{T}}\sup_{n\in\mathbb{N}}\mathbb{E}\bigg{[}|\xi_{x,t}^{m}|^{p}\bigg{]}\leq
K(\alpha,\beta)\sum_{x\in\gamma}e^{-\alpha|x|}b_{x}.$

The latter inequality holds for all finite $\eta\subset\gamma$, which implies
that

$\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}\sup_{n\in\mathbb{N}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}\leq
K(\alpha,\beta)\sum_{x\in\gamma}e^{-\alpha|x|}b_{x}$

and the proof is complete.

###### Theorem 8

The sequence $\\{\Xi^{n}\\}_{n\in\mathbb{N}}$ is Cauchy in
$\mathcal{R}_{\beta}^{p}$ for any $\beta>\alpha$.

Proof. Let us fix $n,m\in\mathbb{N}$ and and assume, without loss of
generality, that $\Lambda_{n}\subset\Lambda_{m}$. We first consider the
situation where $x\in\Lambda_{n}$. It follows from the second part of Lemma 22
in Appendix (with $\xi^{(1)}\equiv\xi^{n}$ and $\xi^{(2)}\equiv\xi^{m}$) that
for all $x\in\Lambda_{n}$ and $t\in\mathcal{T}$ we have

$\displaystyle\mathbb{E}|\bar{\xi}_{x,t}^{n,m}|^{p}$ $\displaystyle\leq
Bn_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}\mathbb{E}|\bar{\xi}_{y,s}^{n,m}|^{p}ds,$
(3.4) $\displaystyle\bar{\xi}_{x,t}^{n,m}$
$\displaystyle=\xi_{x,t}^{n}-\xi_{x,t}^{m}.$ (3.5)

In the case where $x\in\Lambda_{m}\setminus\Lambda_{n}$ we see that for all
$t\in\mathcal{T}$

$|\bar{\xi}_{x,t}^{n,m}|^{p}\leq(|\xi_{x,t}^{n}|+|\xi_{x,t}^{m}|)^{p}\leq
2^{p-1}|\xi_{x,t}^{n}|^{p}+2^{p-1}|\xi_{x,t}^{m}|^{p},$

so that

$\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}^{n,m}|^{p}\bigg{]}\leq
2^{p}\sup_{n\in\mathbb{N}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}\leq
2^{p}1_{\Lambda_{m}\setminus\Lambda_{n}}(x)\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}<\infty$
(3.6)

(cf. Theorem 7). Combining equations (3.4) and (3.6) and taking into account
that $\bar{\xi}_{x,t}^{n,m}=0$ for $x\notin\Lambda_{m}$, we obtain the
inequality

$\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}^{n,m}|^{p}\bigg{]}\leq
B_{1}n_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}\mathbb{E}\bigg{[}|\bar{\xi}_{y,s}^{n,m}|^{p}\bigg{]}ds+2^{p}1_{\Lambda_{m}\setminus\Lambda_{n}}(x)\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}$

for all $x\in\gamma$ and $t\in\mathcal{T}$ . We can now proceed as in the
proof of Theorem 7. Define a measurable map
$\varrho^{n,m}:\mathcal{T}\rightarrow l_{\alpha}^{1}$ via the formula

$\varrho_{x}^{n,m}(t):=\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}^{n,m}|^{p}\bigg{]},\quad
t\in\mathcal{T},$

and set

$b_{x}=2^{p}1_{\Lambda_{m}\setminus\Lambda_{n}}(x)\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}.$

Obviously, $(\bar{\gamma}_{x})_{x\in\gamma}\in l_{\alpha^{\prime}}^{1}$ for
any $\alpha^{\prime}$. Fix $\alpha^{\prime}\in(\alpha,\beta)$. It follows then
from Lemma 23 in Appendix that

$\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}\varrho_{x}^{n,m}(t)\leq
K(\alpha^{\prime},\beta)\sum_{x\in\gamma}e^{-\alpha^{\prime}|x|}b_{x}.$

So we have shown that the following inequality holds:

$\|\Xi^{n}-\Xi^{m}\|_{\mathcal{R}_{\beta}^{p}}^{p}\leq
2^{p}K(\alpha^{\prime},\beta)\sum_{x\in\Lambda_{m}\setminus\Lambda_{n}}e^{-\alpha^{\prime}|x|}\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}\\\
\leq
2^{p}K(\alpha^{\prime},\beta)\sum_{x\in\gamma-\Lambda_{n}}e^{-\alpha^{\prime}|x|}\sup_{n\in\mathbb{N}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\xi_{x,t}^{n}|^{p}\bigg{]}.$
(3.7)

It follows from Theorem 7 that the right hand side of (3.7) is the remainder
of the convergent series (3.2) (with $\alpha^{\prime}$ in place of $\beta$),
which completes the proof.

###  3.2 One Dimensional Special Case

We have shown in the previous section that, for any $\beta>\alpha$, the
sequence $\\{\Xi^{n}\\}_{n\in\mathbb{N}}$ is Cauchy in the Banach space
$\mathcal{R}_{\alpha}^{p}$ and thus converges in this space. So we are now in
a position to define the process

$\overbrace{\ \Xi:=\lim_{n\rightarrow\infty}\Xi^{n}\ }^{\text{in}\
\mathcal{R}_{\beta}^{p}}.$ (3.8)

This process is a candidate for a solution of the system (2.1). A standard way
to show this would be to pass to the limit in both sides of (3.1). This
approach requires however somewhat stronger convergence than that in
$\mathcal{R}_{\beta}^{p}$. We are going to overcome this difficulty by
considering special one-dimensional equations.

Consider an arbitrary $x\in\gamma$. It is convenient to consider elements of
$S^{\gamma}$ as pairs $(\sigma_{x},Z^{(x)})$, where $\sigma_{x}\in S$ and
$Z^{(x)}=\left(z_{y}\right)_{y\in\gamma\backslash x}\in S^{\gamma\backslash
x}$. In these notations, we can write
$\Phi_{x}(\Xi_{s})=\Phi_{x}(\xi_{x,s},\Xi_{s}^{(x)})$ and
$\Psi_{x}(\Xi_{s})=\Psi_{x}(\xi_{x,s},\Xi_{s}^{(x)})$, where
$\Xi^{(x)}:=\left(\xi_{y}\right)_{y\in\gamma\backslash x}$.

Let now fix process $\Xi$ defined by (3.8) and consider the following one-
dimensional equation:

$\eta_{x,t}=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\eta_{x,s},\Xi_{s}^{(x)})ds+\int_{0}^{t}\Psi_{x}(\eta_{x,s},\Xi_{s}^{(x)})dW_{x}(s),\
t\in\mathcal{T}.$ (3.9)

The main goal of this section is to prove that the equation (3.9) has a unique
solution $\eta_{x,t}$.

###### Remark 9

Note that, for a fixed $x\in\gamma$, the principal difference between
equations (3.9) and (2.1) is that the process $\Xi$ is fixed in (3.9) and
defined by the limit (3.8), which makes (3.9) a one-dimentional equation
w.r.t. $\eta_{x}$.

In order to establish existence of a strong solution of equation (3.9) we need
the following auxiliary result.

###### Theorem 10

Let $x\in\gamma$ and $\xi_{x}$ be an $x$-component of the process $\Xi$
defined by (3.8). Then

$\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\xi_{x,t}|^{p}\bigg{]}<\infty.$
(3.10)

Proof. It is sufficient to show that, for a fixed $x\in\gamma$, the sequence
$\left\\{\xi_{x}^{n}\right\\}_{n\in\mathbb{N}}$, is Cauchy in the norm
$\left(\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\cdot|^{p}\right)^{1/p}$.

Fix $\bar{N}\in\mathbb{N}$ such that $x\in\Lambda_{\bar{N}}$ and
$n,m\geq\bar{N}$ and assume, without loss of generality, that $n<m$ so that
$x\in\Lambda_{n}\subset\Lambda_{m}$ Consider the process
$\bar{\xi}_{x,t}^{n,m}$ defined in (3.5) and proceed as in Appendix, Lemma 22,
with $\xi^{(1)}\equiv\xi^{n}$ and $\xi^{(2)}\equiv\xi^{m}$. Taking
$\sup_{t\in\mathcal{T}}$ of both sides of the equality (5.10) we obtain the
bound

$\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\bar{\xi}_{x,t}^{n,m}|^{p}\bigg{]}\leq
K+\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}\int_{0}^{t}p(\bar{\xi}_{x,s}^{n,m})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s)\bigg{]},$
(3.11)

where

$K:=Bn_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{T}\mathbb{E}\bigg{[}|\bar{\xi}_{y,s}^{n,m}|^{p}\bigg{]}ds\leq
Bn_{x}^{2}T\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}_{y,t}^{n,m}|^{p}\bigg{]}$
(3.12)

and

$\Psi_{x}^{n,m}(s):=\left(\Psi_{x}(\xi_{s}^{n})-\Psi_{x}(\xi_{s}^{m})\right).$

Now using first the Burkholder-Davis-Gundy inequality and then the Jensen
inequality we see that the following estimate on the stochastic term from
(3.11) holds.

$\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}\int_{0}^{t}p(\bar{\xi}_{x,s}^{n,m})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s)\bigg{]}\leq\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}\bigg{(}p(\bar{\xi}_{x,s}^{n,m})^{p-1}\Psi_{x}^{n,m}(s)\bigg{)}^{2}ds\bigg{)}^{\frac{1}{2}}\bigg{]}\\\
\leq\bigg{(}\mathbb{E}\bigg{[}\int_{0}^{t}\bigg{(}p(\bar{\xi}_{x,s}^{n,m})^{p-1}\Psi_{x}^{n,m}(s)\bigg{)}^{2}ds\bigg{]}\bigg{)}^{\frac{1}{2}}.$
(3.13)

The integrand in the right-hand side of the above inequality can be estimated
in a similar way as (5.9), so that we obtain

$\bigg{(}(\bar{\xi}_{x,t}^{n,m})^{p-1}\Psi_{x}^{n,m}(t)\bigg{)}^{2}\leq
2M^{2}n_{x}^{2}|\bar{\xi}_{x,t}^{n,m}|^{2p}+2M^{2}n_{x}^{4}\sum_{y\in\bar{\gamma}_{x}}|\bar{\xi}_{y,t}^{n,m}|^{2p}.$

It follows now that inequality (3.13) can be written in the following way:

$\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}\int_{0}^{t}p(\bar{\xi}_{x,s}^{n,m})^{p-1}\Psi_{x}^{n,m}(s)dW_{x}(s)\bigg{]}\leq
C_{1}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}^{n,m}|^{2p}\bigg{]}\\\
+C_{2}\sum_{y\in\bar{\gamma}_{x}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}^{n,m}|^{2p}\bigg{]},$

where

$C_{1}:=2p^{2}M^{2}n_{x}^{2}T\quad\text{and}\quad
C_{2}:=2p^{2}M^{2}n_{x}^{4}T.$

Therefore returning to inequalities (3.11) and (3.12) we see that

$\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\bar{\xi}_{x,t}^{n,m}|^{p}\bigg{]}\leq
Bn_{x}^{2}T\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}_{y,t}^{n,m}|^{p}\bigg{]}\\\
+C_{1}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}^{n,m}|^{2p}\bigg{]}+C_{2}\sum_{y\in\bar{\gamma}_{x}}\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}^{n,m}|^{2p}\bigg{]}.$
(3.14)

Since $\bar{\gamma}_{x}$ is finite we can now use Theorem 8 to conclude that,
with a suitable choice of $n,m\in\mathbb{N}$, the right hand side of the
inequality (3.14) above can be made arbitrary small hence the proof is
complete.

###### Theorem 11

Equation (3.9) admits a unique strong solution.

Proof. By standard arguments, see e.g. [1, Proposition 2.9], we conclude that
equation (3.9) admits a unique local maximal solution $\eta_{x}$ such that for
all $t\in[0,\infty)$

$\eta_{x,t\wedge\tau_{n}}=\zeta_{x}+\int_{0}^{t\wedge\tau_{n}}\Phi_{x}(\eta_{x,s\wedge\tau_{n}},\Xi_{s\wedge\tau_{n}}^{(x)})ds+\int_{0}^{t\wedge\tau_{n}}\Psi_{x}(\eta_{x,s\wedge\tau_{n}},\Xi_{s\wedge\tau_{n}}^{(x)})dW_{x}(s),$

where by construction, for all $n\in\mathbb{N}$, stopping time $\tau_{n}$ is
the first exit time of $\eta_{x}$ from the interval $(-n,\ n)$, defined as

$\tau_{n}=\left\\{\begin{array}[]{c}T,\text{ if
}\left|\eta_{t}\right|<n,~{}t\in[0,T]\\\
\inf\left\\{t\in[0,T]:\left|\eta_{t}\right|\geq n\right\\},\
\text{otherwise}\end{array}\right..$

Hence to complete the proof it is sufficient to establish that almost surely
$\lim_{n\rightarrow\infty}\tau_{n}=T$. We will prove this fact along the lines
of [1, Theorem 3.1], using the bound (3.10). We begin by using the Ito Lemma
to establish the equality

$|\eta_{x,t\wedge\tau_{n}}|^{p}=\int_{0}^{t\wedge\tau_{n}}p(\eta_{x,s\wedge\tau_{n}})^{p-1}\Phi_{x}(\eta_{x,s\wedge\tau_{n}},\Xi_{s\wedge\tau_{n}}^{(x)})ds+\\\
+\int_{0}^{t\wedge\tau_{n}}\frac{p(p-1)}{2}(\eta_{x,s\wedge\tau_{n}})^{p-2}(\Psi_{x}(\eta_{x,s\wedge\tau_{n}},\Xi_{s\wedge\tau_{n}}^{(x)}))^{2}ds+\\\
+\int_{0}^{t\wedge\tau_{n}}p(\eta_{x,s\wedge\tau_{n}})^{p-1}\Psi_{x}(\eta_{x,s\wedge\tau_{n}},\Xi_{s\wedge\tau_{n}}^{(x)})dW_{x}(s),$

for all $t\in\mathcal{T}$. Before proceeding we define for convenience the
following short hand notations:

$\displaystyle\bar{\Phi}_{x}^{p}(\eta,t)$
$\displaystyle:=(\eta_{x,t\wedge\tau_{n}})^{p-1}\Phi_{x}(\eta_{x,t\wedge\tau_{n}},\Xi_{t\wedge\tau_{n}}^{(x)}),$
$\displaystyle\bar{\Psi}_{x}^{p}(\eta,t)$
$\displaystyle:=(\eta_{x,t\wedge\tau_{n}})^{p-2}(\Psi_{x}(\eta_{x,t\wedge\tau_{n}},\Xi_{t\wedge\tau_{n}}^{(x)}))^{2}.$

An application of Lemma 21 in Appendix shows that for all $t\in\mathcal{T}$ we
have

$\bar{\Phi}_{x}^{p}(\eta,t)\leq\left|\eta_{x,t\wedge\tau_{n}}\right|^{p-2}\\\
\times\left((b+\frac{1}{2})|\eta_{x,t\wedge\tau_{n}}|^{2}+\frac{1}{2}\bar{a}^{2}n_{x}\sum_{y\in\gamma_{x}}|\xi_{y,s}|^{2}+\left|\eta_{x,t\wedge\tau_{n}}\phi(0)\right|\right)\\\
\leq(b+\frac{1}{2})|\eta_{x,t\wedge\tau_{n}}|^{p}+\frac{1}{2}\bar{a}^{2}n_{x}(\eta_{x,t\wedge\tau_{n}})^{p-2}\sum_{y\in\gamma_{x}}|\xi_{y,t\wedge\tau_{n}}|^{2}+\left|\eta_{x,t\wedge\tau_{n}}\right|^{p-1}\left|\phi(0)\right|\\\
\leq(b+\frac{1}{2}+2^{p-1}c)|\eta_{x,t\wedge\tau_{n}}|^{p}+\frac{1}{2}\bar{a}^{2}n_{x}\left|\eta_{x,t\wedge\tau_{n}}\right|^{p-2}\sum_{y\in\gamma_{x}}|\xi_{y,t\wedge\tau_{n}}|^{2}\\\
+2^{p-1}c,$ (3.15)

where constants $b$ and $c$ are defined in Assumption 1. In the last
inequality we used the simple estimate
$C^{p-1}\leq(1+C)^{p-1}\leq(1+C)^{p}\leq 2^{p-1}(1+C^{p})$ for any $C>0$,
which holds because $p>1$. We can now use the Hölder inequality and classical
estimate $\left(\sum_{k=1}^{m}a_{k}\right)^{N}\leq
m^{N-1}\sum_{k=1}^{m}a_{k}^{N}$ (see e.g. [20]) in conjunction with inequality
(3.15) above to see that for all $t\in\mathcal{T}$ we have

$\mathbb{E}\bigg{[}\bar{\Phi}_{x}^{p}(\eta,t)\bigg{]}\leq(b+\frac{1}{2}+2^{p-1}c)\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}\\\
+\frac{1}{2}\bar{a}^{2}n_{x}\bigg{(}\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}\bigg{)}^{\frac{p-2}{p}}\bigg{(}\mathbb{E}\bigg{[}\bigg{(}\sum_{y\in\gamma_{x}}|\xi_{y,t\wedge\tau_{n}}|^{2}\bigg{)}^{\frac{p}{2}}\bigg{]}\bigg{)}^{\frac{2}{p}}+2^{p-1}c\\\
\leq(b+\frac{1}{2}+2^{p-1}c)\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}\\\
+\frac{1}{2}\bar{a}^{2}n_{x}\bigg{(}1+\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}\bigg{)}n_{x}^{\frac{p-2}{p}}\bigg{(}\mathbb{E}\bigg{[}\sum_{y\in\gamma_{x}}|\xi_{y,t\wedge\tau_{n}}|^{p}\bigg{]}\bigg{)}^{\frac{2}{p}}+2^{p-1}c.$

In a similar way, we obtain the inequality

$\bar{\Psi}_{x}^{p}(\eta,t)\leq
3M(n_{x}^{2}+3a^{2}2^{p-1})|\eta_{x,t\wedge\tau_{n}}|^{p}\\\
+3M^{2}n_{x}^{2}(\eta_{x,t\wedge\tau_{n}})^{p-2}\sum_{y\in\gamma_{x}}|\xi_{y,t\wedge\tau_{n}}|^{2}+3a^{2}2^{p-1}.$

Setting

$A_{x}:=\max\bigg{\\{}\frac{1}{2}\bar{a}^{2}n_{x}^{1+\frac{p-2}{p}},\
3M^{2}n_{x}^{2}\bigg{\\}}\bigg{(}\sum_{y\in\gamma_{x}}\mathbb{E}\bigg{[}\sup_{t\in\mathcal{T}}|\xi_{y,t}|^{p}\bigg{]}\bigg{)}^{\frac{2}{p}},$

we get the bounds

$\mathbb{E}\bigg{[}\bar{\Phi}_{x}^{p}(\eta,t)\bigg{]}\leq(b+\frac{1}{2}+2^{p-1}c+A_{x})\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}+A_{x}+2^{p-1}c$

and

$\mathbb{E}\bigg{[}\bar{\Psi}_{x}^{p}(\eta,t)\bigg{]}\leq(3M^{2}n_{x}^{2}+3a^{2}2^{p-1}+A_{x})\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}+A_{x}+3a^{2}2^{p-1}.$

Observe that $A_{x}<\infty$ by Theorem 10. Finally letting

$\displaystyle D$
$\displaystyle:=p(b+\frac{1}{2}+2^{p-1}c^{2}+A_{x})+\frac{p(p-2)}{2}(3M^{2}n_{x}^{2}+3a^{2}2^{p-1}+A_{x}),$
$\displaystyle K$
$\displaystyle:=pT(A_{x}+2^{p-1}c^{2})+\frac{p(p-2)}{2}T(A_{x}+3a^{2}2^{p-1}),$

we see that for all $t\in[0,\infty)$ we have

$\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}\leq
D\int_{0}^{t}\mathbb{E}\bigg{[}|\eta_{x,s\wedge\tau_{n}}|^{p}\bigg{]}ds+K.$
(3.16)

Observe that constants $K$ and $D$ are independent of the stopping time
$\tau_{n}$.

The rest of the proof is standard and can be completed along the lines of [1,
Theorem 3.1]. We give its sketch for a convenience of the reader. Using
Gronwall’s inequality together with the inequality (3.16) above we see that
for all $t\in[0,T]$ we have

$\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}\leq Ke^{Dt}.$

It follows from the definition of stopping time $\tau_{n}$ that

$\mathbb{E}\bigg{[}|\eta_{x,t\wedge\tau_{n}}|^{p}\bigg{]}\geq
n^{p}\mathbb{P}(\tau_{n}<t),\ $

so that, for all $t\in[0,T]$,

$\mathbb{P}(\tau_{n}<t)\leq\frac{1}{n^{p}}Ke^{Dt}\rightarrow 0,\
n\rightarrow\infty.$

Now convergence in probability and the fact that
$\\{\tau_{n}\\}_{n\in\mathbb{N}}$ is an increasing sequence imply that almost
surely $\lim_{n\rightarrow\infty}\tau_{n}=T$, hence the proof is complete.

###  3.3 Proof of Existence and Uniqueness

In this section, we are going to prove Theorem 3. We will show that, for any$\
\beta>\alpha,$ the process

$\overbrace{\ \Xi:=\lim_{n\rightarrow\infty}\Xi^{n}\ }^{\text{in}\
\mathcal{R}_{\beta}^{p}}$ (3.17)

solves system (2.1). For this, we will use auxiliary processes $\eta_{x}$
constructed in Theorem 11.

Proof of the existence. According to Theorem 11, for each $x\in\gamma$
equation

$\eta_{x,t}=\zeta_{x}+\int_{0}^{t}\Phi_{x}(\eta_{x,s},\Xi_{s})ds+\int_{0}^{t}\Psi_{x}(\eta_{x,s},\Xi_{s})dW_{x,s},\
t\in\mathcal{T},\ \mathbb{P}-a.s.,$

admits a unique solution. Thus it is sufficient to prove that this solution
coincides with the process $\xi_{x}$. The convergence (3.17) implies that, for
any fixed $x\in\gamma$,

$\lim_{n\rightarrow\infty}\mathbb{E}|\xi_{x,t}^{n}-\xi_{x,t}|^{p}=0,\
t\in\mathcal{T}.$ (3.18)

Therefore in order to conclude this proof it remains to show that, for any $\
t\in\mathcal{T}$,

$\lim_{n\rightarrow\infty}\mathbb{E}|\xi_{x,t}^{n}-\eta_{x,t}|^{p}=0.$ (3.19)

Let us fix $x\in\gamma$ and$\ t\in\mathcal{T}$ and assume without loss of
generality that $x\in\Lambda_{n}\subset\gamma$. Define the following
processes:

$\displaystyle\Phi_{x}^{n}(t)\\!$
$\displaystyle:=\Phi_{x}(\xi_{x,t}^{n},\Xi_{t}^{n})-\Phi_{x}(\eta_{x,t},\Xi_{t}),$
$\displaystyle\Psi_{x}^{n}(t)\\!$
$\displaystyle:=\Psi_{x}(\xi_{x,t}^{n},\Xi_{t}^{n})-\Psi_{x}(\eta_{x,t},\Xi_{t}),\
$ $\displaystyle\mathcal{X}_{x,t}^{n}$
$\displaystyle:=\xi_{x,t}^{n}-\eta_{x,t}.$

The rest of the proof is rather similar to the proof of Theorem 11. The Ito
Lemma shows that for all $t\in\mathcal{T}$ we have

$|\mathcal{X}_{x,t}^{n}|^{p}=\int_{0}^{t}p(\mathcal{X}_{x,s}^{n})^{p-1}\Phi_{x}^{n}(s)ds+\\\
+\int_{0}^{t}\frac{p(p-1)}{2}(\mathcal{X}_{x,s}^{n})^{p-2}(\Psi_{x}^{n}(s))^{2}ds+\\\
+\int_{0}^{t}p(\mathcal{X}_{x,s}^{n})^{p-1}\Psi_{x}^{n}(s)dW_{x}(s).$ (3.20)

Using Lemma 22 in Appendix, we can see that for all $t\in\mathcal{T}$

$\displaystyle(\mathcal{X}_{x,t}^{n})^{p-1}\Phi_{x}^{n}(t)$
$\displaystyle\leq(b+\frac{1}{2})\left|\mathcal{X}_{x,t}^{n}\right|^{p}$
$\displaystyle+\bar{a}^{2}n_{x}\left|\mathcal{X}_{x,t}^{n}\right|^{p-2}\sum_{y\in\gamma_{x}}(\xi_{y,t}^{n}-\xi_{y,t})^{2},$

and

$\displaystyle(\mathcal{X}_{x,t}^{n})^{p-2}\Psi_{x}^{n}(t)^{2}$
$\displaystyle\leq 2M^{2}n_{x}^{2}\left|\mathcal{X}_{x,t}^{n}\right|^{p}$
$\displaystyle+2M^{2}n_{x}\left|\mathcal{X}_{x,t}^{n}\right|^{p-2}\sum_{y\in\gamma_{x}}(\xi_{y,t}^{n}-\xi_{y,t})^{2}.$

As the proof of Theorem 11, we see that for all $t\in\mathcal{T}$

$\mathbb{E}\bigg{[}(\mathcal{X}_{x,t}^{n})^{p-1}\Phi_{x}^{n}(t)\bigg{]}\leq(b+\frac{1}{2}+A_{x}^{n})\mathbb{E}\bigg{[}\left|\mathcal{X}_{x,t}^{n}\right|^{p}\bigg{]}+A_{x}^{n}$
(3.21)

and

$\mathbb{E}\bigg{[}(\mathcal{X}_{x,t}^{n})^{p-2}\bigg{(}\Psi_{x}^{n}(t)\bigg{)}^{2}\bigg{]}\leq(2M^{2}n_{x}^{2}+A_{x}^{n})\mathbb{E}\bigg{[}\left|\mathcal{X}_{x,t}^{n}\right|^{p}\bigg{]}+A_{x}^{n},$
(3.22)

where

$A_{x}^{n}:=\bigg{(}\max\bigg{\\{}\bar{a}^{2}n_{x}^{1+\frac{p-2}{p}},\
2M^{2}n_{x}^{2}\bigg{\\}}\bigg{(}\mathbb{E}\bigg{[}\sum_{y\in\bar{\gamma}_{x}}(\xi_{y,t}^{n}-\xi_{y,t})^{p}\bigg{]}\bigg{)}^{\frac{2}{p}}.$

Now, because $b_{x}$ is finite it is clear from equation (3.18) that

$\bigg{(}\mathbb{E}\sum_{y\in\bar{\gamma}_{x}}(\xi_{y,t}^{n}-\xi_{y,t})^{p}\bigg{)}^{\frac{2}{p}}\rightarrow
0,\ n\rightarrow\infty,$

so that $A_{x}^{n}\rightarrow 0$ as $n\rightarrow\infty$, too. Therefore using
inequality (3.21) and (3.22) above we can conclude from equation (3.20) that
for all $x\in\gamma$ and all $t\in\mathcal{T}$ we have

$\mathbb{E}\bigg{[}|\mathcal{X}_{x,t}^{n}|^{p}\bigg{]}\leq
C_{x}^{n}\int_{0}^{t}\mathbb{E}\bigg{[}|\mathcal{X}_{x,t}^{n}|^{p}\bigg{]}ds+\bar{A}_{x}^{n},$

where

$C_{x}^{n}:=p(b+\frac{1}{2}+A_{x}^{n})+\frac{p(p-1)}{2}(2M^{2}n_{x}^{2}+A_{x}^{n})\text{
and}\ \bar{A}_{x}^{n}:=pTA_{x}^{n}+\frac{p(p-1)}{2}TA_{x}^{n},$

and consequently $C_{x}^{n},\bar{A}_{x}^{n}\rightarrow 0$ on $\mathcal{T}$ as
$n\rightarrow\infty$. Finally using Gronwall inequality we see that for all
$t\in\mathcal{T}$ we have

$\mathbb{E}\bigg{[}|\mathcal{X}_{x,t}^{n}|^{p}\bigg{]}\leq\bar{A}_{x}^{n}e^{(C_{x}^{n})T},$

which shows that for all $x\in\gamma$ and uniformly on $\mathcal{T}$

$\lim_{n\rightarrow\infty}\mathbb{E}\bigg{[}|\mathcal{X}_{x,t}^{n}|^{p}\bigg{]}=0.$

Equation (3.19) now follows immediately hence the proof is complete.

Proof of the uniquenes and continuous dependence. Suppose that
$\Xi_{t}^{1}=\left(\xi_{x,t}^{1}\right)_{x\in\gamma}$ and
$\Xi_{t}^{2}=\left(\xi_{x,t}^{2}\right)_{x\in\gamma}$,
$\Xi^{1},\Xi^{2}\in\mathcal{R}_{\alpha+}^{p}$, are two strong solutions of
system (2.1), with initial values $\xi_{x,0}^{1}$ and $\xi_{x,0}^{2}$,
respectively. Lemma 22 implies that

$\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}|^{p}\bigg{]}\leq\left|\bar{\xi}_{x,0}\right|^{p}+Bn_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}\mathbb{E}\bigg{[}|\bar{\xi}_{y,s}|^{p}\bigg{]}ds.$

Fix an arbitrary $\beta>\alpha$ and $\alpha_{1}\in\left(\alpha,\beta\right)$.
An application of Lemma 23 to a measurable map $\kappa:\mathcal{T}\rightarrow
l_{\alpha_{1}}^{1}$ defined by the formula

$\kappa_{x}(t):=\mathbb{E}\bigg{[}|\bar{\xi}_{x,t}|^{p}\bigg{]}.$

shows that

$\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}\kappa_{x}(t)\leq
K(\alpha_{1},\beta)\sum_{x\in\gamma}e^{-\alpha|x|}|b_{x}|,\ \beta>\alpha_{1},$

where $b_{x}=|\bar{\xi}_{x,0}|^{p}$. Therefore we establish that

$||\Xi^{1}-\Xi^{2}||_{\mathcal{R}_{\beta}^{p}}^{p}\ \equiv\
\sup_{t\in\mathcal{T}}\mathbb{E}\bigg{[}\sum_{x\in\gamma}e^{-\beta|x|}|\bar{\xi}_{x,t}|^{p}\bigg{]}\leq
K(\alpha_{1},\beta)\sum_{x\in\gamma}e^{-\alpha|x|}|\bar{\xi}_{x,0}|^{p},$

which implies both statements.

###  3.4 Markov semigroup

In this section we denote by $\Xi_{t}(\bar{\zeta})$ the solution of equation
(2.1) with initial condition $\bar{\zeta}$. This process generates an operator
family $T_{t}:C_{b}(l_{\beta}^{p})\rightarrow C_{b}(l_{\alpha}^{p})$,
$\alpha<\beta$, by the formula

$T_{t}f(\bar{\zeta})=\mathbb{E}f(\Xi_{t}(\bar{\zeta})).$ (3.23)

Operators $T_{t}$ can be considered as operators in $C_{b}(l_{\alpha+}^{p})$
for any $\alpha\in\mathcal{A}$, where
$l_{\alpha+}^{p}=\cap_{\beta>\alpha}l_{\beta}^{p}$ is a Polish space equipped
with the projective limit topology, see e.g. [17]. The uniqueness of the
solution (cf. Theorem 3) implies in the standard way the evolution property

$T_{t}T_{s}=T_{t+s},\ t,s\geq 0.$

Similarly, the truncated process $\Xi_{t}^{n}(l_{\alpha}^{p})$ generates the
semigroup $T_{t}^{n}:C_{b}(l_{\alpha}^{p})\rightarrow C_{b}(l_{\alpha}^{p})$,
$\alpha\in\mathcal{A}$. It follows from the convergence

$\Xi_{t}^{n}(\bar{\zeta})\rightarrow\Xi_{t}(\bar{\zeta}),\
n\rightarrow\infty,$

in $\mathcal{R}_{\beta}^{p}$ any $\beta>\alpha$ that

$T_{t}^{n}f(\bar{\zeta})\rightarrow T_{t}f(\bar{\zeta}),\
n\rightarrow\infty,\text{ for any }f\in C_{b}(l_{\alpha+}^{p})\text{ and
}\bar{\zeta}\in l_{\alpha+}^{p}.$

Moreover, the dominated convergence theorem implies that

$\int T_{t}^{n}f(\bar{\zeta})\nu(d\bar{\zeta})\rightarrow\int
Tf(\bar{\zeta})\nu(d\bar{\zeta}),\ n\rightarrow\infty,$ (3.24)

for any probability measure $\nu$ on $l_{\alpha+}^{p}$.

##  4 Stochastic dynamics associated with Gibbs measures

As an application of our results, we will present a construction of stochastic
dynamics associated with Gibbs measures on $S^{\gamma}$. Necessary conditions
of the existence of these measures were derived in [12]. For convenience of
the reader, we start with the reminder of the general definition of Gibbs
measures, adopted to our framework.

###  4.1 Construction of Gibbs measures

In the standard Dobrushin-Lanford-Ruelle (DLR) approach in statistical
mechanics [18, 26], Gibbs measures (states) are constructed by means of their
local conditional distributions (constituting the so-called Gibbsian
specification). We are interested in Gibbs measures describing equilibrium
states of a (quenched) system of particles with positions $\gamma\subset
X=\mathbb{R}^{d}$ and spin space $S=\mathbb{R}$, defined by pair and single-
particle potentials $W_{xy}$ and $V$, respectively. We assume the following:

  * •

$W_{xy}:S\times S\rightarrow\mathbb{R}$, $x,y\in X$, are measurable functions
satisfying the polynomial growth estimate

$\left|W_{xy}(u,v)\right|\leq
I_{W}\left(\left|u\right|^{r}+\left|v\right|^{r}\right)+J_{W},\ \ \ u,v\in S,$
(4.1)

and the finite range condition $W_{xy}\equiv 0$ if $\left|x-y\right|\leq\rho$
for all $x,y\in X$ and some constants $I_{W},J_{W},R,r\geq 0$. We assume also
that $W_{xy}(u,v)$ is symmetric with respect to the permutation of $(x,u)$ and
$(y,v)$.

  * •

the single-particle potential $V$ satisfies the bound

$V(u)\geq a_{V}\left|u\right|^{\tau}-b_{V},\ \ u\in S,$ (4.2)

for some constants $a_{V},b_{V}>0$, and $\tau>r$.

###### Example 12

A typical example is given by the pair interaction in of the form

$W_{xy}(u,v)=a(x-y)u~{}v,\ \text{\ }u,v\in S,$

where $a:X\rightarrow\mathbb{R}$ is as in Section 2. In this case, $r=2$ and
so we need $\tau>2$ in (4.2). The method of [12] does not allow us to control
the case of $\tau=2,$ even when the underlying particle configuration $\gamma$
is a typical realisation of a homogeneous Poisson random field on $\Gamma(X)$.

Let $\mathcal{F}(\gamma)$ be the collection of all finite subsets of
$\gamma\in\Gamma(X)$. For any $\eta\in\mathcal{F}(\gamma),$
$\bar{\sigma}_{\eta}=(\sigma_{x})_{x\in\eta}\in S^{\eta}$ and
$\bar{z}_{\gamma}=(z_{x})_{x\in\gamma}\in S^{\gamma}$ define the relative
local interaction energy

$E_{\eta}(\bar{\sigma}_{\eta}\left|\bar{z}_{\gamma}\right.)=\sum_{\\{x,y\\}\subset\eta}W_{xy}(\sigma_{x},\sigma_{y})+\sum_{\begin{subarray}{c}x\in\eta\\\
y\in\gamma\setminus\eta\end{subarray}}W_{xy}(\sigma_{x},z_{y}).$

The corresponding specification kernel
$\Pi_{\eta}(d\bar{\sigma}_{\gamma}\left|\bar{z}_{\gamma}\right.)$ is a
probability measure on $S^{\gamma}$ of the form

$\Pi_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})=~{}\mu_{\eta}(d\bar{\sigma}_{\eta}|\bar{z}_{\eta})\otimes\delta_{\bar{z}_{\gamma\setminus\eta}}(d\bar{\sigma}_{\gamma\setminus\eta}),$
(4.3)

where

$\mu_{\eta}(d\bar{\sigma}_{\eta}|\bar{z}_{\eta}):=Z(\bar{z}_{\eta})^{-1}\mathrm{exp}~{}\left[-E_{\eta}(\bar{\sigma}_{\eta}\left|\bar{z}_{\eta}\right.)\right]\bigotimes\limits_{x\in\eta}e^{-V(\sigma_{x})}d\sigma_{x}$
(4.4)

is a probability measure on $S^{\eta}$. Here $Z(\bar{z}_{\eta})$ is the
normalizing factor and
$\delta_{\bar{z}_{\gamma\setminus\eta}}(d\bar{\sigma}_{\gamma\setminus\eta})$
is the Dirac measure on $S^{\gamma\setminus\eta}$ concentrated on
$\bar{z}_{\gamma\setminus\eta}$. The family
$\left\\{\Pi_{\eta}(d\bar{\sigma}|\bar{z}),\
\eta\in\mathcal{F}(\gamma),\bar{z}\in S^{\gamma}\right\\}$ is called the
Gibbsian specification (see e.g. [18, 26]).

A probability measure $\nu$ on $S^{\gamma}$ is said to be a Gibbs measure
associated with the potentials $W$ and $V$ if it satisfies the DLR equation

$\nu(B)=\int_{S^{\gamma}}\Pi_{\eta}(B|\bar{z})\nu(d\bar{z}),\quad
B\in\mathcal{B}(S^{\gamma}),$ (4.5)

for all $\eta\in\mathcal{F}(\gamma)$. For a given $\gamma\in\Gamma(X)$, by
$\mathcal{G}(S^{\gamma})$ we denote the set of all such measures.

By $\mathcal{G}_{\alpha,p}(S^{\gamma})\subset\mathcal{G}(S^{\gamma})$ we
denote the set of all Gibbs measures on $S^{\gamma}$ associated with $W$ and
$V$, which are supported on $l_{\alpha}^{p}$.

###### Theorem 13

Assume that conditions (4.1) and (4.2) are satisfied and
$p\in\left[r,\tau\right]$. Then the set $\mathcal{G}_{\alpha,p}(S^{\gamma})$
is non-empty for any $\alpha\in\mathcal{A}$.

Proof. It follows in a straightforward manner from condition (2.2) that

$a_{\gamma,\rho}(\gamma)=\sum_{x\in\gamma}e^{-\alpha\left|x\right|}n_{x}^{p_{1}}\sum_{y\in\gamma_{x}}n_{y}^{p_{1}}<\infty$

for any $p_{1},p_{2}\in\mathbb{N}$, which is sufficient for the existence of
$\nu\in\mathcal{G}_{\alpha,p}(S^{\gamma})$ for any $p\in\left[r,\tau\right]$,
see [22] and [12].

###### Remark 14

The result of [22], [12] is more refined and states in addition certain bounds
on exponential moments of $v\in\mathcal{G}_{\alpha,p}(S^{\gamma})$.

###  4.2 Construction of the stochastic dynamics

In this section, we will construct a process $\Xi_{t}$ with invariant measure
$\nu\in\mathcal{G}_{\alpha_{\ast},p}(S^{\gamma})$ defined by interaction
potentials $W$ and $V$ as in Example 12. By Theorem 13, the set
$\mathcal{G}_{\alpha_{\ast},p}(S^{\gamma})$ is not empty if
$p\in\left[2,\tau\right]$. Then, according to the general paradigm, $\Xi_{t}$
will be a solution of the system (2.1) with the coefficients satisfying the
following:

  1. (1)

the drift coefficient has a gradient form, that is, $\phi=-\nabla V$, and
satisfies Conditions (2.4) and (2.5); a typical example is given by

$\phi(\sigma)=-\sigma^{2n+1}\text{ for any}\ n=1,2,..,$

in which case $R=2n+1$ and $\tau=2n+2$, cf. (2.4);

  2. (2)

for each $x\in\gamma$, the noise is additive, that is, $\Psi_{x}=id$.

Thus the system (2.1) obtains the form

$d\xi_{x,t}=\left[\nabla
V(\xi_{x,t})+\sum_{y\in\bar{\gamma}_{x}}a(x-y)\xi_{y,t}\right]dt+dW_{x,t},\
x\in\gamma.$

According to Theorem 3, this system admits a unique strong solution
$\Xi\in\mathcal{R}_{\alpha+}$ for any initial condition
$\bar{\sigma}_{\gamma}\in l_{\alpha}^{p}$ with arbitrary
$\alpha\in\mathcal{A}$ and $p\geq R$.

A standard way of rigorously proving the invariance of $\nu$ would require
dealing with Markov processes and semigroups in nuclear spaces. This
difficulty can be avoided by using the limit transition (3.24).

###### Theorem 15

Assume that $p\in\left[\max\left\\{2,R\right\\},\tau\right]$ and let $T_{t}$
be the semigroup defined by the process $\Xi_{t}$, cf. (3.23). Then

$\int T_{t}f(\bar{\sigma}_{\gamma})\nu(d\bar{\sigma}_{\gamma})=\int
f(\bar{\sigma}_{\gamma})\nu(d\bar{\sigma}_{\gamma})$

for any $\nu\in\mathcal{G}_{\alpha_{\ast},p}(S^{\gamma})$ and $f\in
C_{b}(l_{\alpha_{\ast}+}^{p})$.

Proof. First observe that condition
$p\in\left[\max\left\\{2,R\right\\},\tau\right]$ ensures that
$\mathcal{G}_{\alpha_{\ast},p}(S^{\gamma})\neq\emptyset$ and semigroup $T_{t}$
is well-defined. Consider the specification kernel
$\Pi_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})$, cf. (4.3). We have

$\int
T_{t}^{n}f(\bar{\sigma}_{\gamma})\Pi_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})=\int\mathbb{E}f(\Xi_{t}^{n}(\bar{\sigma}_{\eta}\times\bar{\sigma}_{\bar{\eta}}))~{}\mu_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})\otimes\delta_{\bar{z}_{\bar{\eta}}}(d\bar{\sigma}_{\bar{\eta}})\\\
=\int_{S^{\eta}}\mathbb{E}f(\Xi_{t}^{n}(\bar{\sigma}_{\eta}\times\bar{z}_{\bar{\eta}}))\mu_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})\mathrm{,}$

where $\mu_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})$ is given by (4.4).
By standard theory of finite dimensional SDEs,
$\mu_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})$ is invariant measure of
the process
$\left(\xi_{x}^{n}(\bar{\sigma}_{\eta}\times\bar{z}_{\bar{\eta}},t)\right)_{x\in\eta}$
in $S^{\eta}$, which implies that

$\int_{S^{\eta}}\mathbb{E}f(\Xi^{n}(\bar{\sigma}_{\eta}\times\bar{z}_{\bar{\eta}},t))\mu_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})=\int_{S^{\eta}}f(\bar{\sigma}_{\eta}\times\bar{z}_{\bar{\eta}},t)\mu_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma}).$

Thus we have in turn the equality

$\int
T_{t}^{n}f(\bar{\sigma}_{\gamma})\Pi_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma})=\int
f(\bar{\sigma}_{\gamma})\Pi_{\eta}(d\bar{\sigma}_{\gamma}|\bar{z}_{\gamma}).$

Integrating both sides with respect to $\nu(d\bar{z}_{\gamma})$ and applying
the DLR equation (4.5) we see that

$\int T_{t}^{n}f(\bar{\sigma}_{\gamma})\nu(d\bar{\sigma}_{\gamma})=\int
f(\bar{\sigma}_{\gamma})\nu(d\bar{\sigma}_{\gamma}).$

Passing to the limit as $n\rightarrow\infty$ (cf. (3.24)) we obtain that

$\int T_{t}f(\bar{\sigma}_{\gamma})\nu(d\bar{\sigma}_{\gamma})=\int
f(\bar{\sigma}_{\gamma})\nu(d\bar{\sigma}_{\gamma}),$

as required.

##  5 Appendix

###  5.1 Linear operators in the spaces of sequences

We start with the formulation of a general result from [9] on the existence of
(infinite-time) solutions for a special class of linear differential
equations, which extends the so-called Ovsjannikov method, see e.g. [16].

###### Definition 16

Let $\mathfrak{B}\mathbf{=}\\{B_{\alpha}\\}_{\alpha\in\mathcal{A}}$ be a scale
of Banach spaces. A liner operator
$A:\bigcup\limits_{\alpha\in\mathcal{A}}B_{\alpha}\rightarrow\bigcup\limits_{\alpha\in\mathcal{A}}B_{\alpha}$
is called an Ovsjannikov operator of order $q>0$ if $A(B_{\alpha})\subset
B_{\beta}$ and there exists a constant $L>0$ such that

$||Ax||_{B_{\beta}}\leq\frac{L}{(\beta-\alpha)^{q}}||x||_{B_{\alpha}},\ x\in
B_{\alpha},$

for all $\alpha<\beta\in\mathcal{A}$. The space of such operators will be
denoted by $\mathcal{O}(\mathfrak{B},q)$.

###### Theorem 17

[9, Theorem 3.1 and Remark 3.3] Let $A\in\mathcal{O}(\mathfrak{B},q)$ with
$q<1$. Then, for any $\alpha,\beta\in\mathcal{A}$ such that $\alpha<\beta$ and
$f_{0}\in B_{\alpha}$, there exists a unique continuous function
$f:[0,\infty)\rightarrow B_{\beta}$ with $f(0)=f_{0}$ such that:

  1. (1)

$f$ is continuously differentiable on $(0,\infty)$;

  2. (2)

$Af(t)\in B_{\beta}$ for all $t\in(0,\infty)$;

  3. (3)

$f$ solves the differential equation

$\dfrac{d}{dt}f(t)=Af(t),\quad t>0.$

Moreover,

$\lVert f(t)\rVert_{B_{\beta}}\leq K(\alpha,\beta)\lVert
f_{0}\rVert_{B_{\alpha}},\quad t>0,$

where
$K(\alpha,\beta):=\sum_{n=0}^{\infty}\frac{L^{n}t^{n}}{(\beta-\alpha)^{qn}}\frac{n^{qn}}{n!}<\infty$.

The aim of this section is to give a sufficient condition for an infinite
matrix $\\{Q_{x,y}\\}_{x,y\in\gamma}$ to generate an Ovsjannikov operator in
the scale $\mathcal{L}^{1}$ of spaces of sequences defined by (2.8). Denote by
$Q$ the linear operator generated by the matrix $\\{Q_{x,y}\\}_{x,y\in\gamma}$
in $S^{\gamma}$.

###### Theorem 18

Assume that the matrix $\\{Q_{x,y}\\}_{x,y\in\gamma}$ be an infinite real
matrix is such that for all $x,y\in\gamma$ we have

  * •

$Q_{x,y}=0$ if $\left|x-y\right|>\rho$;

  * •

there exist $C>0$ and $s>0$ such that

$|Q_{x,y}|\leq Cn_{x}^{s}.$ (5.1)

Then $Q\in\mathcal{O}(\mathcal{L}^{1},q)$ for any $q<1$.

Proof. Since $Q$ is linear, it is sufficient to show that

$\|Qz\|_{\beta}\leq\frac{L}{(\beta-\alpha)^{q}}\|z\|_{\alpha}$ (5.2)

for any $\alpha<\beta\in\mathcal{A}$ and $z\in l_{\alpha}^{1}$. By the
definition of the norm in $l_{\beta}^{1}$ we have

$\|Qz\|_{\beta}=\sum_{x\in\gamma}e^{-\beta|x|}\bigg{|}\sum_{y\in\gamma}Q_{x,y}z_{y}\bigg{|}.$

Now, using estimate (5.1) we see that

$\|Qz\|_{\beta}\leq\sum_{x\in\gamma}\sum_{y\in\gamma}|Q_{x,y}|e^{-\beta|x|}|z_{y}|\leq
e^{\beta\rho}\sum_{x\in\gamma}\sum_{y\in\bar{\gamma}_{x}}|Q_{x,y}|e^{-\beta|y|}|z_{y}|\\\
\leq
e^{\beta\rho}\sum_{x\in\gamma}\sum_{y\in\bar{\gamma}_{x}}|Q_{x,y}|e^{-(\beta-\alpha)|y|}e^{-\alpha|y|}|z_{y}|\\\
\leq e^{\alpha^{\ast}\rho}K\|z\|_{\alpha},$ (5.3)

because $Q_{x,y}=0$ for $y\in\bar{\gamma}_{x}$ and $-|x|\leq-|y|+\rho$ for
$y\in\bar{\gamma}_{x}$. Here

$K:=\underset{y\in\gamma}{\sup}\sum_{x\in\gamma}|Q_{x,y}|e^{-(\beta-\alpha)|y|}.$

Our next goal is to estimate the constant $K$. Using condition (5.1) we see
that for all $y\in\gamma$

$\sum_{x\in\gamma}|Q_{x,y}|e^{-(\beta-\alpha)|y|}\leq C\sum_{x\in
B_{y}}n_{x}^{s}e^{-(\beta-\alpha)|y|}.$

Observe that there exist constants $M,N\in\mathbb{N}$ such that

$M<|x|\implies n_{x}\leq N|x|^{q/2s}.$

Then, taking into account that $|x|^{q/2}\leq|y|^{q/2}+\rho^{q/2}$ for $x\in
B_{y},$we obtain

$\sum_{x\in B_{y}}n_{x}^{s}\leq\sum_{\begin{subarray}{c}x\in B_{y}\\\
|x|>M\end{subarray}}N|x|^{q/2}+\sum_{\begin{subarray}{c}x\in\gamma\\\ |x|\leq
M\end{subarray}}n_{x}^{q}\\\ \leq N\sum_{\begin{subarray}{c}x\in B_{y}\\\
|x|>M\end{subarray}}\left(|y|^{q/2}+\rho^{q/2}\right)+P\leq
Nn_{y}\left(|y|^{q/2}+\rho^{q/2}\right)+P\\\ \leq
N^{2}|y|^{q/2}\left(|y|^{q/2}+\rho^{q/2}\right)+P\\\ \leq
2N^{2}|y|^{q}+N^{2}\rho^{q}+P,$

where $P=P(\gamma,M,q):=\sum_{\begin{subarray}{c}x\in\gamma\\\ |x|\leq
M\end{subarray}}n_{x}^{q}<\infty.$ Hence for all $y\in\gamma$ we have

$\sum_{x\in\gamma}|Q_{x,y}|e^{-(\beta-\alpha)|y|}\leq
C\left(N^{2}|y|^{q}+N^{2}\rho^{q}+P\right)e^{-(\beta-\alpha)|y|}\leq
a_{1}+a_{2}|y|^{q}e^{-(\beta-\alpha)|y|}$

with $a_{1}=C\left(N^{2}\rho^{q}+P\right)$ and $a_{2}=CN^{2}$. Now we see that

$K\leq a_{1}+a_{2}\sup\bigg{\\{}|y|^{q}e^{-(\beta-\alpha)|y|}\bigg{|}\
y\in\gamma\bigg{\\}}\\\ \leq
a_{1}+a_{2}\sup\bigg{\\{}\bigg{(}he^{-\frac{\beta-\alpha}{q}h}\bigg{)}^{q}\bigg{|}\
h>0\bigg{\\}}\\\ \leq
a_{1}+a_{2}\bigg{(}\sup\bigg{\\{}he^{-\frac{\beta-\alpha}{q}h}\ \bigg{|}\
h>0\bigg{\\}}\bigg{)}^{q}.$ (5.4)

Hence, we can deduce that function
$he^{-\frac{\beta-\alpha}{q}h}:(0,\infty)\rightarrow\mathbb{R}$ attains its
supremum when $\frac{d}{dh}he^{-\frac{\beta-\alpha}{q}h}=0$ that is when
$h=\frac{q}{(\beta-\alpha)}$. Hence it follows from inequality (5.4) that

$K\leq\frac{a_{1}(\alpha^{\ast}-\alpha_{\ast})^{q}+a_{2}\left(e^{-1}q\right)^{q}}{(\beta-\alpha)^{q}}.$

Now, continuing from equation (5.3) we finally see that (5.2) holds with
$L=e^{\alpha^{\ast}\rho}(a_{1}(\alpha^{\ast}-\alpha_{\ast})^{q}+a_{2}q^{q})$,
and the proof is complete.

###  5.2 Comparison theorem and Gronwall-type inequality

In this section, we prove generalizations of the classical comparison theorem
for differential equations and, as a consequence, a version of the Gronwall
inequality, that work in our scale of Banach spaces of sequences.

Let us consider the linear integral equation

$f(t)=\bar{z}+\int_{0}^{t}Qf(s)ds,\quad t\in\mathcal{T},$ (5.5)

in $l_{\alpha^{\ast}}^{1}$ where $Q\in\mathcal{O}(\mathcal{L}^{1},q)$, $q<1$,
is a linear operator generated by the infinite matrix
$\\{Q_{x,y}\\}_{x,y\in\gamma}$ and $\bar{z}=(z_{x})_{x\in\gamma}\in
l_{\alpha}^{1}$ for some $\alpha<\alpha^{\ast}$. It follows from Theorem 17
that this equation has a unique solution $f\in l_{\alpha+}^{1}$.

The next result is an extension of the classical comparison theorem to our
framework.

######  Theorem 19 (Comparison Theorem)

Suppose that $Q_{x,y}\geq 0$ for all $x,y\in\gamma$ and let
$g:\mathcal{T}\rightarrow l_{\alpha}^{1}$ be a bounded map such that

$g_{x}(t)\leq z_{x}+\bigg{[}\int_{0}^{t}Qg(s)ds\bigg{]}_{x},\quad
t\in\mathcal{T},\ x\in\gamma.$

Then for all $t\in\mathcal{T}$ and all $x\in\gamma$ we have the inequality

$g_{x}(t)\leq f_{x}(t),$

where $f=(f_{x})_{x\in\gamma}$ is the solution of (5.5) .

Proof. Let $\mathcal{B}_{a}:=\mathcal{B}([0,T],l_{\alpha}^{1}),\
a\in\mathcal{A}$, be the Banach space of bounded measurable functions
$\mathcal{T\rightarrow}l_{\alpha}^{1}$. For any $g\in\mathcal{B}_{\alpha}$
define the function

$\mathcal{I}(g)(t):=\bar{z}+\int_{0}^{t}Qg(s)ds.$

It is clear that $\mathcal{I}(g)\in\mathcal{B}_{\alpha+}$, which implies that
the composition power
$\mathcal{I}^{n}:\mathcal{B}_{a}\rightarrow\mathcal{B}_{a\mathfrak{+}}$ is
well-defined. It follows from (the proof of) [9, Theorem 3.1.] that

$\overbrace{\bigg{[}\lim_{n\rightarrow\infty}\mathcal{I}^{n}(g)\bigg{]}}^{\text{in}\
\mathcal{B}([0,T],l_{\beta}^{1})}=f,\ \beta>\alpha.$

We have therefore
$\lim_{n\rightarrow\infty}\mathcal{I}_{x}^{n}(g)(t)=f_{x}(t)$ for all
$x\in\gamma$ and all $t\in\mathcal{T}$. Hence to conclude the proof it is
sufficient to fix $x\in\gamma$ and $t\in\mathcal{T}$ and prove by induction
that

$g_{x}(t)\leq\mathcal{I}_{x}^{n}(g)(t),\ \forall n\in\mathbb{N}.$ (5.6)

The case $n=1$ is satisfied by the initial assumption on $g$. Let us now
assume that (5.6) is true for some $n\geq 1$ and proceed by considering the
following chain of inequalities:

$\mathcal{I}_{x}^{n+1}(g)(t)=z_{x}+\bigg{[}\int_{0}^{t}Q(\mathcal{I}^{n}(g)(s))ds\bigg{]}_{x}\\\
=z_{x}+\sum_{y\in\gamma}Q_{x,y}\int_{0}^{t}\mathcal{I}_{y}^{n}(g)(s)ds\\\ \geq
z_{x}+\sum_{y\in\gamma}Q_{x,y}\int_{0}^{t}g_{y}(s)ds\\\
=z_{x}+\bigg{[}\int_{0}^{t}Q(g(s))ds\bigg{]}_{x}\geq g_{x}(t),$

which completes the proof.

######  Corollary 20 (Generalized Gronwall inequality)

Suppose in addition that $z_{x}\geq 0$ for all $x\in\gamma$. Moreover assume
that components of the map $g$ are non-negative functions, that is,
$g_{x}(t)\geq 0$ for all $x\in\gamma$ and all $t\in\mathcal{T}$. Then for all
$\beta>\alpha$ we have the inequality

$\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}g_{x}(t)\leq
K(\alpha,\beta)\sum_{x\in\gamma}e^{-\alpha|x|}z_{x},$

where
$K(\alpha,\beta)=\sum_{n=0}^{\infty}\frac{L^{n}T^{n}}{(\beta-\alpha)^{qn}}\frac{n^{qn}}{n!}<\infty$.

Proof. Using Theorem 19, we see that for all $x\in\gamma$ and all
$t\in\mathcal{T}$ we have

$g_{x}(t)\leq z_{x}+\bigg{[}\int_{0}^{t}Q(g(s))ds\bigg{]}_{x}\leq
z_{x}+\bigg{[}\int_{0}^{t}Q(f(s))ds\bigg{]}_{x}.$

Since functions $g$ and therefore $f$ are non-negative see that for all
$x\in\gamma$

$\sup_{t\in\mathcal{T}}g_{x}(t)\leq
z_{x}+\bigg{[}\int_{0}^{T}Q(f(s))ds\bigg{]}_{x}=f_{x}(T).$

Hence it follows that

$\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}g_{x}(t)\leq\sum_{x\in\gamma}e^{-\beta|x|}f_{x}(T)\leq\|f(T)\|_{l_{\beta}^{1}}.$

The right-hand side of the inequality above can be estimated using [9, Theorem
3.1], cf. Theorem 17. In particular we get

$\|f(T)\|_{l_{\beta}^{1}}\leq\sum_{n=0}^{\infty}\frac{L^{n}T^{n}}{(\beta-\alpha)^{qn}}\frac{n^{qn}}{n!}\|z_{\alpha_{\ast}}\|_{l_{\alpha}^{1}}<\infty.$

Hence letting
$K(\alpha,\beta)=\sum_{n=0}^{\infty}\frac{L^{n}T^{n}}{(\beta-\alpha)^{qn}}\frac{n^{qn}}{n!}$
we see that the proof is complete.

###  5.3 Estimates of the solutions

We start with the following auxiliary result.

###### Lemma 21

Suppose that $\sigma_{1},\sigma_{2}\in\mathbb{R}$ and $Z_{1},Z_{2}\in
S^{\gamma}$. Then for all $x\in\gamma$ we have the following inequalities:

$\displaystyle|\Psi_{x}(Z_{1})-\Psi_{x}(Z_{2})|$ $\displaystyle\leq
Mn_{x}|z_{1,x}-z_{2,x}|+M\sum_{y\in\gamma_{x}}|z_{1,y}-z_{2,y}|,$
$\displaystyle|\Psi_{x}(0)|$ $\displaystyle\leq a\text{ for some }a>0,$

and

$|\Phi_{x}(Z_{1})|\leq
c(1+|z_{1,x}|^{R})+\bar{a}\sqrt{n_{x}}\left(\sum_{y\in\bar{\gamma}_{x}}z_{1,y}^{2}\right)^{\frac{1}{2}},$
$(z_{1,x}-z_{2,x})(\Phi_{x}(Z_{1})-\Phi_{x}(Z_{2}))\\\
\leq(b+\frac{1}{2})(z_{1,x}-z_{2,x})^{2}+\frac{1}{2}\bar{a}^{2}n_{x}\sum_{y\in\gamma_{x}}(z_{1,y}-z_{2,y})^{2},$

where constants $M,c,b$ and $\bar{a}$ are defined in Assumption 1.

Proof. The proof follows by a direct calculation using assumptions on $\Phi$
and $\Psi$ stated in Section 2.

Let us fix $\alpha\in\mathcal{A}$ and consider two processes
$\Xi_{t}^{(1)}=\left(\xi_{x,t}^{(1)}\right)_{x\in\gamma}$ and
$\Xi_{t}^{(2)}=\left(\xi_{x,t}^{(2)}\right)_{x\in\gamma}$,
$\Xi^{(1)},\Xi^{(2)}\in\mathcal{R}_{\alpha+}^{p}$, with initial values
$\xi_{x,0}^{1},\xi_{x,0}^{2}\in l_{\alpha}^{p}$.

###### Lemma 22

Let $p\geq 2$ and $x\in\gamma$ be fixed and assume that $\mathbb{R}$-valued
processes $\xi_{x,t}^{(1)}$ and $\xi_{x,t}^{(2)}$ are $p$-integrable and
satisfy equation (2.1). Then there exist universal constants $B,C_{1}$and
$C_{2}$ such that

$\mathbb{E}|\xi_{x,t}^{(1)}|^{p}\leq\left|\xi_{x,0}^{(1)}\right|^{p}+C_{1}n_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}\mathbb{E}|\bar{\xi}_{y,s}|^{p}ds+C_{2}$
(5.7)

and

$\mathbb{E}|\bar{\xi}_{x,t}|^{p}\leq\left|\bar{\xi}_{x,0}\right|^{p}+Bn_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}\mathbb{E}|\bar{\xi}_{y,s}|^{p}ds,$
(5.8)

for all$\ t\in\mathcal{T}$, where
$\bar{\xi}_{x,t}:=\xi_{x,t}^{(1)}-\xi_{x,t}^{(2)}$. The constants $B,C_{1}$
and $C_{2}$ are independent of the processes $\Xi^{(1)},\Xi^{(2)}$ and
$x\in\gamma$.

Proof. We start with the proof of inequality (5.7). Using Ito Lemma we see
that if $x\in\Lambda_{n}$ then for all $t\in\mathcal{T}$

$|\xi_{x,t}^{(1)}|^{p}=|\zeta_{x}|^{p}+p\int_{0}^{t}(\xi_{x,s}^{(1)})^{p-1}\Phi_{x}(\Xi_{s}^{(1)})ds\\\
+\frac{(p-1)p}{2}\int_{0}^{t}(\xi_{x,s}^{(1)})^{p-2}(\Psi_{x}(\Xi_{s}^{(1)}))^{2}ds\\\
+p\int_{0}^{t}(\xi_{x,s}^{(1)})^{p-1}\Psi_{x}(\Xi_{s}^{(1)})dW_{x}(s).$

Now from assumptions (2.4) and (2.5) and Lemma 21 we can deduce that for all
$t\in\mathcal{T}$

$(\xi_{x,s}^{(1)})^{p-1}\Phi_{x}(\Xi_{s}^{(1)})=(\xi_{x,s}^{(1)})^{p-2}(\xi_{x,s}^{(1)})\Phi_{x}(\Xi_{s}^{(1)})\\\
\leq\left|\xi_{x,s}^{(1)}\right|^{p-2}\left[(b+\frac{1}{2})|\xi_{x,s}^{(1)}|^{2}+\frac{1}{2}\tilde{a}_{x}^{2}\sum_{y\in\gamma_{x}}|\xi_{y,s}^{(1)}|^{2}+\left|\xi_{x,s}^{(1)}V(0)\right|\right]\\\
\leq(b+\frac{1}{2})|\xi_{x,s}^{(1)}|^{p}+\frac{1}{2}\tilde{a}_{x}^{2}|\xi_{x,s}^{(1)}|^{p-2}\sum_{y\in\gamma_{x}}|\xi_{y,s}^{(1)}|^{2}+\left|\xi_{x,s}^{(1)}\right|^{p-1}c\\\
\leq(b+\frac{1}{2})|\xi_{x,s}^{(1)}|^{p}+\frac{1}{2}\tilde{a}_{x}^{2}n_{x}\max_{y\in\bar{\gamma}_{x}}|\xi_{y,s}^{(1)}|^{p}+(1+|\xi_{x,s}^{(1)}|)^{p}c.$

where $\tilde{a}_{x}:=\bar{a}\sqrt{n_{x}}$ and constants $\bar{a},$ $b$ and
$c$ are defined in Assumption 1. In the last inequality we used the simple
estimate $C^{p-1}\leq(1+C)^{p-1}\leq(1+C)^{p}$ for any $C>0$, which holds
because $p>1$. Taking into account that
$\max_{y\in\bar{\gamma}_{x}}|\xi_{y,s}^{(1)}|^{p}\leq\sum_{y\in\bar{\gamma}_{x}}|\xi_{y,s}^{(1)}|^{p}$
and using inequality $(1+\alpha)^{p}\leq 2^{p-1}(1+\alpha^{p})$ we arrive at
the following:

$(\xi_{x,s}^{(1)})^{p-1}\Phi_{x}(\Xi_{s}^{(1)})\leq(b+\frac{1}{2})|\xi_{x,s}^{(1)}|^{p}\\\
+\frac{1}{2}\tilde{a}_{x}^{2}n_{x}\sum_{y\in\bar{\gamma}_{x}}|\xi_{y,s}^{(1)}|^{p}+2^{p-1}c+2^{p-1}c|\xi_{x,s}^{(1)}|^{p}\\\
\leq(b+\frac{1}{2}+2^{p-1}c)|\xi_{x,s}^{(1)}|^{p}+\frac{1}{2}\bar{a}^{2}n_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}|\xi_{y,s}^{(1)}|^{p}+2^{p-1}c.$

In a similar way, using assumption (2.7) we obtain the estimate

$(\xi_{x,s}^{(1)})^{p-2}(\Psi_{x}(\Xi_{s}^{(1)}))^{2}\\\
\leq(\xi_{x,s}^{(1)})^{p-2}\left[3M^{2}n_{x}^{2}|\xi_{x,s}^{(1)}|^{2}+3M^{2}n_{x}\sum_{y\in\gamma_{x}}|\xi_{y,s}^{(1)}|^{2}+3|\Psi(0)|^{2}\right]\\\
\leq
3M^{2}n_{x}^{2}|\xi_{x,s}^{(1)}|^{p}+3M^{2}n_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}|\xi_{y,s}^{(1)}|^{p}+3m^{2}|\xi_{x,s}^{(1)}|^{p-2}\\\
\leq
3\left(n_{x}^{2}+M^{2}m^{2}2^{p-1}\right)|\xi_{x,s}^{(1)}|^{p}+3M^{2}n_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}|\xi_{y,s}^{(1)}|^{p}+3m^{2}2^{p-1}.$
(5.9)

Observe that $n_{x}\geq 1$. Thus there exist constants $C_{1},C_{2}>0$ such
that

$|\xi_{x,t}^{(1)}|^{p}\leq\left|\xi_{x,0}^{(1)}\right|^{p}+C_{1}n_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}|\bar{\xi}_{y,s}|^{p}ds+C_{2}+p\int_{0}^{t}(\xi_{x,s}^{(1)})^{p-1}\Psi_{x}(\Xi_{s}^{(1)})dW_{x}(s),$

which implies that (5.7) holds.

The proof of inequality (5.8) can be obtained in a similar way. Using the
relation

$\bar{\xi}_{x,t}=\bar{\xi}_{x,0}+\int_{0}^{t}\left(\Phi_{x}(\Xi_{s}^{(1)})-\Phi_{x}(\Xi_{s}^{(2)})\right)ds+\int_{0}^{t}\left(\Psi_{x}(\Xi_{s}^{(1)})-\Psi_{x}(\Xi_{s}^{(2)})\right)dW_{x}(s),$

$\ t\in\mathcal{T},$ and applying the Ito Lemma to $|\bar{\xi}_{x,t}|^{p}$ we
obtain the inequality

$|\bar{\xi}_{x,t}|^{p}\leq\left|\bar{\xi}_{x,0}\right|^{p}+Bn_{x}^{2}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}|\bar{\xi}_{y,s}|^{p}ds\\\
+\int_{0}^{t}p(\bar{\xi}_{x,t})^{p-1}\left(\Psi_{x}(\Xi_{s}^{(1)})-\Psi_{x}(\Xi_{s}^{(2)})\right)dW_{x}(s)$
(5.10)

for some constant $B>0$, which implies the result.

###### Lemma 23

Consider a bounded measurable map
$\varrho:\mathcal{T\rightarrow}l_{\alpha}^{1}$, $\rho\geq 0$, and assume that
its components satisfy the inequality

$\varrho_{x}(t)\leq
Bn_{x}^{k}\sum_{y\in\bar{\gamma}_{x}}\int_{0}^{t}\varrho_{y}(s)ds+b_{x},\
t\in\mathcal{T},\ x\in\gamma,$ (5.11)

for some constants $B,k>0$ and $b:=(b_{x})_{x\in\gamma}\in$ $l_{\alpha}^{1}$,
$b_{x}\geq 0$. Then we have the estimate

$\sum_{x\in\gamma}e^{-\beta|x|}\sup_{t\in\mathcal{T}}\varrho_{x}(t)\leq
K(\alpha,\beta)\sum_{x\in\gamma}e^{-\alpha|x|}b_{x}{}^{p}$ (5.12)

for any $\beta>\alpha$, with
$K(\alpha,\beta)=\sum_{n=0}^{\infty}\frac{L^{n}T^{n}}{(\beta-\alpha)^{qn}}\frac{n^{qn}}{n!}<\infty$,
cf. Theorem 17.

Proof. Inequality (5.11) can be rewritten in the form

$\varrho_{x}(t)\leq\sum_{y\in\gamma}Q_{x,y}\int_{0}^{t}\varrho_{y}(s)ds+b_{x},\
t\in\mathcal{T,}$

where

$Q_{x,y}=\begin{cases}Bn_{x}^{k},&|x-y|\leq\rho,\\\ 0,&|x-y|>\rho\end{cases}$

for all $x\in\gamma.$ We have
$\varrho\in\mathcal{B}(\mathcal{T},l_{\alpha}^{1})$, and $|Q_{x,y}|\leq
Bn_{x}^{k}$. Therefore using Theorem 18 we conclude that for any $q\in(0,1)$
matrix $\left(Q_{x,y}\right)$ generates an Ovsjannikov operator of order $q$
on $\mathcal{L}^{1}$. Therefore we can now use Corollary 20 to conclude that
(5.12) holds.

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