#  Nonlinear Stochastic parabolic partial differential equations with a
monotone operator of the Ladyzenskaya-Smagorinsky type, driven by a Lévy noise

Phuong Nguyen1,2, Krutika Tawri1 and Roger Temam1

###### Abstract.

The aim of this article is to show the global existence of both martingale and
pathwise solutions of stochastic equations with a monotone operator, of the
Ladyzenskaya-Smagorinsky type, driven by a general Lévy noise. The classical
approach based on using directly the Galerkin approximation is not valid.
Instead, our approach is based on using appropriate approximations for the
monotone operator, Galerkin approximations and on the theory of martingale
solutions.

1 Department of Mathematics and Institute for Scientific Computing and Applied
Mathematics, Indiana University, Bloomington, IN 47405, USA.
2 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX
79409, USA.

###### Contents

  1. 1 Introduction
  2. 2 Mathematical framework
  3. 3 Existence results
    1. 3.1 The main results
  4. 4 Galerkin scheme
    1. 4.1 Moment estimates
    2. 4.2 Estimates in fractional Sobolev spaces
    3. 4.3 Tightness
    4. 4.4 Passage to the limit
      1. 4.4.1 Convergence results
  5. 5 Passage to the limit $R\rightarrow\infty$
  6. 6 Passage to the limit $\widetilde{R}\rightarrow\infty$
  7. 7 Global pathwise solution of (3.1)-(3.2)
    1. 7.1 Pathwise uniqueness
    2. 7.2 Pathwise solution
  8. 8 Examples
    1. 8.1 The Ladyzensaya-Smagorinsky equations
    2. 8.2 Equations without the operator $A$ and with $B=0$
      1. 8.2.1 Evolution equations with $p$-Laplacian
      2. 8.2.2
    3. 8.3 Polynomial nonlinearity
    4. 8.4 Other examples
  9. A About the Skorohod representation theorem by Justin Cyr
  10. 2 Acknowledgment

##  1\. Introduction

In this article we study the existence and uniqueness of global martingale and
pathwise solutions to stochastic partial differential equations (SPDEs) with a
nonlinear monotone operator driven by a Lévy noise, not necessarily square-
integrable. As explained below from the physical point of view, the two types
of equations that we consider are firstly the equations of Ladyzenskaya-
Smagorinsky type, that are essentially the Navier Stokes equations with a
viscosity depending on the gradient of velocity. The second type of equations
that we consider are parabolic equations with a linear elliptic part and a
nonlinearity of polynomial type. Such equations were already considered in
[14], [20] with restrictions on the degree of the polynomials but here
polynomials of arbitrary degrees can be considered. Such equations appear in
biology in e.g. population dynamics.
The equation that we consider is written in the abstract form

(1.1) $\begin{cases}\mathrm{d}u(t)+[\gamma
Au(t)+B(u(t),u(t))+F(u(t))]\mathrm{d}t\\\ \hskip
36.135pt=G(u(t-))\mathrm{d}W+\int_{E_{0}}K(u(t-),\xi)\mathrm{d}\widehat{\pi}(t,\xi)+\int_{E\setminus
E_{0}}\mathcal{K}(u(t-),\xi)\mathrm{d}\pi(t,\xi),\\\ u(0)=u_{0},\end{cases}$

where $u$ takes values in a real, separable Hilbert space $H$. On the left-
hand side of (1.1), $A$ is a linear operator on $H$, generally unbounded, $B$
is a bilinear form of the type often encountered in fluid dynamics and $F$ is
a monotone operator on a reflexive Banach space $X$ which is continuously and
densely embedded into $H$ (see e.g. [8] and below). In Section 2 we introduce
the precise assumptions on $F$. The terms on the right-hand side of (1.1) are
stochastic integrals that represent different effects of a Lévy noise, which
is fitting in stochastic models of fluid dynamics as a way to represent
interactions that occur at random times. The term $G(u)\mathrm{d}W$ has
effects that are continuous in time on the system, whereas the two terms
$\int_{E_{0}}K(u(t-),\xi)\mathrm{d}\widehat{\pi}(t,\xi)$, $\int_{E\setminus
E_{0}}\mathcal{K}(u(t-),\xi)\mathrm{d}\pi(t,\xi),$ influence the system
discretely in time.

Interest in SPDEs with Lévy noise has been growing in the recent years, see
e.g., [12], [17], [15], [42], [43]. In [13] and [52] the authors use processes
with jumps to model phase transitions in bursts of wind that contribute to the
dynamics of El Niño. In [48] the authors use processes with jumps to model
phase transitions in precipitation dynamics. Models based on SPDEs with Lévy
noise have been used to describe observational records in paleoclimatology in
[21], with the jump events of the Lévy noise proposed as representing abrupt
triggers for shifts between glacial and warm climate states. See also [24] for
a comparison of Wiener noise and Lévy noise in modeling climate phenomena.

Our main results are Theorem 3.3 and Theorem 3.4 which establish the existence
and uniqueness of martingale and pathwise solutions to the equation (1.1) (see
also Definitions 3.1 and 3.2). We use the classical Galerkin approximation
method to prove Theorem 3.3. However a traditional application of the Galerkin
method fails as the projectors $P_{n}$, defined in (4.1), may not necessarily
be projectors in the Banach space $X$ associated with $F$ (see Section 2).
Hence we approximate the monotone operator $F$ in (1.1) by a family of
monotone operators $F^{R}$ acting in another Hilbert space, continuously and
densely embedded in $H$, so as to recover certain useful properties when
$R\rightarrow\infty$. The a priori estimates for (1.1) also suggest the need
to truncate the bilinear term $B$ in the original system into
$B_{\widetilde{R}}$ (see equations (4.2)), then pass to the limit
$\widetilde{R}\rightarrow\infty$.

The paper is organized as follows. In Section 3 we state the main results. In
Section 2 we layout the functional analytical and probabilistic framework and
the assumptions on the terms appearing in (1.1). In Section 4 we introduce the
Galerkin system for the approximation equations (4.2). We establish a priori
estimates in Section 4.1 and tightness of the laws of the Galerkin
approximations in Section 4.3. Then we use the Skorohod convergence theorem to
obtain a sequence, same in distribution, that converges a.s. on a new
probability space and pass to the limit to obtain a global martingale solution
to the truncated equation (4.2) in Section 4.4. We then pass to the limit
$R\rightarrow\infty$ and $\widetilde{R}\rightarrow\infty$ in Sections 5 and 6
respectively to obtain a global martingale solution to (1.1). In Section 7 we
establish pathwise uniqueness for the equation (1.1) and then obtain a global
pathwise solution to (1.1) using a well-known argument by Gyöngy and Krylov
that generalizes to infinite dimensions the classical Yamada–Watanabe theorem.
In Section 8 we describe examples that fit the setting of our analysis.

##  2\. Mathematical framework

The following is a general framework which we will adapt to the actual problem
under consideration. We consider a separable, real Hilbert space $V\subset H$,
such that the embedding is dense and compact. Similarly, $X$ is a reflexive
Banach space continuously and densely embedded into $H$. We may thus define
the Gelfand inclusions

(2.1) $\displaystyle V\subset H\equiv H^{\prime}\subset V^{\prime},$ (2.2)
$\displaystyle X\subset H\equiv H^{\prime}\subset X^{\prime}.$

where $V^{\prime}$, $X^{\prime}$ and $H^{\prime}$ are the dual spaces of $V$,
$X$ and $H$ respectively. In fact, we will strengthen (2.1) and (2.2) by
requiring that $V\cap X$ is dense in $V$ and $X$, so that, for
$V^{\prime}+X^{\prime}$ denoting the dual of $V\cap X$, we have

(2.3a) $V\cap X\text{}\subset X\subset H\subset X^{\prime}\subset
V^{\prime}+X^{\prime},$ and (2.3b) $V\cap X\text{}\subset V\subset H\subset
V^{\prime}\subset V^{\prime}+X^{\prime}.$

We denote by $(\cdot,\cdot)$, $|\cdot|$, $((\cdot,\cdot))$, and $\|\cdot\|$
the norms and inner products of $H$, and $V$ respectively.
The duality pairing between $V^{\prime}$ and $V$ will be denoted by
$\langle\cdot,\cdot\rangle$.
For the purpose of the abstract analysis in this paper, in continuation with
[49], and keeping our primary examples in mind, we will first assume that, for
some open set $\mathcal{O}\subseteq\mathbb{R}^{d}$, $V$ and $H$ are closed
subspaces of $(H^{1}(\mathcal{O}))^{d}$ and $(L^{2}(\mathcal{O}))^{d}$
respectively and that $X=(W^{k,p}(\mathcal{O}))^{d}\cap V$, for some $p>2$ and
an integer $k\geq 0$. However slightly different settings can be introduced
depending on the properties of the monotone operator (see Remark 2.1 and
Section 8.4).

We now give the precise assumptions on each of the terms appearing in (1.1),
beginning with the linear term $A$. In our analysis we shall assume that
$\gamma>0$ in (1.1). However we show in Section 8.2 (see Remark 8.4), how the
case $\gamma=0$ is handled. We assume that $A:D(A)\subset V\rightarrow H$ is
an unbounded, densely defined, bijective, linear operator such that
$(Av,w)=((v,w))$ for all $v,w\in D(A)$. Under these conditions on $A$, the
space $D(A)$ is complete under the Hilbertian norm $|v|_{D(A)}:=|Av|$. When
$H$ is a subspace of $(L^{2}(\mathcal{O}))^{d}$ and $V$ a subspace of
$(H^{1}(\mathcal{O}))^{d}$, the role of $A$ can be played by the negative
Laplacian associated with suitable boundary conditions. It is clear that for
suitable boundary conditions, the operator $A$ is symmetric. The operator $A$
can be viewed as a linear continuous operator from $V$ to $V^{\prime}$ via

(2.4) $\langle Av,w\rangle:=((v,w)),\qquad\textnormal{for all }v,w\in V.$

Furthermore, since we assume that $V$ is compactly embedded in $H$, it follows
that $A^{-1}$ is a compact operator on $H$. Since $A$ is assumed to be
symmetric and surjective we easily see that $A^{-1}$ is self-adjoint. By the
spectral theory for positive compact operators, there exists an orthonormal
basis $\left(w_{k}\right)_{k=1}^{\infty}$ of $H$ made of eigenvectors of $A$.
The sequence $\left(w_{k}\right)_{k=1}^{\infty}$ lies in $D(A)$ and forms an
orthogonal basis in $D(A),V,V^{\prime}$. Furthermore, each $w_{k}$ is an
eigenvector of $A$ whose corresponding eigenvalue $\lambda_{k}$ is positive
and $\lambda_{k}\rightarrow\infty$ as $k\rightarrow\infty$. Note that

$D(A^{1/2}):=\Big{\\{}\sum_{k=1}^{\infty}\alpha_{k}w_{k}\in
H:\sum_{k=1}^{\infty}\alpha_{k}^{2}\lambda_{k}<\infty\Big{\\}}$

is nothing but the space $V$. Under the conditions given above, one can show
that the embedding $D(A)\subset D(A^{1/2})=V$ is compact (see e.g. [37],
[51]).

Next we assume that $B\colon V\times D(A)\to V^{\prime}$ is bilinear and
continuous. Furthermore, we assume that $B$ maps the subspace $D(A)\times
D(A)$ continuously into $H$ and that $B$ possesses the cancellation property

(2.5) $\langle B(v,w),w\rangle=0,\qquad\textnormal{for all }v\in V\textnormal{
and }w\in D(A).$

We also assume that

(2.6)
$\begin{split}\begin{cases}|{B(v,w)}|_{D(A)^{\prime}}&\lesssim|v|\|w\|\qquad\textnormal{for
all }v\in H,w\in V,\\\ &\lesssim{\|v\||w|}\qquad\textnormal{for all }v\in
V,w\in H,\end{cases}\end{split}$

and,

(2.7) $\displaystyle|{B(v,v)}|_{V^{\prime}}$
$\displaystyle\lesssim|v|^{\frac{1}{2}}\|v\|^{\frac{3}{2}}\qquad\textnormal{for
all }v\in V.$

In Section 7.1, we also use the following assumption:

(2.8) $\displaystyle|B(v,v)|_{X^{\prime}}\lesssim|v|\|v\|\qquad\textnormal{for
all }v\in V.$

We will often write $B(v):=B(v,v)$.
For certain subspaces $V$ and $H$ of $(H^{1}(\mathcal{O}))^{d}$ and
$(L^{2}(\mathcal{O}))^{d}$ respectively, we have in mind for $B$ the bilinear
function $B(v,w):=(v\cdot\nabla)w$, corresponding to the nonlinear terms in
the Navier-Stokes equations (see e.g. [51]).

######  Remark 2.1.

The framework above, which we will use till the end of Section 8.3, covers the
main example that we have in view, namely the Ladyzenskaya-Smagorinsky
equations and equations of the same type. We will see in Section 8.4 a
variation of this framework which covers other related equations.

Before discussing the coefficients $G$ and $K$ of the noise terms in equation
(1.1) we discuss the assumptions on the noise. The noise processes will be
defined on a fixed probability space $(\Omega,\mathcal{F},{\bf P})$. In this
paper, we assume that $W$ is a Wiener process taking values in a real,
separable Hilbert space $U$ (see [6]). We denote by $Q$ the covariance
operator of $W$ and we set $U_{0}:=Q^{1/2}(U)$. We assume that $\pi$ is a
Poisson random measure on $(0,\infty)\times E$ arising from a stationary
Poisson point process $\Pi$ on a measurable space $(E,\mathscr{E})$. The
intensity measure of $\pi$ has the form $\mathrm{d}\nu\otimes\mathrm{d}t$ for
some $\sigma$-finite measure $\nu$ on $(E,\mathscr{E})$. In equation (1.1) we
fix a set $E_{0}\in\mathscr{E}$ such that $\nu(E\setminus E_{0})<\infty$.
Furthermore, we assume that $W$ is independent of $\pi$. Under this condition
it is possible to construct a filtration $\left(\mathcal{F}_{t}\right)_{t\geq
0}$ on $(\Omega,\mathcal{F},{\bf P})$ such that $W$ is an
$\mathcal{F}_{t}-$Wiener process, $\Pi$ is an $\mathcal{F}_{t}-$ Poisson point
process and such that the filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\bf P})$ satisfies the usual
conditions (as explained in e.g. [16]). The motivation behind these
assumptions stems from the Lévy-Khinchin decomposition of a Lévy process $L$
which, informally, gives us the rule for the decomposition of integration with
respect to $L$ as
$\mathrm{d}L=a\mathrm{d}t+\mathrm{d}W+\mathrm{d}\widehat{\pi}+\mathrm{d}\pi$,
where $\pi$ is the jump measure of $L$. Hence as an example, we can set
$E:=U\setminus\\{0\\}$, $E_{0}:=\\{y\in U:0<|y|_{U}<1\\}$ and endow the space
$E$ with a metric such that it is separable, complete, such that $A\subset E$
is bounded if and only if $A$ is separated from 0, and $\nu$ is finite on
bounded subsets of $E$.
Now, let $L_{2}(U_{0};H)$ denote the space of Hilbert-Schmidt operators from
${U_{0}}$ to $H$ equipped with the Hilbert-Schmidt norm
$\|S\|^{2}_{L_{2}(U_{0};H)}:=\sum_{n=1}^{\infty}|S{e_{n}}|^{2}$ where
$\\{e_{n}\\}_{n=1}^{\infty}$ is an orthonormal basis of ${U_{0}}$.

Then for the coefficients $G$ and $K$ in equation (1.1) we assume that,

  * •

$G\colon H\rightarrow L_{2}(U_{0},H)$ and $G$ maps $V$ into $L_{2}(U_{0},V)$,
and

  * •

$K\colon H\times E\rightarrow H$ is a Borel measurable function and $K$ maps
$V\times E$ into $V$.

  * •

And for some $\rho>0$ we have

(2.9)
$\displaystyle\left\|G(v)\right\|^{2}_{L_{2}(U_{0},H)}+\int_{E_{0}}|K(v,\xi)|^{2}\mathrm{d}\nu(\xi)\leq\rho(1+|v|^{2})\qquad\textnormal{for
all }v\in H,$ (2.10)
$\displaystyle\left\|G(v)-G(w)\right\|^{2}_{L_{2}(U_{0},H)}+\int_{E_{0}}|K(v,\xi)-K(w,\xi)|^{2}\mathrm{d}\nu(\xi)\leq\rho|v-w|^{2}\quad\textnormal{for
all }v,w\in H.$

Note that the above conditions and Lemma 3.2 from [14] imply that the
stochastic integrals $\int_{0}^{t}G(u(s-))\mathrm{d}W(s)$ and
$\int_{(0,t]}\int_{E_{0}}K(u(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi)$ appearing
on the right-hand side of equation (1.1) are well-defined for every adapted,
càdlàg, $H$-valued process $u$ that satisfies ${\bf
E}\int_{0}^{T}|u(s)|^{2}\mathrm{d}s<\infty$.
We assume that $\mathcal{K}\colon H\times E\rightarrow V$ is a measurable
function. In contrast to $K$, there will be no growth or Lipschitz assumptions
on $\mathcal{K}$. We introduce the additional term with coefficient
$\mathcal{K}$ on the right-hand side of (1.1) to represent the influence of
large jumps from Lévy noise. If the Lévy noise is not square integrable, then
the influence of large jumps cannot be represented without this additional
term; see e.g., [16] or [45].

Now we introduce the nonlinear term $F:X\rightarrow X^{\prime}$. We assume
that $F$ satisfies the following conditions:

  * •

(F1) (Monotonicity): For all $v,w\in X$ we have

(2.11) $\displaystyle(F(v)-F(w),v-w)_{X,X^{\prime}}\geq 0.$

  * •

(F2) (Hemicontinuity) For all $u,v,w\in X$, the map

(2.12) $\displaystyle\mathbb{R}:\lambda\mapsto({F(u+\lambda
v)},{w})_{X,X^{\prime}}\quad\text{is continuous.}$

In view of using the results of [49] we also assume the existence of a
functional $J(v):X\rightarrow\mathbb{R}$, such that its Gâteaux derivative
$J^{\prime}(v)=F(v)$. Then, given (2.11), we know from [36], [23] that for
given $v\in X$, the map $v\mapsto J(v)$ is convex. For $i\in\mathbb{N}$, we
denote by $D^{i}v$, the ordered tuple of all partial derivatives of $v$ of
order $i$. We also assume, as in [49] that $J$ possesses the integral
representation:

(2.13) $\displaystyle
J(v)=\sum_{i=0}^{k}\int_{\mathcal{O}}\mathscr{J}_{i}(D^{i}v)\mathrm{d}x,$

where for every $i=0,1,...,k$,
$\mathscr{J}_{i}:\mathbb{R}^{d^{i+1}}\rightarrow\mathbb{R}$ is further assumed
to have the following properties:
(i) $\mathscr{J}_{i}$ is convex and attains its minima at
$0\in\mathbb{R}^{d^{i+1}}$ and there exist $c_{1}>0,c_{2}\geq 0$ such that

(2.14) $\displaystyle xD^{2}\mathscr{J}_{i}(x)x^{T}\geq
c_{1}|x|_{l^{p}(\mathbb{R}^{d^{i+1}})}^{p}-c_{2}\quad\forall
x\in\mathbb{R}^{d^{i+1}}.$

(ii) $\mathscr{J}_{i}$ satisfies the following growth conditions: there exists
$c_{3}\geq 0$ such that $\forall x\in\mathbb{R}^{d^{i+1}}$

(2.15) $\begin{split}\mathscr{J}_{i}(x)&\geq
c_{1}|x|_{l^{p}(\mathbb{R}^{d^{i+1}})}^{p}-c_{3},\\\
|\nabla_{x}\mathscr{J}_{i}(x)|_{l^{\frac{{p}}{{p}-1}}(\mathbb{R}^{d^{i+1}})}&\leq
c_{1}|x|_{l^{{p}}(\mathbb{R}^{d^{i+1}})}^{{p}-1}+c_{3},\\\
|x\,D^{2}\mathscr{J}_{i}(x)|_{l^{\frac{{p}}{{p}-1}}(\mathbb{R}^{d^{i+1}})}&\leq
c_{1}|x|_{l^{{p}}(\mathbb{R}^{d^{i+1}})}^{{p}-1}+c_{3},\end{split}$

where $D^{2}\mathscr{J}_{i}$ is the Hessian of $\mathscr{J}_{i}$. Observe that
by assuming the above conditions we have that the hypothesis (2.18) below is
satisfied. Adhering to the notation
$f_{i}(x):\mathbb{R}^{d^{i+1}}\rightarrow\mathbb{R}^{d^{i+1}}$ given by
$f_{i}=\nabla_{x}\mathscr{J}_{i}(x)$ from [49] we have for $v,\varphi\in X$,

(2.16) $\displaystyle(F(v),\varphi)$
$\displaystyle=\sum_{i=0}^{k}\int_{\mathcal{O}}\nabla\mathscr{J}_{i}(D^{i}v)\cdot
D^{i}\varphi\,\mathrm{d}x$
$\displaystyle=\sum_{i=0}^{k}\int_{\mathcal{O}}f_{i}(D^{i}v)\cdot
D^{i}\varphi\,\mathrm{d}x.$

We also assume,

  * •

(F3) (Coercivity) We have $\forall v\in X$,

(2.17) $\displaystyle(F(v),v)_{X,X^{\prime}}\geq
c_{1}\left\|v\right\|^{p}_{X}.$

  * •

(F4) (Boundedness) There exist $c_{4}\geq 0$ such that $\forall v\in X$,

(2.18) $\displaystyle\left\|F(v)\right\|_{X^{\prime}}\leq
c_{1}\left\|v\right\|^{p-1}_{X}+c_{4}.$

Here we would like to point out that the assumptions (F1)-(F4) are not unusual
and that they have physical significance. Examples of such an operator $F$
include the monotone operator appearing in the Ladyzenskaya equations
(Smagorinsky’s model of turbulence) and the $p$-Laplacian. These examples are
discussed at length in Section 8.

##  3\. Existence results

###  3.1. The main results

In order to prove the existence of solutions to (1.1) one can first prove the
existence when $\mathcal{K}=0$ and then add the noise term using the so-called
piecing out argument (see e.g.[29],[30]). So we will first consider the
following stochastic differential equation:

(3.1) $\displaystyle\mathrm{d}u(t)+[\gamma A(u(t))+$ $\displaystyle
B(u(t),u(t))+F(u(t))]\mathrm{d}t$
$\displaystyle=G(u(t-))\mathrm{d}W(t)+\int_{E_{0}}K(u(t-),\xi)\mathrm{d}\widehat{\pi}(t,\xi)\quad{\text{in
}\mathcal{O}\times(0,T)}$ (3.2) $\displaystyle u(0)$
$\displaystyle=u_{0}\quad\text{in }\mathcal{O}.$

The precise meaning of (3.1)-(3.2) appears below where we define
(probabilistically) "weak" and "strong" solutions.

######  Definition 3.1.

A pair $(\widetilde{\mathscr{S}},\widetilde{u})$ is called a martingale
solution to (3.1)-(3.2) if for a stochastic basis
$\widetilde{\mathscr{S}}=(\widetilde{\Omega},\widetilde{\mathcal{F}},(\widetilde{\mathcal{F}}_{t})_{t\geq
0},\widetilde{\bf P},\widetilde{W},\widetilde{\pi})$, the process
$\widetilde{u}\in L^{2}(\widetilde{\Omega},L^{\infty}(0,T;H))\cap
L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};V)\cap
L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};X)$ is $\widetilde{\mathcal{F}}$-adapted with càdlàg sample paths in
$V^{\prime}+X^{\prime}$, $\widetilde{\bf P}$-a.s., and $\widetilde{u}(0)$ is
$\widetilde{\mathcal{F}}_{0}$-measurable with law $\mu_{0}$, and the equation

(3.3) $\displaystyle\widetilde{u}(t)+\int_{0}^{t}[\gamma A(\widetilde{u}(s))+$
$\displaystyle
B(\widetilde{u}(t),\widetilde{u}(t))+F(\widetilde{u}(s))]\mathrm{d}s$
$\displaystyle=\widetilde{u}(0)+\int_{0}^{t}G(\widetilde{u}(s-))\mathrm{d}\widetilde{W}(s)+\int_{(0,t]}\int_{E_{0}}K(\widetilde{u}(s-),\xi)\mathrm{d}\widehat{\widetilde{\pi}}(s,\xi),$

holds $\widetilde{{\bf P}}$-a.s., for all $t\in[0,T]$.

######  Definition 3.2.

We are given a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},{\bf P},W,\pi)$ and a random variable
$u_{0}\in L^{2}(\Omega,\mathcal{F}_{0},{\bf P};H)$ with law $\mu_{0}$. We say
that an $\mathcal{F}_{t}$-adapted process $u\in
L^{2}(\Omega,L^{\infty}(0,T;H))\cap
L^{2}([0,T]\times\Omega,\mathrm{d}t\otimes\mathrm{d}{\bf P};V)\cap
L^{p}([0,T]\times\Omega,\mathrm{d}t\otimes\mathrm{d}{\bf P};X)$ is a (global)
pathwise solution to (3.1)-(3.2) if $u$ has càdlàg sample paths in
$V^{\prime}+X^{\prime}$, ${\bf P}$-a.s., and if the equation

(3.4) $\displaystyle u(t)+\int_{0}^{t}[\gamma A(u(s))+$ $\displaystyle
B(u(t),u(t))+F(u(s))]\mathrm{d}s$
$\displaystyle=u_{0}+\int_{0}^{t}G(u(s-))\mathrm{d}W(s)+\int_{(0,t]}\int_{E_{0}}K(u(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi),$

holds ${\bf P}$-a.s., for all $t\in[0,T]$.

Now we state the main results of this paper:

######  Theorem 3.3.

We assume that we are given an $\mathcal{F}_{0}$-measurable random variable
$u_{0}:\Omega\rightarrow{H}$ with law $\mu_{0}$ that satisfies

(3.5) $\displaystyle\int_{H}|v|^{2}d\mu_{0}(v)<\infty.$

We also assume that $G,K$ satisfy the hypotheses (2.9)-(2.10), the term $B$
satisfies (2.6)-(2.7), $\gamma>0$ and the term $A$ satisfies (2.4), and that
the nonlinear term $F$ satisfies the conditions (2.11)$-$(2.18). Then for any
$p>2$ there exists a martingale solution $\widetilde{u}$ to (3.1)-(3.2) in the
sense of Definition 3.1.

######  Theorem 3.4.

We assume that the hypotheses of Theorem 3.3 hold. We additionally assume that
the term $B$ satisfies (2.8). Then for any $p>2$, there exists a unique global
pathwise solution $u$ to (3.1)-(3.2), in the sense of Definition 3.2, with
initial condition $u_{0}$.

##  4\. Galerkin scheme

We describe here the Galerkin approximation of the problem, then derive moment
estimates, estimates in fractional Sobolev spaces, tightness of the Galerkin
approximation and then pass to the limit to obtain the existence of solutions
in Theorem 3.3.
To construct the Galerkin scheme for equation (3.1)-(3.2) we use the
orthonormal basis $(w_{k})^{\infty}_{k=1}$ of $H$ which consists of the
eigenvectors of $A$ as described in Section 2. Consider the finite dimensional
subspace of $H$:

$H_{n}:=\text{span}\\{w_{1},...,w_{n}\\},n\geq 1.$

Let $P_{n}:H\rightarrow H_{n}$ be the finite dimensional projection of $H$
onto $H_{n}$, i.e.,

(4.1) $P_{n}v:=\sum_{i=1}^{n}(v,w_{i})w_{i},\text{ }v\in H.$

Note that since the $w_{k}$ are eigenvectors of $A$, $P_{n}$ is also an
orthogonal projector in $V$, $D(A)$, $V^{\prime}$ onto $H_{n}$. However this
property is not true for the space $X$ and this is precisely what prevents us
from applying directly the Galerkin scheme to (3.1)-(3.2) in the traditional
way. Hence instead we consider the following truncated stochastic equations:

(4.2) $\begin{split}\begin{cases}&\mathrm{d}u(t)+[\gamma
A(u(t))+\theta_{\widetilde{R}}(|u|)B(u(t),u(t))+F^{R}(u(t))]\mathrm{d}t\\\
&\hskip
108.405pt=G(u(t-))\mathrm{d}W(t)+\int_{E_{0}}K(u(t-),\xi)\mathrm{d}\widehat{\pi}(t,\xi)\\\
&\hskip 86.72377ptu(0)=u_{0}.\end{cases}\end{split}$

Here, for the parameter $\widetilde{R}>0$,
$\theta_{\widetilde{R}}:[0,\infty)\rightarrow[0,1]$ is a Lipschitz function
with constant $1$ such that $\theta_{\widetilde{R}}\equiv 1$ on
$[0,\widetilde{R}]$ and $\theta_{\widetilde{R}}\equiv 0$ on
$[2\widetilde{R},\infty)$. We will frequently denote
$\theta_{\widetilde{R}}(|u|)B(u,u)$ by $B_{\widetilde{R}}(u,u)$. Given another
parameter $R>0$, $F^{R}:V\rightarrow V^{\prime}$ is a monotone, hemicontinuous
and coercive operator on $V$ defined later and approximating $F$ as
$R\rightarrow\infty$ as described in [49] and below.
Like in (2.16), for well-chosen functions
$f^{R}_{i}:\mathbb{R}^{d^{i+1}}\rightarrow\mathbb{R}^{d^{i+1}}$ we consider
$F^{R}:V\rightarrow V^{\prime}$ of the form

(4.3) $\displaystyle(F^{R}(v),\varphi)$
$\displaystyle=\sum_{i=0}^{k}\int_{\mathcal{O}}f^{R}_{i}(D^{i}v)\cdot
D^{i}\varphi\,\mathrm{d}x\quad\text{for }v,\varphi\in V.$

The choice of the functions $f^{R}_{i}$ is made by the following construction
from [49].
For each $i=0,..,k$ we choose
$f_{i}^{R}:\mathbb{R}^{d^{i+1}}\rightarrow\mathbb{R}^{d^{i+1}}$ as follows:
For $x\in\mathbb{R}^{d^{i+1}}$

(4.4) $\displaystyle
f_{i}^{R}(x)=f_{i}(x)\mathbbm{1}_{\\{|x|_{p}<R\\}}+\left[f_{i}\left(\frac{Rx}{|x|_{p}}\right)+\left(1-\frac{R}{|x|_{p}}\right)xD^{2}\mathscr{J}_{i}\left(\frac{Rx}{|x|_{p}}\right)\right]\mathbbm{1}_{\\{|x|_{p}>R\\}},$

where for brevity we use the notation
$|\cdot|_{p}\,:=|\cdot|_{l^{p}(\mathbb{R}^{d^{i+1}})}$ given $x\in
R^{{d^{i+1}}}$ and $i=0,1,...,k$. We also present here in brief a few results
proved in [49] that we will make use of:

  * •

(FR1) (Monotonicity): We have $\forall u,v\in V$,

(4.5) $\displaystyle\langle F^{R}(u)-F^{R}(v),u-v\rangle\geq 0.$

  * •

(FR2) (Hemicontinuity) For all $u,v,w\in V$, the map

(4.6) $\displaystyle\mathbb{R}:\lambda\mapsto\langle F^{R}(u+\lambda
v),w\rangle\quad\text{is continuous.}$

  * •

(FR3) (Equicoercivity) With an adaptation of the proof of Theorem 2.2 in [49]
we can show that there exist $c_{5}=c_{5}(\mathcal{O},d)\geq 0$ independent of
$R$ and some $R_{0}>1$ such that for any $R>R_{0}$ and any $v\in V$,

(4.7) $\displaystyle\langle F^{R}(v),v\rangle\geq
c_{1}\left\|v\right\|^{2}-c_{5}.$

The importance of (4.7) is mentioned in Remark 8.4.

  * •

(FR4) (Boundedness) We have for any $v\in V$,

(4.8) $\displaystyle|{F^{R}(v)}|_{V^{\prime}}\leq
c_{1}R^{p-2}\left\|v\right\|.$

Now, for fixed $R>R_{0}$ and $\widetilde{R}>0$ we can introduce the $n^{th}$
Galerkin approximation of (4.2) $u^{n}:=u^{n,R,\widetilde{R}}$ that satisfies

(4.9) $\begin{cases}&\mathrm{d}u^{n}(t)+\left[\gamma
A(u^{n}(t))+P_{n}B_{\widetilde{R}}(u^{n}(t),u^{n}(t))+{P_{n}F^{R}(u^{n}(t))}\right]\mathrm{d}t\\\
&\hskip
144.54pt=P_{n}G(u^{n}(t-))\mathrm{d}W(t)+\int_{E_{0}}P_{n}K(u^{n}(t-),\xi)\mathrm{d}\widehat{\pi}(t,\xi),\\\
&u^{n}(0)=P_{n}u_{0}=:u^{n}_{0}.\end{cases}$

Since $B_{\widetilde{R}}$ is globally Lipschitz on $H_{n}$, $F^{R}$ satisfies
(4.5)-(4.8) and $G$ and $K$ satisfy the Lipschitz condition (2.10), the
existence of a unique pathwise solution $u^{n}$ to the SDE (4.9) is guaranteed
thanks to Theorem 1.2 in [31] and its generalization in Theorem 3.1 in [1].
(See also Theorem 3.5 in [14] for a local existence result.)

###  4.1. Moment estimates

In order to construct a solution to the system (4.2), we need to perform a
priori estimates for the approximating sequence $u^{n}:=u^{n,R,\widetilde{R}}$
and then pass to the limit $n\rightarrow\infty$, $R\rightarrow\infty$,
$\widetilde{R}\rightarrow\infty$ (in that order).

######  Lemma 4.1.

($L^{2}$-regularity) Suppose that F satisfies the conditions (2.13)-(2.15), so
that the family of operators $F^{R}$ satisfies (FR1)-(FR4). Then there exists
a constant $C(T,\rho,c_{1},c_{5})>0$, independent of $n$ and in fact
independent of $R$ and $\widetilde{R}$ such that for every $n\geq 1$:

(4.10) $\displaystyle{\bf E}\sup_{s\in[0,T]}|u^{n}(s)|^{2}+(\gamma+c_{1}){\bf
E}\int_{0}^{T}\left\|u^{n}(s)\right\|^{2}\mathrm{d}s\leq C{\bf
E}|u_{0}|^{2}=:K_{1}(u_{0},T,\rho,c_{1},c_{5}).$

###### Proof.

We begin by applying the Itô formula to the process in (4.9). To be precise,
here we use the Itô formula in the case of solutions to SDEs with Lévy noise
as stated in Theorem 2.19 in [14] and the special case in which
$\psi(u)=|u|^{2}$ as stated in Corollary 1 in [14]. See also Theorem I.3.1 in
[32] and [38] for the Itô formula in the general context of semi-martingales.
We obtain

(4.11)
$\begin{split}&|{u^{n}}|^{2}+2\int_{0}^{t}\gamma(Au^{n}(s),u^{n}(s))\mathrm{d}s+2\int_{0}^{t}(P_{n}B_{\widetilde{R}}(u^{n}(s)),u^{n}(s))\mathrm{d}s+2\int_{0}^{t}(P_{n}F^{R}(u^{n}(s)),u^{n}(s))\mathrm{d}s\\\
&=|u^{n}_{0}|^{2}+\int_{(0,t]}\int_{E_{0}}2(u^{n}(s),P_{n}K(u^{n}(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi))+\int_{0}^{t}2(u^{n}(s),P_{n}G(u^{n}(s-))\mathrm{d}W(s))\\\
&+\int_{0}^{t}\left\|P_{n}G(u^{n}(s))\right\|_{L_{2}(U_{0},H)}^{2}\mathrm{d}s+\int_{(0,t]}\int_{E_{0}}|{P_{n}K(u^{n}(s-),\xi)}|^{2}\mathrm{d}\pi(s,\xi).\end{split}$

Here we made use of the cancellation property $(B_{\widetilde{R}}(u,u),u)=0$.
Now we use the equicoercivity result (4.7) along with the equation (2.4), we
take the supremum in time over $[0,r]$ for a fixed $r\in[0,T]$ and then take
expectation of both sides; we find

(4.12) $\displaystyle{\bf E}[\sup_{0\leq t\leq r}|u^{n}(t)|^{2}]+$
$\displaystyle 2{\bf
E}\int_{0}^{r}{\left((\gamma+c_{1})\left\|u^{n}(s)\right\|^{2}-c_{5}\right)}\mathrm{d}s$
$\displaystyle\leq{\bf E}[|u^{n}_{0}|^{2}]+{\bf E}\sup_{0\leq t\leq
r}[I(t)+J(t)+K(t)+L(t)],$

where

$\displaystyle I(t)$
$\displaystyle=\int_{0}^{t}\left\|P_{n}G(u^{n}(s))\right\|_{L_{2}(U_{0},H)}^{2}\mathrm{d}s,$
$\displaystyle J(t)$
$\displaystyle=\int_{(0,t]}\int_{E_{0}}|{P_{n}K(u^{n}(s-),\xi)}|^{2}\mathrm{d}\pi(s,\xi),$
$\displaystyle K(t)$
$\displaystyle=\int_{0}^{t}(u^{n}(s),P_{n}G(u^{n}(s-))\mathrm{d}W(s)),$
$\displaystyle L(t)$
$\displaystyle=\int_{(0,t]}\int_{E_{0}}(u^{n}(s),P_{n}K(u^{n}(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi)).$

We estimate the terms $I(t)$ and $J(t)$ based on the assumption (2.9) along
with Young’s inequality as follows

$\displaystyle{\bf E}\sup_{0\leq t\leq
r}\int_{0}^{t}\left\|P_{n}G(u^{n}(s))\right\|_{L_{2}(U_{0},H)}^{2}\mathrm{d}s$
$\displaystyle\lesssim{\bf E}\sup_{0\leq t\leq
r}\int_{0}^{t}\left\|G(u^{n}(s))\right\|_{L_{2}(U_{0},H)}^{2}\mathrm{d}s$
$\displaystyle={\bf
E}\int_{0}^{r}\left\|G(u^{n}(s))\right\|_{L_{2}(U_{0},H)}^{2}\mathrm{d}s$
(4.13) $\displaystyle\lesssim{\bf
E}\int_{0}^{r}\rho(|u^{n}(s)|^{2}+1)\mathrm{d}s.$

Similarly for $J(t)$ we use the isometry property (see e.g. Theorem 2.16 in
[14]) to obtain,

$\displaystyle{\bf E}\sup_{0\leq t\leq
r}\int_{(0,t]}\int_{E_{0}}|P_{n}K(u^{n}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi)$
$\displaystyle\leq{\bf E}\sup_{0\leq t\leq
r}\int_{(0,t]}\int_{E_{0}}|K(u^{n}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi)$
$\displaystyle={\bf
E}\int_{(0,r]}\int_{E_{0}}|K(u^{n}(s-),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s$
(4.14) $\displaystyle\lesssim{\bf
E}\int_{(0,r]}\rho(|u^{n}(s)|^{2}+1)\mathrm{d}s.$

Note that in (4.13) and (4.14) we used the fact that $P_{n}$ is a projector on
$H$. The next two martingale terms $K(t)$ and $L(t)$ are treated by using the
Burkholder-Davis-Gundy (BDG) inequality (see e.g. [45]) and Young’s
inequality. Observe that for some constant $C(\rho)>0$ independent of $n$ we
obtain

$\displaystyle{\bf E}\sup_{0\leq t\leq
r}\left|\int_{0}^{t}(u^{n}(s),P_{n}G(u^{n}(s-))\mathrm{d}W(s))\right|$
$\displaystyle\lesssim{\bf
E}\left(\int_{0}^{r}(u^{n}(t),P_{n}G(u^{n}(t-)))^{2}\mathrm{d}t\right)^{\frac{1}{2}}$
$\displaystyle\lesssim{\bf
E}\left(\int_{0}^{r}\left\|P_{n}G(u^{n}(t))\right\|^{2}_{L_{2}(U_{0},H)}|u^{n}(t)|^{2}\mathrm{d}t\right)^{\frac{1}{2}}$
$\displaystyle\lesssim{\bf E}\sup_{0\leq t\leq
r}|u^{n}(t)|\left(\int_{0}^{r}\left\|G(u^{n}(t))\right\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}t\right)^{\frac{1}{2}},$

which implies for $K(t)$ that,

$\displaystyle{\bf E}\sup_{0\leq t\leq
r}\left|\int_{0}^{t}(u^{n}(s),P_{n}G(u^{n}(s-))\mathrm{d}W(s))\right|$
$\displaystyle\leq\frac{1}{4}{\bf E}\sup_{0\leq t\leq
r}|u^{n}(t)|^{2}+C(\rho){\bf
E}\int_{0}^{r}\left(1+|u^{n}(s)|^{2}\right)\mathrm{d}s.$

Similarly using the Burkholder-Davis-Gundy inequality, (see Proposition 2 pg.
5650 in [14]) we obtain for the term $L(t)$:

$\displaystyle{\bf E}\sup_{0\leq t\leq
r}\left|\int_{(0,t]}\int_{E_{0}}(u^{n}(s),P_{n}K(u^{n}(s-),\xi))\mathrm{d}\widehat{\pi}(s,\xi)\right|$
$\displaystyle\qquad\qquad\qquad\lesssim{\bf
E}\left(\int_{(0,r]}\int_{E_{0}}|u^{n}(s)|^{2}|P_{n}K(u^{n}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi)\right)^{\frac{1}{2}}$
$\displaystyle\qquad\qquad\qquad\leq\frac{1}{4}{\bf E}\sup_{0\leq t\leq
r}|{u^{n}(t)}|^{2}+C{\bf
E}\bigg{[}\int_{(0,r]}\int_{E_{0}}|K(u^{n}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi)\bigg{]}$
$\displaystyle\qquad\qquad\qquad=\frac{1}{4}{\bf E}\sup_{0\leq t\leq
r}|{u^{n}(t)}|^{2}+C{\bf
E}\bigg{[}\int_{(0,r]}\int_{E_{0}}|K(u^{n}(s-),\xi)|^{2}\mathrm{d}\nu(\xi)ds\bigg{]}$
(4.15) $\displaystyle\qquad\qquad\qquad\leq\frac{1}{4}{\bf E}\sup_{0\leq t\leq
r}|u^{n}(t)|^{2}+C(\rho){\bf
E}\int_{0}^{r}\left(1+|u^{n}(s)|^{2}\right)\mathrm{d}s.$

Collecting all relations from (4.13) through (4.15) and applying in (4.12), we
obtain

$\displaystyle{\bf E}\sup_{0\leq s\leq r}|u^{n}(s)|^{2}+{(\gamma+c_{1})}{\bf
E}\int_{0}^{r}\left\|u^{n}(s)\right\|^{2}\mathrm{d}s$ $\displaystyle\hskip
144.54pt\leq\leavevmode\nobreak\ {\bf E}|u^{n}_{0}|^{2}+\frac{1}{2}{\bf
E}\sup_{0\leq s\leq r}|u^{n}(s)|^{2}+C(\rho){\bf
E}\int_{0}^{r}(1+|u^{n}(s)|^{2})\mathrm{d}s.$

Multiplying both sides by 2 and using the fact that ${\bf
E}|u_{0}^{n}|^{2}\leq{\bf E}|u_{0}|^{2}$, we obtain for some constant
$C(T,\rho,c_{1},c_{5})>0$ independent of $n,R,\widetilde{R}$

(4.16) $\displaystyle{\bf E}\sup_{0\leq s\leq
r}|u^{n}(s)|^{2}+{2(\gamma+c_{1})}\int_{0}^{r}{\bf
E}\left\|u^{n}(s)\right\|^{2}\mathrm{d}s\leq\leavevmode\nobreak\ C\left({\bf
E}|u_{0}|^{2}+{\int_{0}^{r}\sup_{0\leq t\leq s}{\bf
E}|u^{n}(t)|^{2}\mathrm{d}s}\right)+C.$

The proof is complete after applying the Grönwall lemma to ${\bf E}\sup_{0\leq
s\leq r}|u^{n}(s)|^{2}$. ∎

######  Remark 4.2.

Observe that the absence of the linear term $A$, that is if $\gamma=0$, does
not change this result thanks to the equicoercivity result stated in (4.7) and
derived in [49] for the operators $F^{R}$. This fact will be used in Section
8. See also Remark 8.4 for an explanation.

###  4.2. Estimates in fractional Sobolev spaces

To obtain compactness, we will use a particular characterization of the
fractional Sobolev space ${W^{\alpha,m}(0,T;Z)}$ where $Z$ is a Banach space.
Given $m>1$ and $\alpha\in(0,1)$, $W^{\alpha,m}(0,T;Z)$ is the Sobolev space
consisting of the $v\in L^{m}(0,T;Z)$ such that the (Gagliardo) seminorm
$\int_{0}^{T}\int_{0}^{T}\frac{|v(t)-v(s)|^{m}}{|t-s|^{1+\alpha
m}}\mathrm{d}t\mathrm{d}s$ is finite. This space is then endowed with the norm

(4.17)
$\displaystyle\|v\|^{m}_{W^{\alpha,m}(0,T;Z)}=\|v\|_{L^{m}(0,T;Z)}^{m}+\int_{0}^{T}\int_{0}^{T}\frac{|v(t)-v(s)|^{m}}{|t-s|^{1+\alpha
m}}\mathrm{d}t\mathrm{d}s.$

We denote $W^{\alpha,2}=:H^{\alpha}$. Now we state the following lemma:

######  Lemma 4.3.

Let $\mathcal{Y}_{0}\subset\mathcal{Y}\subset\mathcal{Y}_{1}$ be Banach
spaces, $\mathcal{Y}_{0}$ and $\mathcal{Y}_{1}$ reflexive with compact
embedding of $\mathcal{Y}_{0}$ in $\mathcal{Y}$. Let $s,m>1$ and
$\alpha\in(0,1)$ be given. Let $\mathcal{G}$ be the space

$\mathcal{G}=L^{\infty}(0,T;\mathcal{Y})\cap L^{s}(0,T;\mathcal{Y}_{0})\cap
W^{\alpha,m}(0,T;\mathcal{Y}_{1}),$

endowed with the natural norm. Then the embedding of $\mathcal{G}$ in
$L^{r}(0,T;\mathcal{Y})$ is compact, $\forall 1\leq r<\infty$.

###### Proof.

Let $q=\min(m,s)>1$ and define $\mathscr{G}_{q}=L^{q}(0,T;\mathcal{Y}_{0})\cap
W^{\alpha,q}(0,T;\mathcal{Y}_{1})$. Then it is shown in [50] (Section 13.3
Theorem 13.2) and [25] (Theorem 2.1) that the injection of $\mathscr{G}_{q}$
in $L^{q}(0,T;\mathcal{Y})$ is compact. Now if $\\{v_{k}\\}_{k\geq 1}$ is a
bounded sequence in $\mathcal{G}$ that converges to $v$ weakly in
$\mathscr{G}_{q}$ and strongly in $L^{q}(0,T;\mathcal{Y})$, then up to the
extraction of a subsequence, $v_{k}(t)\rightarrow v(t)$ strongly in
$\mathcal{Y}$ for a.e. $t\in(0,T)$. Since $v_{k}$ is also bounded in
$L^{\infty}(0,T;\mathcal{Y})$, using the Vitali convergence theorem, we have
that $v_{m}\rightarrow v$ strongly in $L^{r}(0,T;\mathcal{Y})$ for every
$r\in[1,\infty)$. ∎

In what follows, we will apply the above Lemma with $\mathcal{Y}_{0}=V$,
$\mathcal{Y}_{1}=D(A)^{\prime}$, $\mathcal{Y}=H$, $\alpha\in(0,\frac{1}{2})$,
$s=m=2$. We will use the above lemma again in Sections 5 and 6 as well.

We are now in position to show that the approximating sequence
$\\{u^{n}\\}_{n=1}^{\infty}$ is bounded in the space
$L^{2}(\Omega;H^{\alpha}(0,T;D(A)^{\prime}))$ for $\alpha\in(0,\frac{1}{2})$.
For that purpose, we recall the following proposition essentially borrowed
from [25]:

######  Proposition 4.4.

For every $\alpha\in(0,\frac{1}{2})$ and $1\leq\beta\leq 2$ there exists a
constant $C(\alpha,\beta,\rho,K_{1})$, independent of $n,R,\widetilde{R}$,
such that

(4.18) ${\bf
E}\left\|\int_{0}^{t}P_{n}G(u^{n}(s))\mathrm{d}W(s)\right\|^{\beta}_{W^{\alpha,\beta}(0,T;H)}\leq
C,$

and

(4.19) ${\bf
E}\left\|\int_{(0,t]}\int_{E_{0}}P_{n}K(u^{n}(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi)\right\|^{\beta}_{W^{\alpha,\beta}(0,T;H)}\leq
C.$

###### Proof.

Using Lemma 2.1 in [25], the assumption (2.9) and Lemma 4.1 we obtain

(4.20) $\displaystyle\leavevmode\nobreak\ {\bf
E}\left\|\int_{0}^{t}P_{n}G(u^{n}(s))\mathrm{d}W(s)\right\|^{\beta}_{W^{\alpha,\beta}(0,T;H)}$
$\displaystyle\lesssim{\bf
E}\int_{0}^{T}\left\|P_{n}G(u^{n})\right\|^{\beta}_{L_{2}(U_{0},H)}\mathrm{d}s$
$\displaystyle\lesssim\left({\bf
E}\int_{0}^{T}(1+|u^{n}(s)|^{2})\mathrm{d}s\right)^{\frac{\beta}{2}}\leq C.$

A continuity result similar to Lemma 2.1 in [25] holds for the stochastic
integral with respect to a compensated Poisson random measure. For more
details of the case $\beta=2$, we refer the reader to Proposition A.4 in [17].
Using the assumption (2.9) and Lemma 4.1 we thus obtain

$\displaystyle{\bf
E}\left\|\int_{(0,t]}\int_{E_{0}}P_{n}K(u^{n}(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi)\right\|^{\beta}_{W^{\alpha,\beta}(0,T;H)}$
$\displaystyle\lesssim{\bf
E}\int_{0}^{T}\int_{E_{0}}|P_{n}K(u^{n}(s-),\xi)|^{\beta}\mathrm{d}\nu(\xi)\mathrm{d}s$
(4.21) $\displaystyle\lesssim\left({\bf
E}\int_{0}^{T}(1+|u^{n}(s)|^{2})\mathrm{d}s\right)^{\frac{\beta}{2}}\leq C.$

∎

######  Proposition 4.5.

Let $u^{n}$ be the solution to (4.9) with an inital data $u_{0}\in
L^{2}(\Omega,\mathcal{F}_{0},{\bf P};H)$. Then for every
$\alpha\in(0,\frac{1}{2})$ there exists a constant
$C=C(\alpha,\gamma,T,R,\widetilde{R},K_{1})>0$ such that

(4.22) $\sup_{n\geq 1}{\bf
E}\left\|u^{n}\right\|^{2}_{H^{\alpha}(0,T;{{{}D(A)^{\prime}}})}\leq C.$

###### Proof.

In light of Proposition 4.4, it is sufficient to prove that

(4.23) $\sup_{n\geq 1}{\bf
E}\left\|u^{n}-\int_{0}^{\cdot}P_{n}G(u^{n}(s))\mathrm{d}W(s)-\int_{(0,{\cdot}]}\int_{E_{0}}P_{n}K(u^{n}(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi)\right\|^{2}_{H^{\alpha}(0,T;{D(A)^{\prime}})}<\infty.$

Let
$g^{n}:=u^{n}-\int_{0}^{t}P_{n}G(u^{n})\mathrm{d}W(s)-\int_{(0,t]}\int_{E_{0}}P_{n}K(u^{n},\xi)\mathrm{d}\widehat{\pi}(s,\xi)$.
Since $\alpha<1$ we use the fact that the embedding $H^{1}\subset H^{\alpha}$
is continuous to obtain,

$\displaystyle\leavevmode\nobreak\
\left\|g^{n}\right\|^{2}_{H^{\alpha}(0,T;{D(A)^{\prime}})}$
$\displaystyle\lesssim\left\|u_{0}^{n}-\int_{0}^{\cdot}\gamma
Au^{n}(s)\mathrm{d}s-\int_{0}^{\cdot}P_{n}B_{\widetilde{R}}(u^{n}(s))\mathrm{d}s-\int_{0}^{\cdot}P_{n}F^{R}(u^{n}(s))\mathrm{d}s\right\|^{2}_{H^{1}(0,T;{D(A)^{\prime}})}$
$\displaystyle\lesssim\leavevmode\nobreak\
\left\|u_{0}^{n}-\int_{0}^{\cdot}\gamma
Au^{n}(s)\mathrm{d}s-\int_{0}^{\cdot}P_{n}B_{\widetilde{R}}(u^{n}(s))\mathrm{d}s-\int_{0}^{\cdot}P_{n}F^{R}(u^{n}(s))\mathrm{d}s\right\|^{2}_{L^{2}(0,T;{{}D(A)^{\prime}})}$
$\displaystyle\qquad+\int_{0}^{T}\gamma|Au^{n}(s)|_{{D(A)^{\prime}}}^{2}\mathrm{d}s+\int_{0}^{T}|B_{\widetilde{R}}(u^{n}(s))|_{{D(A)^{\prime}}}^{2}\mathrm{d}s+\int_{0}^{T}|F^{R}(u^{n}(s))|_{{{}D(A)^{\prime}}}^{2}\mathrm{d}s$
$\displaystyle\lesssim$ $\displaystyle\leavevmode\nobreak\
|u_{0}|^{2}+\int_{0}^{T}\gamma|Au^{n}(s)|^{2}_{{{}D(A)^{\prime}}}\mathrm{d}s+\int_{0}^{T}|B_{\widetilde{R}}(u^{n}(s))|_{{D(A)^{\prime}}}^{2}\mathrm{d}s+\int_{0}^{T}|F^{R}(u^{n}(s))|^{2}_{{D(A)^{\prime}}}\mathrm{d}s$
(4.24) $\displaystyle=:$ $\displaystyle\leavevmode\nobreak\
|u_{0}|^{2}+\mathcal{K}_{1}+\mathcal{K}_{2}+\mathcal{K}_{3},$

where the hidden constants depend on $T$ but not on $n$. Next, using Lemma 4.1
we obtain the existence of a constant $C=C(T,\gamma,K_{1})>0$, independent of
$n,R,\widetilde{R}$, such that

(4.25) ${\bf E}[\mathcal{K}_{1}]\lesssim{\bf
E}\int_{0}^{T}\gamma|Au^{n}(s)|_{V^{\prime}}^{2}\mathrm{d}s\leq\gamma{\bf
E}\int_{0}^{T}\|u^{n}(s)\|^{2}\mathrm{d}s\leq C.$

We estimate $\mathcal{K}_{2}$ using the available estimates for $B$, (2.6), as
follows.

(4.26) $\displaystyle{\bf
E}\int_{0}^{T}|B_{\widetilde{R}}(u^{n}(s))|_{D(A)^{\prime}}^{2}\mathrm{d}s\lesssim{\bf
E}\int_{0}^{T}\big{(}\theta_{\widetilde{R}}(|u^{n}|)\big{)}^{2}|u^{n}(s)|^{2}\|u^{n}(s)\|^{2}\mathrm{d}s\lesssim\widetilde{R}^{2}{\bf
E}\int_{0}^{T}\|u^{n}(s)\|^{2}\mathrm{d}s.$

The above follows from an application of Hölder’s inequality and the fact that
$\theta_{\widetilde{R}}(x)$ vanishes when $x\geq 2\widetilde{R}$. Then we use
Lemma 4.1 which shows that the right-hand side of (4.26) is bounded by a
constant $C=C(\widetilde{R},T,K_{1})>0$ independent of $n$ and $R$.
We estimate $\mathcal{K}_{3}$ using the property (4.8) and Lemma 4.1 and
obtain for some $C=C(R,T,p,K_{1})>0$ independent of $n$ and $\widetilde{R}$,

(4.27) $\displaystyle{\bf E}[\mathcal{K}_{3}]\lesssim$
$\displaystyle\leavevmode\nobreak\ {\bf
E}\int_{0}^{T}|F^{R}(u^{n}(s))|^{2}_{V^{\prime}}\mathrm{d}s\lesssim
R^{2p-4}{\bf E}\int_{0}^{T}\left\|u^{n}(s)\right\|^{2}\mathrm{d}s\leq C.$

Thus using Proposition 4.4 with $\beta=2$ and (4.25)-(4.27) we obtain the
desired result (4.22). Although there are sharper estimates available for the
term $\mathcal{K}_{3}$, (4.27) suffices at this point. ∎

###  4.3. Tightness

######  Proposition 4.6.

Let $(u^{n})_{n=1}^{\infty}$ be the solution to the equations (4.9) with
initial data
$u_{0}\in L^{2}(\Omega,\mathcal{F}_{0},{\bf P};H)$. Then the laws of
$(u^{n})_{n=1}^{\infty}$ are tight in $L^{2}(0,T;H).$

###### Proof.

The proof follows standard procedure (see e.g. [19]). For $M>0$, let

(4.28) ${\mathcal{B}}_{M}:=\\{u\in L^{2}(0,T;V)\cap
H^{\alpha}(0,T;D(A)^{\prime}):|u|^{2}_{L^{2}(0,T;V)}+|u|^{2}_{H^{\alpha}(0,T;D(A)^{\prime})}\leq
M\\}.$

Thanks to Lemma 4.3, we know that ${\mathcal{B}}_{M}$ is a compact subset of
$L^{2}(0,T;H)$. By using Chebyshev’s inequality, we see that

(4.29) $\displaystyle{\bf P}\left[u^{n}\not\in{\mathcal{B}}_{M}\right]\leq$
$\displaystyle{\bf
P}\left[|u^{n}|^{2}_{L^{2}(0,T;V)}\geq\frac{M}{2}\right]+{\bf
P}\left[|u^{n}|^{2}_{H^{\alpha}(0,T;D(A)^{\prime})}\geq\frac{M}{2}\right]$
$\displaystyle\leq$ $\displaystyle\frac{4}{M^{2}}{\bf
E}\left[|u^{n}|^{2}_{L^{2}(0,T;V)}+|u^{n}|^{2}_{H^{\alpha}(0,T;D(A)^{\prime})}\right].$

By Lemma 4.1 and Proposition 4.5, there exists a constant
$C=C(T,M,R,\widetilde{R},K_{1})>0$ independent of $n$ such that for every
positive integer $n\in\mathbb{N}$,

(4.30) ${\bf
P}\bigg{[}u^{n}\not\in{\mathcal{B}}_{M}\bigg{]}\leq\frac{C}{M^{2}}.$

This proves that the laws of $u^{n}$ are tight in $L^{2}(0,T;H).$∎

However in order to obtain a candidate solution that is càdlàg we will need to
establish tightness of the laws of $u^{n}$ on a certain space of functions
that are càdlàg in time endowed with the Skorohod topology; see below and in
[7].
For that purpose we will show that the Aldous condition, [cf.[2], [3], [39]
(Theorem 3.2, page 29)] is satisfied. The following is an easy way to
guarantee that the Aldous condition is satisfied.

######  Lemma 4.7.

Let $(\Xi,||\cdot||_{\Xi})$ be a separable Banach space and let
$(X_{l})_{l\in\mathbb{N}}$ be a sequence of $\,\Xi$-valued random variables.
Assume that for every sequence $(\tau_{l})_{l\in\mathbb{N}}$ of
$\mathcal{F}$-stopping times with $\tau_{l}\leq T$ and for every $\delta\geq
0$ the following condition holds

(4.31) $\sup_{l\geq 1}{\bf
E}(||X_{l}(\tau_{l}+\delta)-X_{l}(\tau_{l})||^{\alpha}_{\Xi}\leq
C\delta^{\epsilon},$

for some $\alpha,{\epsilon}>0$ and a constant $C>0$. Then the sequence
$(X_{l})_{l\in\mathbb{N}}$ satisfies the Aldous condition in the space $\Xi$
and thus the laws of $X_{l}$ form a tight sequence on $\mathcal{D}([0,T];\Xi)$
endowed with the Skorohod topology.

See also Theorem 13.2 of [50] for a compactness result in the deterministic
setting using a condition analogous to the condition (4.31).

We will show that the laws of the approximate solutions $u^{n}$ denoted by
$\mu_{u}^{n}$ are tight as probability measures on the phase space
$\mathcal{Z}$ where

(4.32) $\mathcal{Z}=L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime}),$

and

(4.33) $\mu_{u}^{n}(\cdot)={\bf P}(u^{n}\in\cdot)\in
Pr\left(L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime})\right).$

Here $Pr(\mathcal{S})$ denotes the set of probability measures on the metric
space $\mathcal{S}$. For a complete, separable metric space $\mathcal{S}$ we
denote by $\mathcal{D}([0,T];\mathcal{S})$ the space of functions
$u:[0,T]\rightarrow\mathcal{S}$ that are right-continuous on $[0,T)$ and have
left-limits at every point in $(0,T]$. The space
$\mathcal{D}([0,T];\mathcal{S})$ is endowed with the Skorokhod topology, which
makes $\mathcal{D}([0,T];\mathcal{S})$ separable and metrizable by a complete
metric (see e.g. Chapter 3 in [7]).

######  Proposition 4.8.

Suppose ${\bf E}|u_{0}|^{2}<\infty$. Then the laws $\\{\mu_{u}^{n}\\}_{n\geq
1}$ of the Galerkin approximations form a tight sequence of probability
measures on $\mathcal{D}([0,T];V^{\prime})$ endowed with the Skorohod
topology.

###### Proof.

Let $(\tau_{n})_{n\geq 1}$ be a sequence of stopping times where
$0\leq\tau_{n}\leq T$ and let us integrate the equation (4.9) from 0 to t for
$t\in[0,T]$; we obtain

(4.34) $\displaystyle u^{n}(t)$ $\displaystyle=u^{n}_{0}-\int_{0}^{t}\gamma
A(u^{n}(s))\mathrm{d}s-\int_{0}^{t}P_{n}B_{\widetilde{R}}(u^{n}(s),u^{n}(s))\mathrm{d}s-\int_{0}^{t}P_{n}F^{R}(u^{n}(s))\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}G(u^{n}(s-))\mathrm{d}W(s)+\int_{(0,t]}\int_{E_{0}}K(u^{n}(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi)$
$\displaystyle=:u^{n}_{0}+\mathcal{M}_{1}(t)+\mathcal{M}_{2}(t)+\mathcal{M}_{3}(t)+\mathcal{M}_{4}(t)+\mathcal{M}_{5}(t).$

Let $\delta>0$ be given. We need to show that a suitable Aldous condition is
satisfied and for that purpose we need to show that all the terms in the above
equation satisfy (4.31) with suitable $\alpha,\epsilon$ and $\Xi=V^{\prime}$.
We first treat $\mathcal{M}_{1}$ by using the Hölder inequality

(4.35) $\displaystyle{\bf
E}\left\|\mathcal{M}_{1}(\tau_{n}+\delta)-\mathcal{M}_{1}(\tau_{n})\right\|_{V^{\prime}}$
$\displaystyle\leq\gamma{\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}|A(u^{n}(s))|_{V^{\prime}}\mathrm{d}s$
$\displaystyle\leq\gamma{\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}\left\|u^{n}(s)\right\|\mathrm{d}s\leq\gamma\delta^{\frac{1}{2}}\left({\bf
E}\int_{0}^{T}\left\|u^{n}(s)\right\|^{2}\mathrm{d}s\right)^{\frac{1}{2}}.$

The last term is bounded independently of $n$ thanks to Lemma 4.1. Hence,
$\mathcal{M}_{1}(t)$ satisfies (4.31) with $\alpha=1,\epsilon=\frac{1}{2}$.
Next, we deal with $\mathcal{M}_{2}(t)$. By applying Hölder’s inequality and
the assumption (2.7), we obtain for some $C=C(T,K_{1})>0$ independent of $n$

$\displaystyle{\bf
E}\left\|\mathcal{M}_{2}(\tau_{n}+\delta)-\mathcal{M}_{2}(\tau_{n})\right\|_{V^{\prime}}\leq$
$\displaystyle C{\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}|B_{\widetilde{R}}(u^{n}(s),u^{n}(s))|_{V^{\prime}}\mathrm{d}s$
$\displaystyle\leq$ $\displaystyle C{\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}{\theta_{\widetilde{R}}(|u^{n}(s)|)}|u^{n}(s)|^{\frac{1}{2}}\left\|u^{n}(s)\right\|^{\frac{3}{2}}\mathrm{d}s$
$\displaystyle\leq$ $\displaystyle C\left({\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}{\theta_{\widetilde{R}}(|u^{n}(s)|)}|u^{n}(s)|^{2}\mathrm{d}s\right)^{\frac{1}{4}}\left({\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}||u^{n}(s)||^{2}\mathrm{d}s\right)^{\frac{3}{4}}$
$\displaystyle\leq$ $\displaystyle C\delta^{\frac{1}{4}}\left({\bf
E}\sup_{s\in[0,T]}{|u^{n}(s)|}^{2}\right)^{\frac{1}{4}}\left({\bf
E}\int_{0}^{T}||u^{n}(s)||^{2}\mathrm{d}s\right)^{\frac{3}{4}}$ (4.36)
$\displaystyle\leq$ $\displaystyle C\delta^{\frac{1}{4}}.$

The last inequality above comes from Lemma 4.1. Hence, $\mathcal{M}_{2}(t)$
satisfies the condition (4.31) for $\alpha=1,\epsilon=\frac{1}{4}$. Next, by
making use of (4.8), we readily obtain that for some $C=C({R},T,p,K_{1})>0$
independent of $n$

(4.37) $\displaystyle{\bf
E}\left\|\mathcal{M}_{3}(\tau_{n}+\delta)-\mathcal{M}_{3}(\tau_{n})\right\|_{V^{\prime}}$
$\displaystyle\lesssim{\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}|F^{R}(u^{n}(s))|_{V^{\prime}}\mathrm{d}s$
$\displaystyle\lesssim{\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}R^{p-2}\|u^{n}(s)\|\mathrm{d}s$
$\displaystyle\lesssim R^{p-2}\delta^{\frac{1}{2}}\left({\bf
E}\int_{0}^{T}\left\|u^{n}(s)\right\|^{2}\mathrm{d}s\right)^{\frac{1}{2}}$
$\displaystyle\leq C\delta^{\frac{1}{2}}.$

Thus, $\mathcal{M}_{3}(t)$ satisfies (4.31) with
$\alpha=1,\epsilon=\frac{1}{2}$. Note that for (4.37) (and earlier for (4.24))
we used the fact that $P_{n}$ is a projector operator in $V^{\prime}$ (as well
as in $H$). This is an essential point motivating the introduction of the
approximation $F^{R}$ of $F$ (see [49]).
Next by using the Itô isometry and (2.9) and noticing that $V\subset H$, we
find

(4.38) $\displaystyle{\bf
E}\left[\left\|\mathcal{M}_{4}(\tau_{n}+\delta)-\mathcal{M}_{4}(\tau_{n})\right\|^{2}_{V^{\prime}}\right]\lesssim{\bf
E}\left[|\mathcal{M}_{4}(\tau_{n}+\delta)-\mathcal{M}_{4}(\tau_{n})|^{2}\right]$
$\displaystyle=$ $\displaystyle{\bf
E}\left|\int_{\tau_{n}}^{\tau_{n}+\delta}P_{n}G(u^{n}(s))\mathrm{d}W(s)\right|^{2}={\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}|P_{n}G(u^{n}(s))|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s$
$\displaystyle\lesssim$ $\displaystyle\leavevmode\nobreak\ {\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}\rho(|u^{n}(s)|^{2}+1)\mathrm{d}s\lesssim\rho\delta{\bf
E}\left[\sup_{s\in[0,T]}(1+|u^{n}(s)|^{2})\right]\leq C(\rho,K_{1})\delta.$

Hence, $\mathcal{M}_{4}(t)$ satisfies (4.31) with $\alpha=2,\epsilon=1$.

Finally, we consider the $\mathcal{M}_{5}(t)$ term. Using the Itô isometry and
(2.9), we write

(4.39) $\displaystyle\leavevmode\nobreak\ {\bf
E}\left[\left\|\mathcal{M}_{5}(\tau_{n}+\delta)-\mathcal{M}_{5}(\tau_{n})\right\|^{2}_{V^{\prime}}\right]\lesssim{\bf
E}\left[|\mathcal{M}_{5}(\tau_{n}+\delta)-\mathcal{M}_{5}(\tau_{n})|^{2}\right]$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\ {\bf
E}\left|\int_{(\tau_{n},\tau_{n}+\delta]}\int_{E_{0}}K(u^{n}(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi)\right|^{2}={\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}\int_{E_{0}}|K(u^{n}(s-),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s$
$\displaystyle\lesssim$ $\displaystyle\leavevmode\nobreak\ {\bf
E}\int_{\tau_{n}}^{\tau_{n}+\delta}\rho(|u^{n}(s)|^{2}+1)\mathrm{d}s\lesssim\rho\delta{\bf
E}\left[\sup_{s\in[0,T]}(1+|u^{n}(s)|^{2})\right]\leq C(\rho,K_{1})\delta.$

Thanks to Lemma 4.1, we see that $\mathcal{M}_{5}(t)$ satisfies the relation
(4.31) with $\alpha=2,\epsilon=1$ and $\Xi=V^{\prime}$. Collecting all the
estimates from (4.35) to (LABEL:M5), we conclude that the laws of
$(u^{n})_{n=1}^{\infty}$ are tight in $\mathcal{D}([0,T];V^{\prime})$ endowed
with the Skorohod topology. ∎

######  Proposition 4.9.

Let $(u^{n})_{n=1}^{\infty}$ be the solution to the equations (4.9) with
$u_{0}\in L^{2}(\Omega,\mathcal{F}_{0},{\bf P};H)$. Then the laws of
$(u^{n})_{n=1}^{\infty}$ are tight in the space $\mathcal{Z}$, which was
defined in (4.32).

###### Proof.

For every $\varepsilon>0$, by Propositions 4.6 and 4.8 and the definition of
tightness, there exist compact sets
$K_{1}\subset\mathcal{D}([0,T];V^{\prime})$ and $K_{2}\subset L^{2}(0,T;H)$
such that

(4.40) $\sup_{n\geq 1}{\bf P}\left[u^{n}\not\in
K_{1}\right]\leq\frac{\varepsilon}{2}\quad\text{ and }\quad\sup_{n\geq 1}{\bf
P}\left[u^{n}\not\in K_{2}\right]\leq\frac{\varepsilon}{2}.$

Since $K_{1}\cap K_{2}$ is sequentially compact in $\mathcal{Z}$ and

(4.41) $\sup_{n\geq 1}{\bf P}\left[u^{n}\in(K_{1}\cap
K_{2})^{C}\right]\leq\sup_{n\geq 1}{\bf P}\left[u^{n}\not\in
K_{1}\right]+\sup_{n\geq 1}{\bf P}\left[u^{n}\not\in
K_{2}\right]\leq\varepsilon,$

we conclude that the laws of $(u^{n})_{n=1}^{\infty}$ are tight on
$\mathcal{Z}$. ∎

###  4.4. Passage to the limit

####  4.4.1. Convergence results

Let $\mathcal{N}^{\\#*}_{[0,\infty)\times E}$ denote the space of counting
measures on $[0,\infty)\times E$ that are finite on bounded sets and let
$\Upsilon:=H\times\mathcal{Z}\times
C([0,T];U)\times\mathcal{N}^{\\#*}_{[0,\infty)\times E}$. It is an established
fact that the spaces $\mathcal{N}^{\\#*}_{[0,\infty)\times E}$ and
$C([0,T];U)$ (cf. [18]) are separable and metrizable by a complete metric.
Thus we see that the sequence of probability measures,
$\mu_{W}^{n}(\cdot):={\bf P}(W\in\cdot)$, being constantly equal to one
element, is tight on $C([0,T];U)$. Similarly we find out that the law of $\pi$
is tight on $\mathcal{N}^{\\#*}_{[0,\infty)\times E}$. It is also clear that
$u^{n}_{0}\rightarrow u_{0}$ ${\bf P}$-a.s. in $H$, that is the laws of
$(u^{n}_{0})_{n=1}^{\infty}$ are tight on $H$ and that their weak limit is the
measure $\mu_{0}={\bf P}(u_{0}\in\cdot)$. Finally, recalling Proposition 4.9
and using Tychonoff’s theorem it can be shown that the sequence of the joint
probability measures $\mu^{n}$ associated with the approximating sequence
${(u^{n}_{0},u^{n},W,\pi)_{n=1}^{\infty}}$ is tight on the space $\Upsilon$.
Hence, by virtue of the Prohorov theorem, $({\mu^{n}})_{n=1}^{\infty}$ is
weakly compact over $\Upsilon$. This implies that there exists a probability
measure $\mu^{\infty}$ on $\Upsilon$ such that for some subsequence $\mu^{n}$
converges weakly to $\mu^{\infty}$. Since we want to show the existence of a
solution to the truncated equations (4.2), the weak convergence of $u^{n}$ to
$u$ in law and the weak convergence along a further subsequence invoking Lemma
4.1 will not be enough. Instead, we will use the Skorohod representation
theorem as stated in Theorem A.1 to upgrade the weak convergence to a.s.
convergence in $\Upsilon$ for random variables defined on a new probability
space.

######  Theorem 4.10.

There exists a probability space
$(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\bf P})$, with the
associated expectation denoted by $\widetilde{\bf E}$, and $\Upsilon-$valued
random variables $(\bar{u}_{0}^{n},\bar{u}^{n},\bar{W}^{n},\bar{\pi}^{n})$ and
$(\bar{u}_{0},\bar{u},\bar{W},\bar{\pi})$, such that

  1. i)

The probability laws of
$(\bar{u}_{0}^{n},\bar{u}^{n},\bar{W}^{n},\bar{\pi}^{n})$ and
$(\bar{u}_{0},\bar{u},\bar{W},\bar{\pi})$ are $\mu^{n}$ and $\mu^{\infty}$,
respectively,

  2. ii)

$(\bar{W}^{n},\bar{\pi}^{n})=(\bar{W},\bar{\pi})$ everywhere on
$\widetilde{\Omega}$,

  3. iii)

$(\bar{u}_{0}^{n},\bar{u}^{n},\bar{W}^{n},\bar{\pi}^{n})$ converges almost
surely in $\Upsilon$ to $(\bar{u}_{0},\bar{u},\bar{W},\bar{\pi})$, i.e. since
$\bar{W}^{n}=W$ and $\bar{\pi}^{n}=\bar{\pi}$:

(4.42) $\displaystyle\bar{u}_{0}^{n}\rightarrow\bar{u}_{0}\text{ in
}H,\,\widetilde{\bf P}\textnormal{-a.s.},$ (4.43)
$\displaystyle\bar{u}^{n}\rightarrow\bar{u}\text{ in
}L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime}),\widetilde{\bf
P}\textnormal{-a.s.},$

  4. iv)

$\bar{\pi}$, which is the Poisson random measure over
$(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\bf P})$ with
intensity measure $\mathrm{d}t\otimes\mathrm{d}\nu$, is time invariant.

  5. v)

$\bar{u}^{n}(0)=\bar{u}^{n}_{0}$ $\widetilde{\bf P}$-a.s. and
$(\bar{u}^{n}_{0},\bar{u}^{n},\bar{W},\widehat{\bar{\pi}})$ satisfies the
following equation in $H_{n}$, $\widetilde{\bf P}$\- a.s. for every
$t\in[0,T]$

(4.44) $\displaystyle\bar{u}^{n}(t)+\int_{0}^{t}\left[\gamma
A(\bar{u}^{n}(s))+P_{n}B_{\widetilde{R}}(\bar{u}^{n}(s),\bar{u}^{n}(s))+P_{n}F^{R}(\bar{u}^{n}(s))\right]\mathrm{d}s$
$\displaystyle\qquad\qquad=\bar{u}^{n}_{0}+\int_{0}^{t}P_{n}G(\bar{u}^{n}(s))\mathrm{d}\bar{W}(s)+\int_{(0,t]}\int_{E_{0}}P_{n}K(\bar{u}^{n}(s-),\xi)\mathrm{d}\widehat{\bar{\pi}}(s,\xi),$

with respect to some filtration $\left(\bar{\mathcal{F}}^{n}_{t}\right)_{t\geq
0}$.

Let $\mathcal{F}_{t}^{\prime}$ be the $\sigma$-field generated by the random
variables $\bar{W}(s),\bar{u}(s)$, $\bar{\pi}((0,s]\times\Gamma)$ for all
$s\leq t$ and all Borel subsets $\Gamma$ of $E_{0}$. Then we define

(4.45) $\displaystyle\mathcal{N}$
$\displaystyle:=\\{\mathcal{A}\in\widetilde{\mathcal{F}}\ |\ \widetilde{\bf
P}(\mathcal{A})=0\\},\qquad\bar{\mathcal{F}}^{0}_{t}:=\sigma(\mathcal{F}_{t}^{\prime}\cup\mathcal{N}),\qquad\bar{\mathcal{F}}_{t}:=\bigcap_{s\geq
t}\bar{\mathcal{F}}^{0}_{s}.$

This gives a complete, right-continuous filtration
$(\bar{\mathcal{F}}_{t})_{t\geq 0}$, dependent on $R$ and $\widetilde{R}$, on
the new probability space
$(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\bf P})$, to which the
noise processes and solutions are adapted. For each $n$ we define a filtration
$\left(\bar{\mathcal{F}}^{n}_{t}\right)_{t\geq 0}$ in $t$ on
$(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\bf P})$ the same way
as above but using $\bar{u}^{n}$ in place of $\bar{u}$. Each $\bar{u}^{n}$ is
adapted to $\left(\bar{\mathcal{F}}^{n}_{t}\right)_{t\geq 0}$.

######  Remark 4.11.

We note here the point appearing in the proof of the Skorohod representation
theorem, as it appears in Theorem A.1, that the space
$(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\bf P})$, unlike the
variables $(\bar{u}_{0}^{n},\bar{u}^{n},\bar{W}^{n},\bar{\pi}^{n})$, does not
depend on $R$ and $\widetilde{R}$. Even though the analysis that follows does
not require the new probability space to be independent of $R$ and
$\widetilde{R}$, as tightness and weak convergence are properties of the law,
the preceding observation keeps the notation concise.

Since $u^{n}_{0}$ converges to $u_{0}$ as $n\rightarrow\infty$, ($P_{n}$
converges to $I$) ${\bf P}$-a.s. in $H$, $\bar{u}_{0}$ has the same law as
that of $u_{0}$, which is independent of $R$ and $\widetilde{R}$. However
$\bar{u}_{0}$ itself might depend on $R$ and $\widetilde{R}$, but this is not
relevant in the analysis below.

The proof of $v)$ can be obtained by adapting the argument of [5] from the
Wiener case to the Lévy noise case. For more details concerning the
utilization of [5], the reader is referred to e.g. Proposition B.1 in [17].
Now we define the new stochastic basis
$\bar{\mathscr{S}}:=(\widetilde{\Omega},\widetilde{\mathcal{F}},(\bar{\mathcal{F}}_{t})_{t\geq
0},\widetilde{\bf P},\bar{W},\bar{\pi})$. We will show in Theorem 4.13 that
$(\bar{\mathscr{S}},\bar{u})$ is a global martingale solution of the system
(4.2). Observe that $\bar{u}^{n}$ also satisfies the estimates in Lemma 4.1
and thus the following lemma holds true.

######  Lemma 4.12.

For $\bar{u}^{n}$ and $\bar{u}$ as in (4.43), Lemma 4.1 implies that
$\bar{u}\in L^{\infty}([0,T];L^{2}(\widetilde{\Omega};H))\cap
L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};V)$ and that there exists a subsequence, still denoted by
$n\rightarrow\infty$ such that

(4.46) $\displaystyle\bar{u}^{n}$ $\displaystyle\rightharpoonup\bar{u}\text{
weak* in }L^{\infty}([0,T];L^{2}(\widetilde{\Omega};H)),$ (4.47)
$\displaystyle\bar{u}^{n}$ $\displaystyle\rightharpoonup\bar{u}\text{ weakly
in
}L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};V),$ and thanks to (4.8) (4.48) $\displaystyle
Y^{n}:=P_{n}F^{R}(\cdot,\bar{u}^{n})$ $\displaystyle\rightharpoonup
Y^{R,\widetilde{R}}\text{ weakly in
}L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};V^{\prime}).$

Now we state the main result of this section.

######  Theorem 4.13.

Suppose that ${\bf E}[|u_{0}|^{2}]<\infty$. Then $\bar{u}$ solves (4.2) with
respect to the stochastic basis
$(\widetilde{\Omega},\widetilde{\mathcal{F}},(\bar{\mathcal{F}}_{t})_{t\geq
0},\widetilde{\bf P},\bar{W},\bar{\pi})$ with initial condition $\bar{u}_{0}$,
where $\bar{u}_{0}$ has the same law as $u_{0}$.

###### Proof.

We are going to prove this theorem in two steps.
Step 1: We will first show that $\bar{u}$ solves the following equation

(4.49) $\displaystyle\bar{u}(t)+\int_{0}^{t}[\gamma
A(\bar{u}(s))+B_{\widetilde{R}}(\bar{u}(s),\bar{u}(s))+Y^{R,\widetilde{R}}]$
$\displaystyle\mathrm{d}s$
$\displaystyle=\bar{u}_{0}+\int_{0}^{t}G(\bar{u}(s))\mathrm{d}\bar{W}(s)+\int_{(0,t]}\int_{E_{0}}K(\bar{u}(s-),\xi)\mathrm{d}\widehat{\bar{\pi}}(s,\xi),$

where $Y^{R,\widetilde{R}}$ was defined in (4.48). We fix $\phi\in D(A)$ and
take the inner product of equation (4.44) in $H$ with $\phi$ on both sides and
let $n\rightarrow\infty$.
First, observe that from (4.43) we have that $\bar{u}^{n}\rightarrow\bar{u}$,
$\widetilde{\bf P}$-a.s. in $L^{2}(0,T;H)$ and thus in $L^{1}(0,T;H)$. Thanks
to Lemma 4.1 we also know that the sequence $\\{\bar{u}^{n}\\}_{n=1}^{\infty}$
is bounded in $L^{2}(\widetilde{\Omega};L^{2}(0,T;H))$ independently of $n$.

Thus an application of the Vitali convergence theorem gives us that

(4.50) $\displaystyle\bar{u}^{n}\rightarrow\bar{u}\quad\text{in
}L^{1}(\widetilde{\Omega};L^{1}(0,T;H)).$

We also know from (4.42) that
$(\bar{u}^{n}_{0},\phi)\rightarrow(\bar{u}_{0},\phi)$ in
$L^{1}(\widetilde{\Omega}).$
We will next show that, as $n\rightarrow\infty$,

(4.51)
$\int_{0}^{t}(A\bar{u}^{n}(s),\phi)\mathrm{d}s\rightarrow\int_{0}^{t}(A\bar{u}(s),\phi)\mathrm{d}s\quad\text{
in }L^{1}(\widetilde{\Omega}\times[0,T]).$

First, we note that for almost every $\omega\in\widetilde{\Omega}$ and for
every $t\in[0,T]$,

(4.52)
$\displaystyle\left|\int_{0}^{t}(A\bar{u}^{n}(s)-A\bar{u}(s),\phi)\mathrm{d}s\right|$
$\displaystyle=\left|\int_{0}^{t}(\bar{u}^{n}(s)-\bar{u}(s),A\phi)\mathrm{d}s\right|$
$\displaystyle\lesssim|A\phi|\left(\int_{0}^{T}|\bar{u}^{n}(s)-\bar{u}(s)|^{2}\mathrm{d}s\right)^{\frac{1}{2}}$
$\displaystyle\rightarrow 0\text{ as }n\rightarrow\infty\text{ from
\eqref{un_convH}}.$

Furthermore, we infer from Lemma 4.1 the existence of a constant
$C=C(T,\phi,K_{1})>0$ independent of $n$ such that,

$\displaystyle\sup_{n\geq 1}\widetilde{\bf
E}\int_{0}^{T}\left|\int_{0}^{t}(A\bar{u}^{n}(s),\phi)\mathrm{d}s\right|^{2}\mathrm{d}t$
$\displaystyle\leq T\sup_{n\geq 1}\widetilde{\bf
E}\int_{0}^{T}\left|(\bar{u}^{n}(s),A\phi)\right|^{2}\mathrm{d}s$
$\displaystyle\leq C(T)\sup_{n\geq 1}\widetilde{\bf
E}\int_{0}^{T}|\bar{u}^{n}(s)|^{2}|A\phi|^{2}\mathrm{d}s$ $\displaystyle\leq
C(T)\|\phi\|_{D(A)}^{2}\,\,\sup_{n\geq 1}\widetilde{\bf
E}\sup_{s\in[0,T]}|\bar{u}^{n}(s)|^{2}$ (4.53) $\displaystyle\leq C.$

From (4.52)-(4.53) and by means of the Vitali convergence theorem, we obtain
(4.51).

We now consider the term $B_{\widetilde{R}}$ as introduced in Section 2. We
will show that as $n\rightarrow\infty$,

(4.54)
$\int_{0}^{t}(P_{n}B_{\widetilde{R}}(\bar{u}^{n},\bar{u}^{n}),\phi)\mathrm{d}s\rightarrow\int_{0}^{t}(B_{\widetilde{R}}(\bar{u},\bar{u}),\phi)\mathrm{d}s\quad\text{
in $L^{1}(\widetilde{\Omega}\times[0,T])$.}$

For this purpose, we write

$\displaystyle\left|\int_{0}^{t}(P_{n}B_{\widetilde{R}}(\bar{u}^{n},\bar{u}^{n})-B_{\widetilde{R}}(\bar{u},\bar{u}),\phi)\mathrm{d}s\right|$
$\displaystyle\leq\int_{0}^{t}|(\theta_{\widetilde{R}}(|\bar{u}^{n}|)-\theta_{\widetilde{R}}(|\bar{u}|))(B(\bar{u},\bar{u}),P_{n}\phi)|\mathrm{d}s$
$\displaystyle+\int_{0}^{t}\theta_{\widetilde{R}}(|\bar{u}^{n}|)|((I-P_{n})B(\bar{u},\bar{u}),\phi)|\mathrm{d}s$
$\displaystyle+\int_{0}^{t}\theta_{\widetilde{R}}(|u^{n}|)|(B(\bar{u}^{n}-\bar{u},\bar{u}^{n})+B(\bar{u},\bar{u}^{n}-\bar{u}),P_{n}\phi)|\mathrm{d}s$
(4.55) $\displaystyle:=I_{B1}^{n}(t)+I_{B2}^{n}(t)+I_{B3}^{n}(t).$

Using the continuity of $\theta_{\widetilde{R}}$ and (4.43) together with the
Lebesgue dominated convergence theorem we obtain that $\widetilde{\bf
E}\int_{0}^{T}I_{B1}^{n}(t)\mathrm{d}t\rightarrow 0$ as $n\rightarrow\infty$.
For a subsequence this implies,

$\displaystyle I_{B1}^{n}\rightarrow 0\text{ a.e.
}(\omega,t)\in\widetilde{\Omega}\times[0,T]\text{ as }n\rightarrow\infty.$

Next observe that
$\|(I-P_{n})\phi\|\leq\frac{1}{\sqrt{\lambda_{n}}}\|\phi\|_{D(A)}$ where the
$\lambda_{n}$ are the eigenvalues of the operator $A$. Thus using (2.7) and
that $V\subset H$ is continuous we see that

$\displaystyle|I_{B2}^{n}(t)|=\int_{0}^{t}\theta_{\widetilde{R}}(|\bar{u}^{n}|)|((I-P_{n})B(\bar{u},\bar{u}),\phi)|\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}|(B(\bar{u},\bar{u}),(I-P_{n})\phi)|\mathrm{d}s$
$\displaystyle\leq\frac{1}{\sqrt{{\lambda_{n}}}}\|\phi\|_{D(A)}\int_{0}^{t}\|\bar{u}(s)\|^{2}\mathrm{d}s.$

Since we know that $\lambda_{n}\rightarrow\infty$ as $n\rightarrow\infty$ we
have $\widetilde{\bf E}\int_{0}^{T}I_{B2}^{n}\mathrm{d}t\rightarrow 0$ as
$n\rightarrow\infty$. For a subsequence this implies,

$I_{B2}^{n}\rightarrow 0\text{ a.e.
}(\omega,t)\in\widetilde{\Omega}\times[0,T]\text{ as }n\rightarrow\infty.$

Next using the fact that $\theta_{\widetilde{R}}\leq 1$ and the assumption
(2.6) we obtain

$\displaystyle|I_{B3}^{n}(t)|$
$\displaystyle\leq\int_{0}^{t}\theta_{\widetilde{R}}(|\bar{u}^{n}|)|\bar{u}^{n}-\bar{u}|\left(\|\bar{u}^{n}\|+\|\bar{u}\|\right)\|\phi\|_{D(A)}\mathrm{d}s$
$\displaystyle\leq\|\phi\|_{D(A)}\left(\|\bar{u}^{n}\|_{L^{2}(0,T;V)}+\|\bar{u}\|_{L^{2}(0,T;V)}\right)\,|\bar{u}^{n}-\bar{u}|_{L^{2}(0,T;H)}.$

We know that $\widetilde{\bf E}\int_{0}^{T}\|\bar{u}^{n}\|^{2}\mathrm{d}s$ is
bounded by a constant independent of $n$, thus thanks to (4.43) we obtain that
$I_{B3}^{n}\rightarrow 0$ for a subsequence $n\rightarrow\infty$ for almost
every $(\omega,t)\in\widetilde{\Omega}\times[0,T]$.
Next, recall that $\theta_{\widetilde{R}}(x)$ vanishes for $x\geq
2\widetilde{R}$. Thus using the assumption (2.6) and Lemma 4.1 we obtain,

$\displaystyle\sup_{n\geq 1}\widetilde{\bf
E}\int_{0}^{T}\bigg{|}\int_{0}^{t}(P_{n}B_{\widetilde{R}}(\bar{u}^{n},\bar{u}^{n}),\phi)\mathrm{d}s\bigg{|}^{2}\mathrm{d}t$
$\displaystyle=\sup_{n\geq 1}\widetilde{\bf
E}\int_{0}^{T}\bigg{|}\int_{0}^{t}\theta_{\widetilde{R}}(|u^{n}|)(B(\bar{u}^{n},\bar{u}^{n}),P_{n}\phi)\mathrm{d}s\bigg{|}^{2}\mathrm{d}t$
$\displaystyle\leq\widetilde{R}^{2}T\|\phi\|_{D(A)}^{2}\,\,\sup_{n\geq
1}\widetilde{\bf E}\int_{0}^{T}\|\bar{u}^{n}(s)\|^{2}\mathrm{d}s$ (4.56)
$\displaystyle\leq C(\phi,T,\widetilde{R},K_{1}).$

Thus using once more the Vitali convergence theorem we obtain (4.54).
The noise terms are treated differently.

######  Lemma 4.14.

The processes
$(\int_{0}^{t}P_{n}G(\bar{u}^{n}(s))\mathrm{d}\bar{W}(s))_{t\in[0,T]}$
converge to $(\int_{0}^{t}G(\bar{u}(s))\mathrm{d}\bar{W}(s))_{t\in[0,T]}$ in
$L^{1}(\widetilde{\Omega};L^{1}([0,T];H))$ as $n\rightarrow\infty$.

###### Proof.

We will first show that along a subsequence, $\widetilde{\bf P}$-a.s.

(4.57) $\displaystyle P_{n}G(\bar{u}^{n})\rightarrow G(\bar{u})\qquad\text{in
$L^{2}([0,T];L_{2}(U_{0},H))$}.$

Indeed, utilizing the Poincaré inequality, the hypothesis (2.10) and the fact
that $P_{n}$ is a projector in $H$ we obtain $\widetilde{\bf P}$-a.s.

$\displaystyle\int_{0}^{T}\|P_{n}G(\bar{u}^{n})-G(\bar{u})\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s$
$\displaystyle\leq\int_{0}^{T}(\|P_{n}G(\bar{u}^{n})-P_{n}G(\bar{u})\|^{2}_{L_{2}(U_{0},H)}$
$\displaystyle\hskip
144.54pt+\|(I-P_{n})G(\bar{u})\|^{2}_{L_{2}(U_{0},H)})\mathrm{d}s$ (4.58)
$\displaystyle\leq\rho\int_{0}^{T}|\bar{u}^{n}-\bar{u}|^{2}+\|(I-P_{n})G(\bar{u})\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s\rightarrow
0.$

The convergence in (4.58) follows from (4.43) and the Lebesgue Dominated
convergence theorem since we have that
$\|(I-P_{n})G(\bar{u})\|^{2}_{L_{2}(U_{0},H)}$ converges to $0$ a.e. on
$[0,T]$ and is dominated by $\|G(\bar{u})\|^{2}_{L_{2}(U_{0},H)}\in
L^{1}[0,T]$.
Due to Lemma 2.1 of [19] (see also [5]), the convergence (4.57) implies that

(4.59)
$\displaystyle\int_{0}^{t}P_{n}(G(\bar{u}^{n}))\mathrm{d}\bar{W}(s)\rightarrow\int_{0}^{t}G(\bar{u}(s))\mathrm{d}\bar{W}(s)\text{
in probability in $L^{2}(0,T;H)$. }$

Additionally, for some $C=C(T,\rho,K_{1})>0$ independent of $n$, thanks to the
Itô isometry we obtain

$\displaystyle\widetilde{\bf
E}\int_{0}^{T}|\int_{0}^{t}P_{n}G(\bar{u}^{n}(s))\mathrm{d}\bar{W}(s)|^{2}\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\widetilde{\bf
E}\int_{0}^{t}\|G(\bar{u}^{n})\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s\mathrm{d}t$
$\displaystyle\leq\rho T\sup_{n\geq 1}\widetilde{\bf
E}\left(\int_{0}^{T}(|\bar{u}^{n}|^{2}+1)\mathrm{d}s\right)$ (4.60)
$\displaystyle\leq\rho T\sup_{n\geq 1}\widetilde{\bf
E}\sup_{t\in[0,T]}(|\bar{u}^{n}|^{2}+1)\leq C.$

Making use of the Vitali convergence theorem again, we conclude the proof of
Lemma 4.14. ∎

######  Lemma 4.15.

The processes
$(\int_{(0,t]}\int_{E_{0}}{P_{n}K(\bar{u}^{n}(s-),\xi)}{}\mathrm{d}\widehat{\bar{\pi}}(s,\xi))_{t\in[0,T]}$
converge to
$(\int_{(0,t]}\int_{E_{0}}{K(\bar{u}(s-),\xi)}{}\mathrm{d}\widehat{\bar{\pi}}(s,\xi))_{t\in[0,T]}$
in $L^{1}(\widetilde{\Omega};L^{1}([0,T];H))$ as $n\rightarrow\infty$.

###### Proof.

First we have $\widetilde{\bf P}$-a.s.

$\displaystyle\int_{0}^{T}\int_{0}^{t}\int_{E_{0}}$
$\displaystyle\left|{P_{n}K(\bar{u}^{n}(s-),\xi)-K(\bar{u}(s-),\xi)}{}\right|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\int_{0}^{T}\int_{0}^{t}\int_{E_{0}}\left|{P_{n}K(\bar{u}^{n}(s-),\xi)-P_{n}K(\bar{u}(s-),\xi)}{}\right|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\mathrm{d}t$
$\displaystyle\hskip
144.54pt+\int_{0}^{T}\int_{0}^{t}\int_{E_{0}}\left|{(I-P_{n})K(\bar{u}(s-),\xi)}{}\right|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\int_{0}^{T}\int_{0}^{t}\int_{E_{0}}\left|{K(\bar{u}^{n}(s-),\xi)-K(\bar{u}(s-),\xi)}{}\right|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\mathrm{d}t$
$\displaystyle\hskip
144.54pt+\int_{0}^{T}\int_{0}^{t}\int_{E_{0}}\left|{(I-P_{n})K(\bar{u}(s-),\xi)}{}\right|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
T\int_{0}^{T}|\bar{u}^{n}(s)-\bar{u}(s)|^{2}\mathrm{d}s+T\int_{0}^{T}\int_{E_{0}}\left|{(I-P_{n})K(\bar{u}(s-),\xi)}{}\right|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s$
(4.61) $\displaystyle\rightarrow$ $\displaystyle\,0\,\text{ as
}n\rightarrow\infty.$

The last line follows thanks to the fact that $\bar{u}^{n}\rightarrow\bar{u}$
in $L^{2}(0,T;H)$ $\widetilde{\bf P}$-a.s. and using the Lebesgue dominated
convergence theorem since we also see that as $n\rightarrow\infty$ the
sequence $|(I-P_{n})K(\bar{u}(s-),\cdot)|^{2}_{L^{2}(E_{0},\nu;H)}$ converges
to $0$ a.e. on $[0,T]$ and is dominated by
$|K(\bar{u}(s-),\cdot)|^{2}_{L^{2}(E_{0},\nu;H)}\in L^{1}([0,T]).$
This implies that ${P_{n}K(\bar{u}^{n}(s-),\xi)}{}$ converges to
${K(\bar{u}(s-),\xi}{})$ in $L^{2}([0,T]\times
E_{0},\mathrm{d}t\otimes\mathrm{d}\nu;H)$ $\widetilde{\bf P}-$a.s. It is then
easy to see that the above almost sure convergence implies the following
convergence

(4.62)
$\int_{(0,t]}\int_{E_{0}}{P_{n}K(\bar{u}^{n}(s-),\xi)}\mathrm{d}\widehat{\bar{\pi}}(s,\xi)\rightarrow\int_{(0,t]}\int_{E_{0}}{K(\bar{u}(s-),\xi)}{}\mathrm{d}\widehat{\bar{\pi}}(s,\xi)$

in probability as $L^{2}(0,T;H)$-valued random variables as
$n\rightarrow\infty$.

Also observe that, using the isometry property (e.g. [30] or Theorem 2.16 in
[14]), we obtain

$\displaystyle\widetilde{\bf
E}\int_{0}^{T}\bigg{|}{\int_{(0,t]}\int_{E_{0}}P_{n}K(\bar{u}^{n}(s-),\xi)}\mathrm{d}\widehat{\bar{\pi}}(s,\xi)\bigg{|}^{2}\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\left(\widetilde{\bf
E}\int_{0}^{t}\int_{E_{0}}|{P_{n}K(\bar{u}^{n}(s-),\xi)}{}|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\right)\mathrm{d}t$
$\displaystyle\leq\rho T\,\sup_{n\geq 1}\widetilde{\bf
E}\int_{0}^{T}(1+|\bar{u}^{n}(s)|^{2})\mathrm{d}s$ (4.63) $\displaystyle\leq
C(T,\rho,K_{1}).$

From (4.61)-(4.4.1), and an application of the Vitali convergence Theorem we
finish the proof of Lemma 4.15. ∎

Gathering all the convergence results (4.50), (4.51), (4.54), Lemma 4.14 and
Lemma 4.15, along with the weak convergence obtained in (4.48), we see that
for any $\phi\in D(A)$ and any measurable set
$\mathcal{S}\subset\widetilde{\Omega}\times[0,T]$, we have

$\displaystyle\widetilde{\bf
E}\int_{0}^{T}\chi_{\mathcal{S}}\bigg{[}(\bar{u}(s),\phi)\mathrm{d}s+\int_{0}^{t}\gamma(A\bar{u}(s),\phi)\mathrm{d}s+\int_{0}^{t}(B_{\widetilde{R}}(\bar{u}(s),\bar{u}(s)),\phi)\mathrm{d}s+\int_{0}^{t}(Y^{R,\widetilde{R}},\phi)\mathrm{d}s\bigg{]}\mathrm{d}t$
(4.64) $\displaystyle=\widetilde{\bf
E}\int_{0}^{T}\chi_{\mathcal{S}}\bigg{[}(\bar{u}_{0},\phi)\mathrm{d}s+\int_{0}^{t}(G(\bar{u}(s)),\phi)\mathrm{d}\bar{W}(s)+\int_{(0,t]}\int_{E_{0}}(K(\bar{u}(s-),\xi),\phi)\mathrm{d}\widehat{\bar{\pi}}(s,\xi)\bigg{]}\mathrm{d}t.$

Since $D(A)$ is dense in $V$, and $\mathcal{S}$ is arbitrary it follows that
$\mathrm{d}\widetilde{\bf P}\otimes\mathrm{d}t$-a.e. we have

$\displaystyle\bar{u}(t)+\int_{0}^{t}\gamma
A\bar{u}(s)\mathrm{d}s+\int_{0}^{t}B_{\widetilde{R}}(\bar{u}(s),\bar{u}(s))\mathrm{d}s+\int_{0}^{t}Y^{R,\widetilde{R}}\mathrm{d}s$
(4.65) $\displaystyle\hskip
72.26999pt=\bar{u}_{0}+\int_{0}^{t}G(\bar{u}(s))\mathrm{d}\bar{W}(s)+\int_{(0,t]}\int_{E_{0}}K(\bar{u}(s-),\xi)\mathrm{d}\widehat{\bar{\pi}}(s,\xi)\text{
in }{V^{\prime}}.$

See Lemma 5.13 in [17] for details. In particular we have
$\bar{u}_{0}=\bar{u}(0)$ $\widetilde{\bf P}$-a.s.
Step 2: In order to show that $\bar{u}$ is a solution to the equations (4.2),
it remains to verify that

(4.66) $\displaystyle
Y^{R,\widetilde{R}}=F^{R}(\bar{u}),\quad\mathrm{d}\widetilde{\bf
P}\otimes\mathrm{d}t\textnormal{-a.e.}$

This follows from a classical monotonicity argument due to G. Minty [40], [41]
(see also [8], [10], [12],[35], [46] [51]). First using the product rule we
have

(4.67) $\displaystyle e^{-\rho
t}|\bar{u}(t)|^{2}=|\bar{u}_{0}|^{2}+\int_{0}^{t}e^{-\rho
s}\left(-\rho|\bar{u}(s)|^{2}\mathrm{d}s+\mathrm{d}|\bar{u}(s)|^{2}\right).$

Now we apply Itô’s formula to the process in (4.4.1) using the function
$\psi(u)=|\bar{u}|^{2}$ as in (4.11):

$\displaystyle|\bar{u}(t)|^{2}-|\bar{u}_{0}|^{2}=\int_{0}^{t}\Bigg{[}-2\gamma(A\bar{u}(s),\bar{u}(s))-2(Y^{R,\widetilde{R}}(s),\bar{u}(s))+\|G(\bar{u}(s))\|^{2}_{L_{2}(U_{0},H)}\Bigg{]}\mathrm{d}s$
$\displaystyle+\int_{(0,t]}\int_{E_{0}}|K(\bar{u}(s-),\xi)|^{2}\mathrm{d}{\bar{\pi}}(\xi,s)+\int_{0}^{t}(\bar{u}(s),G(\bar{u}(s))\mathrm{d}W(s))+\int_{(0,t]}\int_{E_{0}}(K(\bar{u}(s-),\xi),\bar{u}(s))\mathrm{d}\widehat{\bar{\pi}}(s,\xi).$

Combining the above two equations we obtain

$\displaystyle|\bar{u}(t)|^{2}-|\bar{u}_{0}|^{2}$
$\displaystyle=\int_{0}^{t}e^{-\rho
s}\bigg{[}-\rho|\bar{u}(s)|^{2}-2\gamma(A\bar{u}(s),\bar{u}(s))-2(Y^{R,\widetilde{R}}(s),\bar{u}(s))+\|G(\bar{u}(s))\|^{2}_{L_{2}(U_{0},H)}\bigg{]}\mathrm{d}s$
$\displaystyle+\int_{(0,t]}\int_{E_{0}}e^{-\rho
s}|K(\bar{u}(s-),\xi)|^{2}\mathrm{d}{\bar{\pi}}(\xi,s)+\int_{(0,t]}\int_{E_{0}}e^{-\rho
s}(K(\bar{u}(s-),\xi),\bar{u}(s))\mathrm{d}\widehat{\bar{\pi}}(s,\xi)$ (4.68)
$\displaystyle+\int_{0}^{t}e^{-\rho
s}(\bar{u}(s),G(\bar{u}(s))\mathrm{d}W(s)).$

We take expectation of both sides of (4.68). Observing that the last two terms
are martingales and their expectations vanish and we obtain

$\displaystyle\widetilde{\bf E}\left(e^{-\rho
t}|\bar{u}(t)|^{2}-|\bar{u}_{0}|^{2}\right)$ $\displaystyle=\widetilde{\bf
E}\Big{(}\int_{0}^{t}e^{-\rho
s}\bigg{[}-2\gamma(A\bar{u}(s),\bar{u}(s))-2(Y^{R,\widetilde{R}}(s),\bar{u}(s))$
(4.69)
$\displaystyle+\|G(\bar{u}(s))\|^{2}_{L_{2}(U_{0},H)}-\rho|\bar{u}(s)|^{2}\bigg{]}\mathrm{d}s+\int_{(0,t]}\int_{E_{0}}e^{-\rho
s}|K(\bar{u}(s-),\xi)|^{2}\mathrm{d}{\bar{\pi}}(\xi,s)\Big{)}.$

In a similar fashion, we carry out (4.67) for $\bar{u}^{n}$, then apply Itô’s
formula to the process in (4.44) and take expectation of both sides. Using
that $P_{n}(\bar{u}^{n}(t))=\bar{u}^{n}(t)$, we obtain:

$\displaystyle\widetilde{\bf E}\left(e^{-\rho
t}|\bar{u}^{n}(t)|^{2}-|\bar{u}^{n}_{0}|^{2}\right)$
$\displaystyle=\widetilde{\bf E}\big{(}\int_{0}^{t}e^{-\rho
s}\bigg{[}-2\gamma(A\bar{u}^{n}(s),\bar{u}^{n}(s))-2(F^{R}(\bar{u}^{n}(s)),\bar{u}^{n}(s))$
(4.70)
$\displaystyle+\|G(\bar{u}^{n}(s))\|^{2}_{L_{2}(U_{0},H)}-\rho|\bar{u}^{n}(s)|^{2}\bigg{]}\mathrm{d}s+\int_{(0,t]}\int_{E_{0}}e^{-\rho
s}|K(\bar{u}^{n}(s-),\xi)|^{2}\mathrm{d}{\bar{\pi}}(\xi,s)\big{)}.$

Next, we will use the monotonicity of $A$ that comes from its linearity and
that of $F^{R}$ as specified in (4.5). That is, for an arbitrary $\Phi\in
L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};V)$, we write
$(F^{R}(\bar{u}^{n}(s)),\bar{u}^{n})\geq(F^{R}(\bar{u}^{n}(s)),\Phi(s))+(F^{R}(\Phi(s)),\bar{u}^{n}(s)-\Phi(s))$.
Treating the terms involving $A$ similarly we obtain,

$\displaystyle\widetilde{\bf E}\left(e^{-\rho
t}|\bar{u}^{n}(t)|^{2}-|\bar{u}^{n}_{0}|^{2}\right)$
$\displaystyle\leq\widetilde{\bf E}\int_{0}^{t}e^{-\rho
s}\bigg{(}-2\gamma(A\bar{u}^{n}(s),\Phi(s))-2\gamma(A\Phi(s),\bar{u}^{n}(s)-\Phi(s))\mathrm{d}s$
$\displaystyle-2(F^{R}(\bar{u}^{n}(s)),\Phi(s))\mathrm{d}s-2(F^{R}(\Phi(s)),\bar{u}^{n}(s)-\Phi(s))\mathrm{d}s$
(4.71)
$\displaystyle+\|G(\bar{u}^{n}(s))\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s+\int_{E_{0}}|K(\bar{u}^{n}(s-),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s-\rho|\bar{u}^{n}|^{2}\mathrm{d}s\bigg{)}.$

Furthermore, using the equation $|a|^{2}=|a-b|^{2}+2(a,b)-|b|^{2}$ we see that
the right-hand side of the equation above is equal to

$\begin{split}&\widetilde{\bf E}\int_{0}^{t}e^{-\rho
s}\bigg{[}-2\gamma(A\bar{u}^{n}(s),\Phi(s))\mathrm{d}s+2\gamma(A\Phi(s),\Phi(s))\mathrm{d}s-2\gamma(A\Phi(s),\bar{u}^{n}(s))\mathrm{d}s\\\
&+2(F^{R}(\Phi(s)),\bar{u}^{n}(s))\mathrm{d}s-2(F^{R}(\Phi(s)),\Phi(s))\mathrm{d}s-2(F^{R}(\bar{u}^{n}(s)),\Phi(s))\mathrm{d}s\\\
&-\rho\left(|\bar{u}^{n}(s)-\Phi(s)|^{2}-|\Phi(s)|^{2}+2(\bar{u}^{n},\Phi)\right)\mathrm{d}s-\int_{E_{0}}|K(\Phi(s),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s-\|G(\Phi(s))\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s\\\
&+2\int_{E_{0}}(K(\Phi(s),\xi),K(\bar{u}^{n}(s),\xi))\mathrm{d}\nu(\xi)\mathrm{d}s+2(G(\Phi(s)),G(\bar{u}^{n}(s)))_{L_{2}(U_{0},H)}\mathrm{d}s\\\
&+\|G(\bar{u}^{n}(s))-G(\Phi(s))\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s+\int_{E_{0}}|K(\bar{u}^{n}(s-),\xi)-K(\Phi(s),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\bigg{]}.\end{split}$

We next use (2.10) on the last two terms of the above equation and bound them
from above by $\rho|\bar{u}^{n}(s)-\Phi(s)|^{2}$. This gives us the following
upper bounds

$\displaystyle\widetilde{\bf E}\left(e^{-\rho
t}|\bar{u}^{n}(t)|^{2}-|\bar{u}^{n}_{0}|^{2}\right)$
$\displaystyle\leq\widetilde{\bf E}\bigg{[}\int_{0}^{t}e^{-\rho
s}\bigg{[}-2\gamma(A\bar{u}^{n}(s),\Phi(s))\mathrm{d}s+2\gamma(A\Phi(s),\Phi(s))\mathrm{d}s-2\gamma(A\Phi(s),\bar{u}^{n}(s))\mathrm{d}s$
$\displaystyle+2(F^{R}(\Phi(s)),\bar{u}^{n}(s))\mathrm{d}s-2(F^{R}(\Phi(s)),\Phi(s))\mathrm{d}s-2(F^{R}(\bar{u}^{n}(s)),\Phi(s))\mathrm{d}s$
$\displaystyle-\rho\left(|\bar{u}^{n}(s)-\Phi(s)|^{2}-|\Phi(s)|^{2}+2(\bar{u}^{n}(s),\Phi)\right)\mathrm{d}s+{{}\rho|\bar{u}^{n}(s)-\Phi(s)|^{2}\mathrm{d}s}$
$\displaystyle-\|G(\Phi(s))\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s+2(G(\Phi(s)),G(\bar{u}^{n}(s)))_{L_{2}(U_{0},H)}\mathrm{d}s$
(4.72)
$\displaystyle-\int_{E_{0}}|K(\Phi(s),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s+2\int_{E_{0}}(K(\Phi(s),\xi),K(\bar{u}^{n}(s),\xi))\mathrm{d}\nu(\xi)\mathrm{d}s\bigg{]}.$

Now we multiply each side of the above equation by $\varphi\in
L^{\infty}([0,T],\mathrm{d}t;\mathbb{R}),$ $\varphi\geq 0$ and integrate in
time. Finally we let $n\rightarrow\infty$ using the convergence results
(4.43)-(4.47) and obtain

$\displaystyle\widetilde{\bf E}\int_{0}^{T}\varphi(t)$
$\displaystyle\left(e^{-\rho
t}|\bar{u}(t)|^{2}-|\bar{u}_{0}|^{2}\right)\mathrm{d}t$
$\displaystyle\leq\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\bigg{(}\int_{0}^{t}e^{-\rho
s}\bigg{[}-2\gamma(A\bar{u}(s),\Phi(s))\mathrm{d}s+2\gamma(A\Phi(s),\Phi(s))\mathrm{d}s-2\gamma(A\Phi(s),\bar{u}(s))\mathrm{d}s$
$\displaystyle+2(F^{R}(\Phi(s)),\bar{u}(s))\mathrm{d}s-2(F^{R}(\Phi(s)),\Phi(s))\mathrm{d}s-2(Y^{R,\widetilde{R}}(s),\Phi(s))\mathrm{d}s$
$\displaystyle-\rho(-|\Phi(s)|^{2}+2(\bar{u},\Phi))\mathrm{d}s-\left\|G(\Phi(s))\right\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s+2(G(\bar{u}(s)),G(\Phi(s)))_{L_{2}(U_{0},H)}\mathrm{d}s$
$\displaystyle-\int_{E_{0}}|K(\Phi(s),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s+2\int_{E_{0}}(K(\Phi(s),\xi),K(\bar{u}(s-),\xi))\mathrm{d}\nu(\xi)\mathrm{d}s\bigg{]}\bigg{)}\mathrm{d}t.$

Combining this with (4.69), we obtain

$\displaystyle\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\bigg{(}\int_{0}^{t}e^{-\rho
s}\bigg{[}-2\gamma(A\bar{u}(s),\bar{u}(s))-2(Y^{R,\widetilde{R}}(s),\bar{u}(s))+\|G(\bar{u}(s))\|^{2}_{L_{2}(U_{0},H)}{-\rho|\bar{u}(s)|^{2}}\bigg{]}\mathrm{d}s$
$\displaystyle\hskip
144.54pt+\int_{(0,t]}\int_{E_{0}}|K(\bar{u}(s-),\xi)|^{2}\mathrm{d}{\bar{\pi}}(s,\xi)\bigg{)}\mathrm{d}t$
$\displaystyle\leq\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\bigg{(}\int_{0}^{t}e^{-\rho
s}\bigg{[}-2\gamma(A\bar{u}(s),\Phi(s))\mathrm{d}s+2\gamma(A\Phi(s),\Phi(s))\mathrm{d}s-2\gamma(A\Phi(s),\bar{u}(s))\mathrm{d}s$
$\displaystyle\qquad+2(F^{R}(\Phi(s)),\bar{u}(s))\mathrm{d}s-2(F^{R}(\Phi(s)),\Phi(s))\mathrm{d}s-2(Y^{R,\widetilde{R}}(s),\Phi(s))\mathrm{d}s$
$\displaystyle{+\rho\left(|\Phi(s)|^{2}-2(\bar{u}(s),\Phi(s))\right)\mathrm{d}s}-\left\|G(\Phi(s))\right\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s+2(G(\bar{u}(s)),G(\Phi(s)))_{L_{2}(U_{0},H)}\mathrm{d}s$
$\displaystyle-\int_{E_{0}}|K(\Phi(s),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s+2\int_{E_{0}}(K(\Phi(s),\xi),K(\bar{u}(s-),\xi))\mathrm{d}\nu(\xi)\mathrm{d}s\bigg{]}\bigg{)}\mathrm{d}t.$

Using again $|a|^{2}=|a-b|^{2}+2(a,b)-|b|^{2}$ we obtain

$\displaystyle\widetilde{\bf E}\int_{0}^{T}\varphi(t)\int_{0}^{t}$
$\displaystyle e^{-\rho
s}\bigg{(}2\gamma(A\bar{u}(s)-A\Phi(s),\bar{u}(s)-\Phi(s))\mathrm{d}s+2(F^{R}(\Phi(s))-Y^{R,\widetilde{R}}(s),\Phi(s)-\bar{u}(s))\mathrm{d}s$
$\displaystyle\qquad\qquad\qquad+2|\bar{u}(s)-\Phi(s)|^{2}\mathrm{d}s-\|G(\bar{u}(s))-G(\Phi(s))\|^{2}_{L_{2}(U_{0},H)}\mathrm{d}s$
$\displaystyle\qquad\qquad\qquad-\int_{E_{0}}|K(\bar{u}(s),\xi)-K(\Phi(s),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\bigg{)}\mathrm{d}t\geq
0.$

Since the last two terms in the inequality above are negative we have,

$\displaystyle\widetilde{\bf E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}\bigg{(}2\gamma(A\bar{u}(s)-A\Phi(s),\bar{u}(s)-\Phi(s))+2(F^{R}(\Phi(s))-Y^{R,\widetilde{R}}(s),\Phi(s)-\bar{u}(s))$
(4.73)
$\displaystyle+\rho|\bar{u}(s)-\Phi(s)|^{2}\bigg{)}\mathrm{d}s\mathrm{d}t\geq
0.$

We now let $\Phi=\bar{u}\pm\lambda\zeta{\bf v}$ for $\lambda\geq 0$, ${\bf
v}\in V$ and $\zeta\in
L^{\infty}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};\mathbb{R})$ and use the linearity of the operator $A$ . Then dividing both
sides by $\lambda$, letting $\lambda\rightarrow 0$ and using the
hemicontinuity property of the operator $F^{R}$, we obtain

(4.74) $\widetilde{\bf
E}\left(\int_{0}^{T}\varphi(t)\left(\int_{0}^{t}e^{-\rho s}\zeta(s)\langle
F^{R}(\bar{u}(s))-Y^{R,\widetilde{R}}(s),{\bf
v}\rangle\mathrm{d}s\right)\mathrm{d}t\right)=0.$

Since $\varphi$, $\zeta$ and ${\bf v}$ are all arbitrary, we conclude that

(4.75) $Y^{R,\widetilde{R}}=F^{R}(\bar{u}(s))\quad\mathrm{d}\widetilde{\bf
P}\otimes\mathrm{d}t-\text{a.e.}$

Therefore, we have now proven that $\bar{u}$ solves (4.2) with respect to the
basis
$(\widetilde{\Omega},\widetilde{\mathcal{F}},(\bar{\mathcal{F}}_{t})_{t\geq
0},\widetilde{\bf P},\bar{W},\bar{\pi})$ which marks the end of the passage to
the limit $n\rightarrow\infty$ (with $R$ and $\widetilde{R}$ fixed) and thus
concludes the proof of Theorem 4.13. ∎

######  Remark 4.16.

Note that in Lemma 4.14 and in Lemma 4.15 we do not obtain convergence of the
mentioned processes in $L^{2}(\widetilde{\Omega};L^{2}([0,T];H))$ as in e.g.
[15] or [19]. The lack of such a result could pose a problem while passing to
the limit $n\rightarrow\infty$ in Step 2. However this is taken care of by
first carrying out (4.67) and then applying the Itô formula using the function
$\psi(u)=|u(t)|^{2}$, as can be observed in (4.72).

##  5\. Passage to the limit $R\rightarrow\infty$

For this section, to emphasize the dependence on the parameter $R$, we will
use the notation
$(\bar{u}_{0}^{R},\bar{u}^{R},\bar{W}^{R},\bar{\pi}^{R}):=(\bar{u}_{0},\bar{u},\bar{W},\bar{\pi})$
to denote the solution found in the previous section and thus omiting for the
moment the dependence on $\widetilde{R}$, e.g.
$\bar{u}^{R}=\bar{u}^{R,\widetilde{R}}$, etc. Now, for fixed $\widetilde{R}$
we will pass to the limit $R\rightarrow\infty$ and show the existence of a
martingale solution to the stochastic equation

(5.1) $\begin{split}u(t)+\int_{0}^{t}[\gamma
Au(s)+&\theta_{\widetilde{R}}(|u|)B(u(s),u(s))+F(u(s))]\mathrm{d}s\\\
&=u(0)+\int_{0}^{t}G(u(s))\mathrm{d}W(s)+\int_{(0,t]}\int_{E_{0}}K(u(s-),\xi)\mathrm{d}\widehat{\pi}(s,\xi).\end{split}$

Recall that $X\subset(W^{k,p}(\mathcal{O}))^{d}$ for some $p>2$ and integer
$k\geq 0$. Recall also that in Lemma 4.1 we used the equicoercivity result
(4.7) for the family $\\{F^{R}\\}_{R>R_{0}}$, derived in [49], to obtain
bounds that were independent of the parameter $R$. This readily gives us the
following energy estimates.

######  Lemma 5.1.

Let $\bar{u}^{R}$ denote a martingale solution to (4.2) with respect to the
basis
$(\widetilde{\Omega},\widetilde{\mathcal{F}},{(\bar{\mathcal{F}}^{R}_{t})_{t\geq
0}},\widetilde{\bf P},{\bar{W}^{R},\bar{\pi}^{R}})$ as found in Theorem 4.13
in the previous section. Assuming all the hypotheses in Lemma 4.1, for large
enough $R>R_{0}$ we have

(5.2) $\displaystyle\widetilde{\bf
E}\sup_{s\in[0,T]}|\bar{u}^{R}(s)|^{2}+(\gamma+c_{1})\widetilde{\bf
E}\int_{0}^{T}\|{\bar{u}^{R}(s)}\|^{2}\mathrm{d}s\leq K_{1}.$

Furthermore, for some $K_{2}(T,\gamma,\rho,k,p,K_{1})>0$ independent of $R$,
we obtain the following bounds,

(5.3) $\displaystyle\sum_{i=0}^{k}\left(\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}|D^{i}\bar{u}^{R}|_{p}^{p}\mathbbm{1}_{\\{|D^{i}\bar{u}^{R}|_{p}<R\\}}\mathrm{d}x\mathrm{d}t+R^{p-2}\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}|D^{i}\bar{u}^{R}|_{p}^{2}\mathbbm{1}_{\\{|D^{i}\bar{u}^{R}|_{p}>R\\}}\,\mathrm{d}x\mathrm{d}t\right)$
$\displaystyle\leq K_{2}.$

######  Remark 5.2.

Note that the bound for the second term on the left hand side of (5.3) further
implies for the same $K_{2}$ as in (5.3),

(5.4) $\displaystyle\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}\mathbbm{1}_{\\{|D^{i}\bar{u}^{R}|_{p}>R\\}}\mathrm{d}x\mathrm{d}t\leq\frac{K_{2}}{R^{p}}.$

###### Proof.

The estimate (5.2) is readily obtained by using the convergence results (4.46)
and (4.47) and Lemma 4.1 along with the lower semicontinuity property of the
norms involved.
We will now very briefly discuss the results from [49] that lead to (5.3);
this discussion will be useful below. First it can be seen that the
assumptions (2.11)-(2.15) give us, for our construction (4.4), the existence
of some constant $c_{6}>0$ independent of $R$ such that

(5.5) $\displaystyle f_{i}(x)\cdot{x}\geq c_{1}|x|_{p}-c_{6}\qquad\forall
x\in\mathbb{R}^{d^{i+1}},i=0,1,...,k.$

These assumptions also give us that for $x\in\mathbb{R}^{d^{i+1}}$, if
$|x|_{p}>R>1$, then for ${c_{7}\geq\max\\{c_{2},c_{6}\\}}>0$ (independent of
$R$) we have

(5.6) $\displaystyle f_{i}\left({\frac{Rx}{|x|_{p}}}\right)\cdot
x+\left(x-{\frac{Rx}{|x|_{p}}}\right)\,{D}^{2}\mathscr{J}_{i}\left({{\frac{Rx}{|x|_{p}}}}\right)\cdot
x\geq(c_{1}R^{p-2}-c_{5})|x|_{p}^{2}.$

Then the definitions (4.3)-(4.4) along with a direct application of (5.5) and
(5.6), give us the following lower bound for any $v\in V$,

(5.7)
$\displaystyle(F^{R}(v),v)\geq\sum_{i=0}^{k}\int_{\mathcal{O}}\left(|D^{i}v|_{p}^{p}\mathbbm{1}_{\\{|D^{i}v|_{p}<R\\}}+R^{p-2}|D^{i}v|_{p}^{2}\mathbbm{1}_{\\{|D^{i}v|_{p}>R\\}}\,\right)\mathrm{d}x-C,$

where the constant $C=C(p,k)>0$ is independent of $R$ and $v$.
For detailed proofs of the inequalities (5.5) and (5.6) and thus of (5.7), we
refer the reader to [49] wherein these inequalities are derived in the
deterministic setting. We now apply Itô’s formula to (4.2) with respect to the
basis
$(\widetilde{\Omega},\widetilde{\mathcal{F}},{(\bar{\mathcal{F}}^{R}_{t})_{t\geq
0}},\widetilde{\bf P},{\bar{W}^{R},\bar{\pi}^{R}})$ using the function
$|\bar{u}^{R}|^{2}$, then take supremum in time and expectations of both
sides, as in (4.11). However unlike in (4.12) we use (5.7) instead of (4.7).
Note that the noise terms are adapted to the filtration
$(\bar{\mathcal{F}}^{R}_{t})_{t\geq 0}$ and we obtain bounds like in (4.13)
-(4.15).

This results in the following bounds,

$\displaystyle\widetilde{\bf
E}\sup_{t\in[0,T]}\sum_{i=0}^{k}\int_{0}^{t}\int_{\mathcal{O}}\left(|D^{i}\bar{u}^{R}|_{p}^{p}\mathbbm{1}_{\\{|D^{i}\bar{u}^{R}|_{p}<R\\}}+R^{p-2}|D^{i}\bar{u}^{R}|_{p}^{2}\mathbbm{1}_{\\{|D^{i}\bar{u}^{R}|_{p}>R\\}}\,\right)\mathrm{d}x\mathrm{d}t-C(T,p,k)$
$\displaystyle\qquad\qquad\qquad\leq{{\bf E}}|u_{0}|^{2}+\widetilde{\bf
E}\sup_{t\in[0,T]}|{\bar{u}^{R}(t)}|^{2}+C(\rho)\widetilde{\bf
E}\int_{0}^{T}(1+|\bar{u}^{R}(s)|^{2})\mathrm{d}s$
$\displaystyle\qquad\qquad\qquad\leq K_{2}\quad\text{ thanks to
\eqref{apriori_gencase}}.$

∎

######  Proposition 5.3.

Let $\bar{u}^{R}$ be a martingale solution to (4.2) with respect to the
stochastic basis
$(\widetilde{\Omega},\widetilde{\mathcal{F}},{(\bar{\mathcal{F}}^{R}_{t})_{t\geq
0}},\widetilde{\bf P},{\bar{W}^{R},\bar{\pi}^{R}})$ as given by Theorem 4.13,
then for every $\alpha\in(0,\frac{1}{2})$ there exists a constant
$C=C(\alpha,T,\widetilde{R},K_{1})>0$ such that

(5.8) $\sup_{R>R_{0}}\widetilde{\bf
E}\left\|\bar{u}^{R}\right\|^{\frac{p}{p-1}}_{W^{\alpha,\frac{p}{p-1}}(0,T;{{D(A)^{\prime}}+X^{\prime}})}\leq
C.$

Furthermore, due to Lemma 4.3 and an argument like in Proposition 4.6, the
estimate (5.8) implies that the laws of $\bar{u}^{R}$ are tight in
$L^{2}(0,T;H)$.

###### Proof.

First observe that Proposition 4.4 holds for $\beta=\frac{p}{p-1}$ and
$P_{n}(G(u^{n}))$ and $P_{n}(K(u^{n}))$ replaced by $G(\bar{u}^{R})$ and
$K(\bar{u}^{R})$ respectively.
Thus the rest of the proof follows the proof of Proposition 4.5 with slight
modifications to the estimates found in (4.26) and (4.27) as described below.
First since $\alpha<1$,
$W^{\alpha,\frac{p}{p-1}}(0,T;D(A)^{\prime}+X^{\prime})$ is continuously
embedded in $W^{1,\frac{p}{p-1}}(0,T;D(A)^{\prime}+X^{\prime})$ and we have,
as in (4.24),

(5.9) $\displaystyle\leavevmode\nobreak\
\left\|g^{R}\right\|^{\frac{p}{p-1}}_{W^{\alpha,\frac{p}{p-1}}(0,T;{D(A)^{\prime}+X^{\prime}})}$
$\displaystyle:=\left\|\bar{u}_{0}^{R}-\int_{0}^{\cdot}\gamma
A\bar{u}^{R}(s)\mathrm{d}s-\int_{0}^{\cdot}B_{\widetilde{R}}(\bar{u}^{R}(s))\mathrm{d}s-\int_{0}^{\cdot}F^{R}(\bar{u}^{R}(s))\mathrm{d}s\right\|^{\frac{p}{p-1}}_{W^{\alpha,\frac{p}{p-1}}(0,T;D(A)^{\prime}+X^{\prime})}$
$\displaystyle\lesssim\leavevmode\nobreak\
|\bar{u}^{R}_{0}|^{\frac{p}{p-1}}+\int_{0}^{T}\gamma|A\bar{u}^{R}(s)|^{\frac{p}{p-1}}_{{D(A)^{\prime}}}\mathrm{d}s+\int_{0}^{T}|B_{\widetilde{R}}(\bar{u}^{R}(s))|_{{D(A)^{\prime}}}^{\frac{p}{p-1}}\mathrm{d}s+\int_{0}^{T}|F^{R}(\bar{u}^{R}(s))|^{\frac{p}{p-1}}_{{X^{\prime}}}\mathrm{d}s,$

where the hidden constants depend on $T,p$ but not on $R$.
Recall that the law of $u^{R}_{0}$ is the same as that of $u_{0}$. Thus, using
the fact that $\frac{p}{p-1}<2$, the expected values of the first two terms
above are treated using the Hölder inequality. For example,

$\displaystyle\widetilde{\bf
E}\int_{0}^{T}\gamma|A\bar{u}^{R}(s)|_{D(A)^{\prime}}^{\frac{p}{p-1}}\mathrm{d}s$
$\displaystyle\lesssim\gamma T^{\frac{p+1}{2p}}\left(\widetilde{\bf
E}\int_{0}^{T}|A\bar{u}^{R}(s)|_{V^{\prime}}^{2}\mathrm{d}s\right)^{\frac{p-1}{2p}}$
$\displaystyle\leq C(T,p,\gamma)\left(\widetilde{\bf
E}\int_{0}^{T}\|\bar{u}^{R}(s)\|^{2}\mathrm{d}s\right)^{\frac{p-1}{2p}}$
(5.10) $\displaystyle\leq C(T,p,\gamma,K_{1}),$

where the constant $C(T,p,\gamma,K_{1})>0$ is independent of $R$.
Next, using (2.6) and remembering that $\theta_{\widetilde{R}}(x)$ vanishes
for $x\geq 2\widetilde{R}$, we find:

$\displaystyle\widetilde{\bf
E}\int_{0}^{T}|B_{\widetilde{R}}(\bar{u}^{R}(s),\bar{u}^{R}(s))|_{D(A)^{\prime}}^{\frac{p}{p-1}}\mathrm{d}s$
$\displaystyle\lesssim\widetilde{\bf
E}\int_{0}^{T}\big{(}\theta_{\widetilde{R}}(|\bar{u}^{R}|)\big{)}^{\frac{p}{p-1}}|\bar{u}^{R}(s)|^{{}^{\frac{p}{p-1}}}\|\bar{u}^{R}(s)\|^{\frac{p}{p-1}}\mathrm{d}s$
(5.11)
$\displaystyle\lesssim\widetilde{R}^{\frac{p}{p-1}}T^{\frac{p-2}{2p-2}}\left(\widetilde{\bf
E}\int_{0}^{T}\|\bar{u}^{R}(s)\|^{2}\mathrm{d}s\right)^{\frac{p}{2p-2}}.$

Thanks to Lemma 5.1 the last line in (5.11) is bounded by a constant for
$C=C(p,T,\widetilde{R},K_{1})>0$ independent of $R$.
Next, as proven in Theorem 4.1 in [49], using the Hölder inequality on the
expressions in (4.3)-(4.4) we obtain

$\displaystyle|F^{R}(\bar{u}^{R})|_{L^{\frac{p}{p-1}}(0,T;X^{\prime})}^{\frac{p}{p-1}}$
$\displaystyle\leq\||D^{i}\bar{u}^{R}|_{p}\mathbbm{1}_{\\{|D^{i}\bar{u}^{R}|_{p}\leq
R\\}}\|^{p}_{L^{p}(0,T;L^{p}(\mathcal{O}))}$
$\displaystyle{}+\|R^{\frac{p-2}{2}}|D^{i}\bar{u}^{R}|_{p}\mathbbm{1}_{\\{|D^{i}\bar{u}^{R}|_{p}\geq
R\\}}\|^{{\frac{p}{p-1}}}_{L^{2}(0,T;L^{2}(\mathcal{O}))}\|R^{p}\mathbbm{1}_{\\{|D^{i}\bar{u}^{R}|_{p}\geq
R\\}}\|^{{\frac{p-2}{2(p-1)}}}_{L^{1}(0,T;L^{1}(\mathcal{O}))}.$

Then we take expectation of both sides and apply the Hölder inequality
observing that $\frac{p}{2(p-1)}+\frac{p-2}{2(p-1)}=1$. Using (5.3) and (5.4),
for a constant $C=C(T,p,K_{2})>0$ independent of $R$, we then obtain

(5.12) $\displaystyle\widetilde{\bf
E}\int_{0}^{T}|F^{{R}}(\bar{u}^{R}(s))|_{X^{\prime}}^{\frac{p}{p-1}}\mathrm{d}s\leq
C(T,p,K_{2}).$

Thus we obtain the desired result (5.8). ∎

Next define, (compare to (4.32)-(4.33)),

(5.13) $\displaystyle\mathcal{Z}_{1}$
$\displaystyle=L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime}+X^{\prime}),$
(5.14) $\displaystyle\mu_{u}^{R}(\cdot)$ $\displaystyle=\widetilde{\bf
P}(\bar{u}^{R}\in\cdot)\in
Pr\left(L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime}+X^{\prime})\right).$

######  Proposition 5.4.

Let $\bar{u}^{R}$ be a solution to (4.2) with respect to the basis
$(\widetilde{\Omega},\widetilde{\mathcal{F}},{(\bar{\mathcal{F}}^{R}_{t})_{t\geq
0}},\widetilde{\bf P},{\bar{W}^{R},\bar{\pi}^{R}})$ as given by Theorem 4.13.
Then the laws $\\{\mu_{u}^{R}\\}_{R>R_{0}}$ of $\\{\bar{u}^{R}\\}_{R>R_{0}}$
are tight in the space $\mathcal{Z}_{1}$.

###### Proof.

First we will show that the sequence of laws of $\bar{u}^{R}$ is tight in the
space $\mathcal{D}([0,T];V^{\prime}+X^{\prime})$ by showing that a suitable
version of the Aldous condition (4.31) is satisfied. Observe that
$\bar{u}^{R}$ satisfies:

(5.15) $\displaystyle\bar{u}^{R}(t)=$ $\displaystyle\leavevmode\nobreak\
\bar{u}^{R}_{0}-\int_{0}^{t}\gamma
A\bar{u}^{R}(s)\mathrm{d}s-\int_{0}^{t}B_{\widetilde{R}}(\bar{u}^{R}(s),\bar{u}^{R}(s))\mathrm{d}s-\int_{0}^{t}F^{R}(\bar{u}^{R}(s))\mathrm{d}s$
$\displaystyle\leavevmode\nobreak\
+\int_{0}^{t}G(\bar{u}^{R}(s-))\mathrm{d}{\bar{W}^{R}}(s)+\int_{(0,t]}\int_{E_{0}}K(\bar{u}^{R}(s-),\xi){\mathrm{d}\widehat{\bar{\pi}}^{R}}(s,\xi)$
$\displaystyle=:$ $\displaystyle\leavevmode\nobreak\
\bar{u}^{R}_{0}+\mathcal{M}_{1}(t)+\mathcal{M}_{2}(t)+\mathcal{M}_{3}(t)+\mathcal{M}_{4}(t)+\mathcal{M}_{5}(t).$

Let $(\tau_{R})_{R>R_{0}}$ be a sequence of stopping times where
$0\leq\tau_{R}\leq T$. Thanks to the similarity of the bounds in Lemma 5.1 to
that in Lemma 4.1, the proof that the terms
$\mathcal{M}_{1},\mathcal{M}_{2},\mathcal{M}_{4}$ and $\mathcal{M}_{5}$
satisfy the Aldous condition follows from the proof of Proposition 4.8
identically. And so we obtain, for some $C=C(\widetilde{R},T,\gamma,K_{1})>0$,
independent of $R$

(5.16) $\displaystyle\widetilde{\bf
E}\left\|\mathcal{M}_{i}(\tau_{R}+\delta)-\mathcal{M}_{i}(\tau_{R})\right\|_{V^{\prime}}\leq
C\delta^{\frac{1}{2}}\quad\text{ for }i=1,2,4,5.$

We will discuss the term $\mathcal{M}_{3}$ that is treated slightly
differently here than in the previous section.

Using (5.12) we obtain for some $C(T,p,K_{1})>0$ independent of $R$,

(5.17) $\displaystyle\widetilde{\bf
E}\left[\left\|\mathcal{M}_{3}(\tau_{R}+\delta)-\mathcal{M}_{3}(\tau_{R})\right\|_{X^{\prime}}\right]$
$\displaystyle\leq C\widetilde{\bf
E}\int_{\tau_{R}}^{\tau_{R}+\delta}|F^{R}(\bar{u}^{R}(s))|_{X^{\prime}}\mathrm{d}s$
$\displaystyle\leq C\delta^{\frac{1}{p}}\left(\widetilde{\bf
E}\int_{\tau_{R}}^{\tau_{R}+\delta}|F^{R}(\bar{u}^{R}(s))|^{\frac{p}{p-1}}_{X^{\prime}}\mathrm{d}s\right)^{\frac{p-1}{p}}$
$\displaystyle\leq C\delta^{\frac{1}{p}}.$

This proves that the Aldous condition (4.31) is satisfied by $\bar{u}^{R}$
with $\Xi=V^{\prime}+X^{\prime}$, $\alpha=1$ and $\epsilon=\frac{1}{p}$. This
implies that the laws of $(\bar{u}^{R})_{\\{R>R_{0}\\}}$ are tight in the
space $\mathcal{D}([0,T];V^{\prime}+X^{\prime})$. Thus combining this result
with Proposition 5.3 and proceeding as in Proposition 4.9 we conclude the
proof of Proposition 5.4. ∎

Let $\Upsilon_{1}:=H\times\mathcal{Z}_{1}\times
C([0,T];U)\times\mathcal{N}^{\\#*}_{[0,\infty)\times E}$. Recall that due to
Theorem 4.10 we know that the laws of $\bar{W}^{R}$ (and similarly the laws of
$\bar{\pi}^{R}$) are equal to one element. Thus the tightness result in
Proposition 5.4 together with an application of Prohorov’s theorem gives us
the existence of a measure $\mu^{\infty}_{1}$ on $\Upsilon_{1}$ such that the
sequence $\mu^{R}$, of the joint probability laws of
$({\bar{u}}_{0}^{R},{\bar{u}}^{R},{\bar{W}}^{R},{\bar{\pi}}^{R})$ indexed by
some infinite set $\Lambda\subseteq\mathbb{N}$, converges weakly to
$\mu^{\infty}_{1}$. Next we apply Theorem A.1 to obtain the following strong
convergence result.

######  Theorem 5.5.

For any $R\in\Lambda$ there exist $\Upsilon_{1}$-valued random variables
$(\bar{\bar{u}}_{0}^{R},\bar{\bar{u}}^{R},\bar{\bar{W}}^{R},\bar{\bar{\pi}}^{R})$
and $(\bar{\bar{u}}_{0},\bar{\bar{u}},\bar{\bar{W}},\bar{\bar{\pi}})$,
depending on $\widetilde{R}$, such that

  1. i)

The probability laws of
$(\bar{\bar{u}}_{0}^{R},\bar{\bar{u}}^{R},\bar{\bar{W}}^{R},\bar{\bar{\pi}}^{R})$
and $(\bar{\bar{u}}_{0},\bar{\bar{u}},\bar{\bar{W}},\bar{\bar{\pi}})$ are
$\mu^{R}$ and $\mu_{1}^{\infty}$, respectively,

  2. ii)

$(\bar{\bar{W}}^{R},\bar{\bar{\pi}}^{R})=(\bar{\bar{W}},\bar{\bar{\pi}})$
everywhere on $\widetilde{\Omega}$,

  3. iii)

$(\bar{\bar{u}}_{0}^{R},\bar{\bar{u}}^{R},\bar{\bar{W}}^{R},\bar{\bar{\pi}}^{R})$
converges almost surely to
$(\bar{\bar{u}}_{0},\bar{\bar{u}},\bar{\bar{W}},\bar{\bar{\pi}})$ in the
topology of $\Upsilon_{1}$, i.e.,

(5.18) $\displaystyle{\bar{\bar{u}}_{0}^{R}\rightarrow\bar{\bar{u}}_{0}\text{
in }H,\quad\widetilde{\bf P}\textnormal{-a.s.},}$ (5.19)
$\displaystyle\bar{\bar{u}}^{R}\rightarrow\bar{\bar{u}}\text{ in
}L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime}+X^{\prime}),\quad\widetilde{\bf
P}\textnormal{-a.s.},$

  4. iv)

$\bar{\bar{\pi}}$, is a time homogeneous Poisson random measure over
$(\widetilde{\Omega},\widetilde{{\mathcal{F}}},\widetilde{\bf P})$ with
intensity measure $\mathrm{d}t\otimes\mathrm{d}\nu$.

  5. v)

$\bar{\bar{u}}^{R}(0)=\bar{\bar{u}}_{0}^{R}$ $\widetilde{\bf P}$-a.s. and
$(\bar{\bar{u}}^{R},\bar{\bar{W}},\widehat{\bar{\bar{\pi}}})$ satisfies the
following equation $\widetilde{\bf P}$-a.s. in $V^{\prime}$ for every
$t\in[0,T]$

$\displaystyle\bar{\bar{u}}^{R}(t)+\int_{0}^{t}\left[\gamma
A(\bar{\bar{u}}^{R}(s))+B_{\widetilde{R}}(\bar{\bar{u}}^{R}(s),\bar{\bar{u}}^{R}(s))+F^{R}(\bar{\bar{u}}^{R}(s))\right]\mathrm{d}s$
(5.20)
$\displaystyle\qquad\qquad=\bar{\bar{u}}^{R}_{0}+{\int_{0}^{t}G(\bar{\bar{u}}^{R}(s-))\mathrm{d}\bar{\bar{W}}(s)+\int_{(0,t]}\int_{E_{0}}K(\bar{\bar{u}}^{R}(s-),\xi)\mathrm{d}\widehat{\bar{\bar{\pi}}}(s,\xi)},$

with respect to some filtration $(\bar{\bar{\mathcal{F}}}^{R}_{t})_{t\geq 0}$.

The filtrations $(\bar{\bar{\mathcal{F}}}^{R}_{t})_{t\geq 0}$ and
$(\bar{\bar{\mathcal{F}}}_{t})_{t\geq 0}$ are defined like in (4.45) and
depend on $\widetilde{R}$. We note that $\bar{\bar{u}}_{0}$ might depend on
$\widetilde{R}$ but the only fact relevant to our analysis is that it has the
same law as $u_{0}$.

######  Remark 5.6.

The random variable
$(\bar{\bar{u}}_{0}^{R},\bar{\bar{u}}^{R},\bar{\bar{W}}^{R},\bar{\bar{\pi}}^{R})$
may or may not be the same as the random variable
$({\bar{u}}_{0}^{R},{\bar{u}}^{R},{\bar{W}}^{R},{\bar{\pi}}^{R})$. This solely
depends on the measure $\mu_{1}^{\infty}$ which decides the refined cover of
the space $\Upsilon_{1}$, in the proof of the Skorohod representation theorem
(see Theorem A.1).

In what follows, we will consider $R$ belonging to the denumerable set
$\Lambda$. Comparing equations (4.2) and (v) we can see that
$\bar{\bar{u}}^{R}$ also satisfies the uniform bounds found in equation (5.2)
in Lemma 5.1. Thus we obtain that $\bar{\bar{u}}\in
L^{\infty}([0,T];L^{2}(\widetilde{\Omega},H))\cap
L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};V)$ and there exists some subsequence $R\rightarrow\infty$, such that

(5.21) $\displaystyle\bar{\bar{u}}^{R}$
$\displaystyle\rightharpoonup\bar{\bar{u}}\text{ weakly in
}L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};V),$ (5.22) $\displaystyle\bar{\bar{u}}^{R}$
$\displaystyle\rightharpoonup\bar{\bar{u}}\text{ weak${}^{*}$ in
}L^{\infty}([0,T];L^{2}(\widetilde{\Omega},H)).$

Bounds in (5.12) also give us, for a subsequence,

(5.23) $Y^{R,\widetilde{R}}=F^{R}(\bar{\bar{u}}^{R})\rightharpoonup
Y^{\widetilde{R}}\text{ weakly in
}L^{\frac{p}{p-1}}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};X^{\prime}).$

We denote
$\bar{\bar{\mathscr{S}}}=({\widetilde{\Omega}},\widetilde{\mathcal{F}},(\bar{\bar{\mathcal{F}}}_{t})_{t\geq
0},\widetilde{\bf P},\bar{\bar{W}},\bar{\bar{\pi}})$. Note that our aim now is
to show that $(\bar{\bar{\mathscr{S}}},\bar{\bar{u}})$ is a martingale
solution of (5.1) which includes showing that
$Y^{\widetilde{R}}=F({\bar{\bar{u}}})$. For that purpose we begin by defining
the truncated functions
$\xi^{R}_{i}:\widetilde{\Omega}\times[0,\infty)\times\mathcal{O}\rightarrow\mathbb{R}^{d^{i+1}}$
for $i=0,...,k$ as follows,

(5.24)
$\displaystyle\xi_{i}^{R}(\omega,t,x)=D^{i}\bar{\bar{u}}^{R}\mathbbm{1}_{\\{|D^{i}\bar{\bar{u}}^{R}|_{p}\leq
R\\}}.$

Here we recall the notation
$|\cdot|_{p}\,:=|\cdot|_{l^{p}(\mathbb{R}^{d^{i+1}})}$ given
$x\in\mathbb{R}^{d^{i+1}}$ and $i=0,1,...,k$. Due to the availability of
bounds independent of $R$ for the first left hand side term in (5.3) we can
see that for some subsequence $R\rightarrow\infty$ and some $\eta_{i}$,
$i=0,...,k$, we have that

(5.25) $\displaystyle\xi^{R}_{i}\rightharpoonup\eta_{i}\text{ weakly in
}L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};(L^{p}(\mathcal{O}))^{d^{i+1}}).$

Observe also that (5.4) implies,

(5.26)
$\displaystyle\mathbbm{1}_{\\{|D^{i}\bar{\bar{u}}^{R}(x,t)|_{p}<R\\}}\rightarrow
1\text{ in probability for almost every $t\in[0,T]$ and $x\in\mathcal{O}$ as
}R\rightarrow\infty.$

An application of the Lebesgue dominated convergence theorem and (5.21),
(5.25) and (5.26) tells us that $\xi^{R}_{i}\rightharpoonup
D^{i}\bar{\bar{u}}\text{ weakly in
}L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};(L^{2}(\mathcal{O}))^{d^{i+1}})$. This, due to (5.25), further implies that
in fact $\eta_{i}=D^{i}\bar{\bar{u}}$ for $i=0,...,k$ and thus

(5.27) $\displaystyle\bar{\bar{u}}$ $\displaystyle\in
L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};X)\cap
L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};V),$ (5.28) $\displaystyle\xi^{R}_{i}$ $\displaystyle\rightharpoonup
D^{i}\bar{\bar{u}}\text{ weakly in
}L^{p}([0,T]\times\bar{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};(L^{p}(\mathcal{O}))^{d^{i+1}})\text{ as }R\rightarrow\infty.$

Now we can prove the main result of this section:

######  Theorem 5.7.

Suppose that ${\bf E}|u_{0}|^{2}<\infty$. Then $\bar{\bar{u}}$ (denoted by
$\bar{\bar{u}}^{\widetilde{R}}$ in the later section) solves (5.1), with
respect to the stochastic basis
$({\widetilde{\Omega}},\widetilde{\mathcal{F}},(\bar{\bar{\mathcal{F}}}_{t})_{t\geq
0},\widetilde{\bf P},\bar{\bar{W}},\bar{\bar{\pi}})$ with initial condition
$\bar{\bar{u}}_{0}$ that has the same law as $u_{0}$.

###### Proof.

Here again as for Theorem 4.13 we will pass to the limit $R\rightarrow\infty$
and prove Theorem 5.7 in two steps. We will skip some details very similar to
the proof of Theorem 4.13.
Step 1:
We first show, as done before in (4.49), that $\bar{\bar{u}}$ solves the
following equation $\widetilde{\bf P}$-a.s. and for a.e. $t\in[0,T]$

$\displaystyle\bar{\bar{u}}(t)+\int_{0}^{t}[\gamma
A(\bar{\bar{u}}(s))+B_{\widetilde{R}}(\bar{\bar{u}}(s),\bar{\bar{u}}(s))+Y^{\widetilde{R}}(s)]\mathrm{d}s$
(5.29) $\displaystyle\hskip
108.405pt=\bar{\bar{u}}_{0}+\int_{0}^{t}G(\bar{\bar{u}}(s))\mathrm{d}\bar{\bar{W}}(s)+\int_{(0,t]}\int_{E_{0}}K(\bar{\bar{u}}(s-),\xi)\mathrm{d}\widehat{\bar{\bar{\pi}}}(s,\xi),$

where $Y^{\widetilde{R}}$ was defined in (5.23). The proof of (5) follows that
of (4.49) in Theorem 4.13.
Note also that, thanks to Theorem 4.10 and Theorem 5.5, we have that
$\bar{\bar{u}}_{0}$ has the same law as $u_{0}$.

Step 2:
In this step we will show that

$\displaystyle
Y^{\widetilde{R}}=F(\bar{\bar{u}}),\quad\mathrm{d}\widetilde{\bf
P}\otimes\mathrm{d}t\textnormal{-a.e.}$

For that purpose we prove the following lemma for the truncated functions
defined in (5.24):

######  Lemma 5.8.

The following four convergences hold true for any $\phi\in
L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};X)$, for any $\varphi\in L^{\infty}([0,T],\mathrm{d}t;\mathbb{R})$ such
that $\varphi\geq 0$ and for $\rho>0$ appearing in (2.9)-(2.10),

(5.30a) $\displaystyle\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}\,({f^{R}_{i}(\xi_{i}^{R}(s))},{\xi^{R}(s)})\mathrm{d}s\mathrm{d}t\leq\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(Y^{\widetilde{R}},\bar{\bar{u}}(s))\mathrm{d}s\mathrm{d}t,$ (5.30b)
$\displaystyle\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}\,(f^{R}_{i}(D^{i}\phi(s)),\xi_{i}^{R}(s))\mathrm{d}s\mathrm{d}t=\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(F(\phi(s)),\bar{\bar{u}}(s))\mathrm{d}s\mathrm{d}t,$ (5.30c)
$\displaystyle\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}\,(f^{R}_{i}(\xi_{i}^{R}(s)),D^{i}\phi(s))\mathrm{d}s\mathrm{d}t=\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(Y^{\widetilde{R}},\phi(s))\mathrm{d}s\mathrm{d}t,$ (5.30d)
$\displaystyle\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}\,(f^{R}_{i}(D^{i}\phi(s)),D^{i}\phi(s))\mathrm{d}s\mathrm{d}t=\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(F(\phi(s)),\phi(s))\mathrm{d}s\mathrm{d}t.$

###### Proof.

Here we describe the outline of the proofs of the above four claims. As
mentioned earlier, the reader is referred to [49] for an in depth description
in the deterministic setting.

Proof of claim (5.30a): First we can observe that for some $K_{3}>0$
independent of $R$ we have

$\displaystyle|\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}),\xi_{i}^{R})\mathrm{d}s\mathrm{d}t|\leq
C(T,\|\varphi\|_{L^{\infty}})\widetilde{\bf
E}\sum_{i=0}^{k}\|\xi^{R}_{i}\|^{p}_{L^{p}([0,T]\times\widetilde{\Omega};\,\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};\,(L^{p}(\mathcal{O}))^{d^{i+1}})}\leq K_{3}.$

This implies that, up to a subsequence if necessary, the limit
$\lim_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}),\xi_{i}^{R})\mathrm{d}s\mathrm{d}t$ exists. Next we
can see that for any $R>1$ we have

(5.31) $\displaystyle\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}),\xi_{i}^{R})\mathrm{d}s=\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}),D^{i}\bar{\bar{u}}^{R})\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(D^{i}\bar{\bar{u}}^{R}),D^{i}\bar{\bar{u}}^{R})\mathrm{d}s.$

Summing both sides in $i$ we obtain that,

(5.32) $\displaystyle\sum_{i=0}^{k}\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}(s)),\xi_{i}^{R}(s))\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}e^{-\rho
s}(F^{R}(\bar{\bar{u}}^{R}(s)),\bar{\bar{u}}^{R}(s))\mathrm{d}s.$

Now we form an equation similar to (4.67), then apply Itô’s formula to the
process in (v), take expectation of both sides and finally use (5.31) to
obtain,

$\displaystyle\sum_{i=0}^{k}\widetilde{\bf E}\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}(s)),\xi_{i}^{R}(s))\mathrm{d}s\leq\widetilde{\bf
E}\int_{0}^{t}e^{-\rho
s}(F^{R}(\bar{\bar{u}}^{R}(s)),\bar{\bar{u}}^{R}(s))\mathrm{d}s$
$\displaystyle\qquad\qquad=\widetilde{\bf E}\Big{[}\int_{0}^{t}e^{-\rho
s}\left({\frac{1}{2}\|G(\bar{\bar{u}}^{R}(s))\|^{2}_{L_{2}(U_{0},H)}}\mathrm{d}s-\gamma({A(\bar{\bar{u}}^{R}(s))},{\bar{\bar{u}}^{R}(s)})\mathrm{d}s-\frac{1}{2}{\rho}|{\bar{\bar{u}}^{R}(s)}|^{2}\mathrm{d}s\right)$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{(0,t]}{\int_{E_{0}}e^{-\rho
s}|K(\bar{\bar{u}}^{R}(s),\xi)|^{2}}\mathrm{d}{\bar{\bar{\pi}}}(s,\xi)\Big{]}-\frac{1}{2}\widetilde{\bf
E}\left(e^{-\rho t}|\bar{\bar{u}}^{R}(t)|-|\bar{\bar{u}}^{R}_{0}|^{2}\right).$

Using $|a|^{2}=|a-b|^{2}+2(a,b)-|b|^{2}$ we obtain that the following is valid
for any $0\leq\varphi\in L^{\infty}([0,T],\mathrm{d}t;\mathbb{R})$

$\displaystyle\widetilde{\bf E}\int_{0}^{T}\varphi(t)$
$\displaystyle\int_{0}^{t}e^{-\rho
s}(F^{R}(\bar{\bar{u}}^{R}(s)),\bar{\bar{u}}^{R}(s))\mathrm{d}s\mathrm{d}t$
$\displaystyle=\widetilde{\bf E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}\Big{[}{\frac{1}{2}\|G(\bar{\bar{u}}^{R}(s))-G(\bar{\bar{u}}(s))\|^{2}_{L_{2}(U_{0},H)}}\mathrm{d}s-\frac{1}{2}{\|G(\bar{\bar{u}}(s))\|^{2}_{L_{2}(U_{0},H)}}\mathrm{d}s$
$\displaystyle+\frac{1}{2}{\int_{E_{0}}|K(\bar{\bar{u}}^{R}(s),\xi)-K(\bar{\bar{u}}(s),\xi)|^{2}}\mathrm{d}\nu(\xi)\mathrm{d}s-\frac{1}{2}{\int_{E_{0}}|K(\bar{\bar{u}}(s),\xi)|^{2}}\mathrm{d}\nu(\xi)\mathrm{d}s$
$\displaystyle+(G(\bar{\bar{u}}(s)),G(\bar{\bar{u}}^{R}(s)))_{L_{2}(U_{0},H)}\mathrm{d}s+(K(\bar{\bar{u}}(s),\xi),K(\bar{\bar{u}}^{R}(s),\xi))\mathrm{d}\nu(\xi)\mathrm{d}s$
$\displaystyle+\gamma({A(\bar{\bar{u}}(s))},\bar{\bar{u}}(s))\mathrm{d}s-\gamma({A(\bar{\bar{u}}^{R}(s))-A(\bar{\bar{u}}(s))},{\bar{\bar{u}}^{R}(s)}-\bar{\bar{u}}(s))\mathrm{d}s-\gamma(A(\bar{\bar{u}}^{R}(s)),\bar{\bar{u}}(s))\mathrm{d}s$
$\displaystyle-\gamma(A(\bar{\bar{u}}(s)),\bar{\bar{u}}^{R}(s))\mathrm{d}s-\frac{1}{2}\rho\left(|\bar{\bar{u}}^{R}(s)-\bar{\bar{u}}(s)|^{2}-|\bar{\bar{u}}(s)|^{2}+2(\bar{\bar{u}}^{R}(s),\bar{\bar{u}}(s))\right)\mathrm{d}s\Big{]}\mathrm{d}t$
$\displaystyle-\frac{1}{2}\widetilde{\bf E}\int_{0}^{T}\varphi(t)(e^{-\rho
t}|\bar{\bar{u}}^{R}(t)|-|\bar{\bar{u}}^{R}_{0}|^{2})\mathrm{d}t.$

Using the fact that
$-({A(\bar{\bar{u}}^{R}(s))-A(\bar{\bar{u}}(s))},{\bar{\bar{u}}^{R}(s)}-\bar{\bar{u}}(s))\leq
0$ and applying an argument that follows the equation (LABEL:nk) we obtain the
following as $R\rightarrow\infty$

$\displaystyle{\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}),\xi_{i}^{R})\mathrm{d}s\mathrm{d}t}$
$\displaystyle\qquad\qquad\leq\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}\Big{[}-\frac{1}{2}{\|G(\bar{\bar{u}}(s))\|^{2}_{L_{2}(U_{0},H)}}\mathrm{d}s-\frac{1}{2}{\int_{E_{0}}|K(\bar{\bar{u}}(s),\xi)|^{2}}\mathrm{d}\nu(\xi)\mathrm{d}s$
$\displaystyle\qquad\qquad+(G(\bar{\bar{u}}(s)),G(\bar{\bar{u}}(s)))_{L_{2}(U_{0},H)}\mathrm{d}s+(K(\bar{\bar{u}}(s),\xi),K(\bar{\bar{u}}(s),\xi))\mathrm{d}\nu(\xi)\mathrm{d}s$
$\displaystyle\qquad\qquad-\gamma({A(\bar{\bar{u}}(s))},\bar{\bar{u}}(s))\mathrm{d}s-\frac{1}{2}\rho(-|\bar{\bar{u}}(s)|^{2}+2(\bar{\bar{u}}(s),\bar{\bar{u}}(s)))\mathrm{d}s\Big{]}\mathrm{d}t$
(5.33) $\displaystyle\qquad\qquad-\frac{1}{2}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)(e^{-\rho
t}|\bar{\bar{u}}(t)|-|\bar{\bar{u}}_{0}|^{2})\mathrm{d}t.$

That is,

$\displaystyle{\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}),\xi_{i}^{R})\mathrm{d}s\mathrm{d}t}$
$\displaystyle\leq\widetilde{\bf E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}\Big{[}\frac{1}{2}{\|G(\bar{\bar{u}}(s))\|^{2}_{L_{2}(U_{0},H)}}\mathrm{d}s+\frac{1}{2}{\int_{E_{0}}|K(\bar{\bar{u}}(s),\xi)|^{2}}\mathrm{d}\nu(\xi)\mathrm{d}s$
$\displaystyle-\gamma({A(\bar{\bar{u}}(s))},\bar{\bar{u}}(s))\mathrm{d}s-\frac{1}{2}\rho|\bar{\bar{u}}(s)|^{2}\mathrm{d}s\Big{]}\mathrm{d}t-\frac{1}{2}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)(e^{-\rho
t}|\bar{\bar{u}}(t)|-|\bar{\bar{u}}_{0}(t)|^{2})\mathrm{d}t$ (5.34)
$\displaystyle=\widetilde{\bf E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(Y^{\widetilde{R}},\bar{\bar{u}})\mathrm{d}s\mathrm{d}t.$

The last step follows from similarly applying, to (5), the argument in (4.67)
and the Itô formula using the function $|\bar{\bar{u}}|^{2}$.
Thus, as claimed in (5.30a), we have

(5.35) $\displaystyle
0\leq\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}),\xi_{i}^{R})\mathrm{d}s\mathrm{d}t\leq\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(Y^{\widetilde{R}},\bar{\bar{u}})\mathrm{d}s\mathrm{d}t.$

Proof of claim (5.30b): Going back to the definition of $f^{R}_{i}$ and
$\xi^{R}_{i}$ in (4.4) and (5.24) we see that

$\displaystyle\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}|(f^{R}_{i}(D^{i}\phi),\xi_{i}^{R})-(f_{i}(D^{i}\phi),D^{i}\bar{\bar{u}})|\mathrm{d}s\mathrm{d}t$
$\displaystyle\qquad\leq\sum_{i=0}^{k}T|\varphi|_{L^{\infty}}\widetilde{\bf
E}\int_{0}^{T}|(f^{R}_{i}(D^{i}\phi),\xi_{i}^{R})-(f_{i}(D^{i}\phi),D^{i}\bar{\bar{u}})|\mathrm{d}s$
$\displaystyle\qquad\leq\sum_{i=0}^{k}T|\varphi|_{L^{\infty}}\widetilde{\bf
E}\int_{0}^{T}|\left(f_{i}(D^{i}\phi),\xi_{i}^{R}-D^{i}\bar{\bar{u}}\right)|+|\left(f_{i}^{R}(D^{i}\phi)-f_{i}(D^{i}\phi),\xi_{i}^{R}\right)|\mathrm{d}s$
$\displaystyle\qquad=:I^{b}_{1}+I^{b}_{2}.$

We can see from (2.15)2 that $f_{i}(D^{i}\phi)\in
L^{p^{\prime}}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};(L^{p^{\prime}}(\mathcal{O}))^{d^{i+1}})$ since $\phi\in
L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};X)$. And thus $I^{b}_{1}\rightarrow 0$ as $R\rightarrow\infty$ thanks to
(5.28).
Next using (2.15) we observe that

$\displaystyle I^{b}_{2}$ $\displaystyle\leq
T|\varphi|_{L^{\infty}}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}|D^{i}\phi|_{p}^{p-1}|\xi_{i}^{R}|_{p}\mathbbm{1}_{\\{|D^{i}\phi|_{p}>R\\}}\mathrm{d}x\mathrm{d}s.$
Since $|\xi^{R}_{i}|_{p}<R$ by definition, we obtain the following bounds on
the sets ${\\{|D^{i}\phi|_{p}>R\\}}$,
$\displaystyle\lim_{R\rightarrow\infty}I^{b}_{2}$ $\displaystyle\leq
T|\varphi|_{L^{\infty}}\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}|D^{i}\phi|_{p}^{p}\mathbbm{1}_{\\{|D^{i}\phi|_{p}>R\\}}\mathrm{d}x\mathrm{d}s.$

Observe that $|D^{i}\phi|_{p}^{p}\mathbbm{1}_{\\{|D^{i}\phi|_{p}>R\\}}$ is
dominated by $|D^{i}\phi|_{p}^{p}\,\in
L^{1}(\widetilde{\Omega}\times[0,T]\times\mathcal{O})$ and
$\mathbbm{1}_{\\{|D^{i}\phi|_{p}>R\\}}\rightarrow 0$ a.e. on
$\widetilde{\Omega}\times[0,T]\times\mathcal{O}$. Therefore, using the
Lebesgue dominated convergence theorem we obtain $I^{b}_{2}\rightarrow 0$.
Hence we have,

(5.36) $\displaystyle\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}|(f^{R}_{i}(D^{i}\phi),\xi_{i}^{R})-(f_{i}(D^{i}\phi),D^{i}\bar{\bar{u}})|\mathrm{d}s\mathrm{d}t=0.$

Recall from (2.16) that since $\phi,\bar{\bar{u}}\in
L^{{p}{}}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};X)$ we have
$(F(\phi),\bar{\bar{u}})=\sum_{i=0}^{k}(f_{i}(D^{i}\phi),D^{i}\bar{\bar{u}})$
and thus (5.30b) follows.
Proof of claim (5.30c): We write

(5.37) $\displaystyle\sum_{i=0}^{k}(f^{R}_{i}(\xi_{i}^{R}),D^{i}\phi)$
$\displaystyle=(F^{R}(\bar{\bar{u}}^{R}),\phi)-I^{c}_{1}(s)$

where $I^{c}_{1}(s)$ is given by,

$\displaystyle\sum_{i=0}^{k}\int_{\mathcal{O}}\Big{(}f_{i}\left({\frac{RD^{i}\bar{\bar{u}}^{R}}{|D^{i}\bar{\bar{u}}^{R}|_{p}}}\right)\cdot
D^{i}\phi+\left(D^{i}\bar{\bar{u}}^{R}-{\frac{RD^{i}\bar{\bar{u}}^{R}}{|D^{i}\bar{\bar{u}}^{R}|_{p}}}\right)D^{2}\mathscr{J}_{i}\left({\frac{RD^{i}\bar{\bar{u}}^{R}}{|D^{i}\bar{\bar{u}}^{R}|_{p}}}\right)\cdot
D^{i}\phi\Big{)}\mathbbm{1}_{\\{|D^{i}\bar{\bar{u}}^{R}|_{p}>R\\}}\mathrm{d}x.$

Using the Hölder inequality and (2.15) we obtain

$\displaystyle\widetilde{\bf E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}|I^{c}_{1}(s)|\mathrm{d}s\mathrm{d}t\leq$ $\displaystyle
T|\varphi|_{L^{\infty}}\sum_{i=0}^{k}R^{\frac{p-2}{2}}\Bigg{(}\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}R^{p-2}|D^{i}\bar{\bar{u}}^{R}|_{p}^{2}\mathbbm{1}_{\\{|D^{i}\bar{\bar{u}}^{R}|_{p}>R\\}}\mathrm{d}x\mathrm{d}s\Bigg{)}^{\frac{1}{2}}$
$\displaystyle\Bigg{(}\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}|D^{i}\phi|_{p}^{p}\mathbbm{1}_{\\{|D^{i}\bar{\bar{u}}^{R}|_{p}>R\\}}\mathrm{d}x\mathrm{d}s\Bigg{)}^{\frac{1}{p}}\Bigg{(}\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}\mathbbm{1}_{\\{|D^{i}\bar{\bar{u}}^{R}|_{p}>R\\}}\mathrm{d}x\mathrm{d}s\Bigg{)}^{\frac{p-2}{2p}}.$

Using (5.2), (5.3) and (5.4) we obtain, for some constant $C=C(p,K_{2})>0$,
that

$\displaystyle\widetilde{\bf E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}|I^{c}_{1}(s)|\mathrm{d}s\mathrm{d}t\leq
C|\varphi|_{\infty}T\sum_{i=0}^{k}\Bigg{(}\widetilde{\bf
E}\int_{0}^{T}\int_{\mathcal{O}}|D^{i}\phi|_{p}^{p}\mathbbm{1}_{\\{|D^{i}\bar{\bar{u}}^{R}|_{p}>R\\}}\mathrm{d}x\mathrm{d}s\Bigg{)}^{\frac{1}{p}}.$

Observe
that$|D^{i}\phi|_{p}^{p}\mathbbm{1}_{\\{|D^{i}\bar{\bar{u}}^{R}|_{p}>R\\}}$ is
dominated by $|D^{i}\phi|_{p}^{p}\in
L^{1}(\mathcal{O}\times\widetilde{\Omega}\times[0,T])$. Thanks to (5.26) we
can apply the Lebesgue dominated convergence theorem and, thinning the
sequence if required, we obtain that

(5.38) $\displaystyle\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}|I^{c}_{1}(s)|\mathrm{d}s\mathrm{d}t\rightarrow 0\text{ as
}R\rightarrow\infty.$

Hence, as claimed, from (5.37) and (5.38) we obtain

(5.39) $\displaystyle\limsup_{R\rightarrow\infty}\sum_{i=0}^{k}\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(f^{R}_{i}(\xi_{i}^{R}),D^{i}\phi)\mathrm{d}s\mathrm{d}t=\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\int_{0}^{t}e^{-\rho
s}(Y^{\widetilde{R}},\phi)\mathrm{d}s\mathrm{d}t.$

The proof of (5.30d) follows closely the proof of (5.30b) and so we omit the
details. ∎

We now observe that because of the monotonicity of $f^{R}$, we have for any
$R>1$, any $\phi\in
L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};X)$ and any nonnegative funtion $\varphi\in
L^{\infty}([0,T],\mathrm{d}t;\mathbb{R})$,

(5.40) $\sum_{i=0}^{k}\widetilde{\bf
E}\left(\int_{0}^{T}\varphi(t)\left(\int_{0}^{t}e^{-\rho
s}({f}^{R}_{i}(\xi_{i}^{R})-{f^{R}_{i}}(D^{i}\phi),\xi_{i}^{R}-D^{i}\phi)\mathrm{d}s\right)\mathrm{d}t\right)\geq
0.$

With Lemma 5.8 in hand, we take $\limsup_{R\rightarrow\infty}$ on both sides
of (5.40) and apply (5.30a)-(5.30d). We find that for all $\phi\in
L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};X)$ and nonnegative $\varphi\in L^{\infty}([0,T],\mathrm{d}t;\mathbb{R})$,

$\displaystyle\widetilde{\bf
E}\left(\int_{0}^{T}\varphi(t)\left(\int_{0}^{t}e^{-\rho
s}(Y^{\widetilde{R}}(s)-{F}(\phi(s)),\bar{\bar{u}}(s)-\phi(s))\mathrm{d}s\right)\mathrm{d}t\right)\geq
0.$

We use the Minty method once again and choose
$\phi=\bar{\bar{u}}\pm\lambda\zeta{\bf v}$ with $\lambda>0$, ${\bf v}\in X$
and $\zeta\in
L^{\infty}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{\bf
P};\mathbb{R})$. We then divide both sides by $\lambda$ and let
$\lambda\rightarrow 0$. We obtain, thanks to the hemicontinuity of the
operator $F$

(5.41) $\widetilde{\bf
E}\left(\int_{0}^{T}\varphi(t)\left(\int_{0}^{t}\zeta(s)(Y^{\widetilde{R}}(s)-F(\bar{\bar{u}}(s)),{\bf
v})\mathrm{d}s\right)\mathrm{d}t\right)=0.$

Since $\varphi$, $\zeta$ and ${\bf v}$ are arbitrary, we conclude that

(5.42) $Y^{\widetilde{R}}=F(\bar{\bar{u}})\quad\mathrm{d}\widetilde{\bf
P}\otimes\mathrm{d}t\textnormal{-a.e.}$

Thus we have proven that $\bar{\bar{u}}$ solves (5.1) with respect to the
basis
$({\widetilde{\Omega}},\widetilde{\mathcal{F}},(\bar{\bar{\mathcal{F}}}_{t})_{t\geq
0},\widetilde{\bf P},\bar{\bar{W}},\bar{\bar{\pi}})$ and this concludes the
passage to the limit $R\rightarrow\infty$ (with $\widetilde{R}$ fixed).
Theorem 5.7 is thus proven.∎

##  6\. Passage to the limit $\widetilde{R}\rightarrow\infty$

In this section we pass to the limit $\widetilde{R}\rightarrow\infty$ and
finally show the existence of a martingale solution to (3.1)-(3.2). Except for
the term $B$, many of the steps will be similar to the passages to the limit
$n\rightarrow\infty$, $R\rightarrow\infty$ and we will skip the details.
From the previous section we know that $\bar{\bar{u}}$, now denoted by
$\bar{\bar{u}}^{\widetilde{R}}$ to emphasize its dependence on
$\widetilde{R}$, solves (5.1) with respect to the basis
$({\widetilde{\Omega}},\widetilde{\mathcal{F}},{(\bar{\bar{\mathcal{F}}}^{\widetilde{R}}_{t})_{t\geq
0}},\widetilde{\bf
P},{\bar{\bar{W}}^{\widetilde{R}},{\bar{\bar{\pi}}}^{\widetilde{R}}})$.
We can see that there exists a constant $C(\gamma,\rho,c_{1},c_{2})>0$,
independent of $\widetilde{R}$, such that

(6.1) $\displaystyle\widetilde{\bf
E}\sup_{s\in[0,T]}|\bar{\bar{u}}^{\widetilde{R}}(s)|^{2}+\gamma\widetilde{\bf
E}\int_{0}^{T}\|{\bar{\bar{u}}^{\widetilde{R}}(s)}\|^{2}\mathrm{d}s+\widetilde{\bf
E}\int_{0}^{T}\|{\bar{\bar{u}}^{\widetilde{R}}(s)}\|_{X}^{p}\mathrm{d}s\leq
C{{\bf E}}|{u}_{0}|^{2}=:K_{3}.$

We will omit the details of the proof of the above energy estimate since it is
derived exactly as in Lemma 4.1 using (2.17) instead of (4.7). Next we will
prove the following bounds in fractional Sobolev spaces.

######  Proposition 6.1.

Let $\bar{\bar{u}}^{\widetilde{R}}$ solve the SPDE (5.1) with respect to the
basis
$({\widetilde{\Omega}},\widetilde{\mathcal{F}},{(\bar{\bar{\mathcal{F}}}^{\widetilde{R}}_{t})_{t\geq
0}},\widetilde{\bf
P},{\bar{\bar{W}}^{\widetilde{R}},{\bar{\bar{\pi}}}^{\widetilde{R}}})$. Since
$p>2$ we can choose $1<m<\min\\{\frac{p}{p-1},\frac{4p}{2+3p}\\}<\frac{4}{3}$.
Then for every $\alpha\in(0,\frac{1}{2})$ there exists a constant
$C=C(\alpha,\gamma,\rho,T,m,p)>0$, independent of $\widetilde{R}$, such that

(6.2) $\widetilde{\bf
E}\|{{\bar{\bar{u}}}^{\widetilde{R}}}\|^{m}_{W^{\alpha,m}(0,T;V^{\prime}+X^{\prime})}\leq
C.$

Furthermore, due to Lemma 4.3 and an argument like in Proposition 4.6, the
estimate (6.2) implies that the laws of $\bar{\bar{u}}^{\widetilde{R}}$ are
tight in $L^{2}(0,T;H)$.

###### Proof.

The proof follows that of Proposition 5.3 and Proposition 4.5 with slight
modifications to the estimates (4.26) and (4.27) as described below. First for
$\alpha<1$ we have, as for (4.24), (5.9):

$\displaystyle\leavevmode\nobreak\
\|{g^{\widetilde{R}}}\|^{m}_{W^{\alpha,m}(0,T;{V^{\prime}+X^{\prime}})}$
$\displaystyle:=\|{\bar{\bar{u}}}_{0}^{\widetilde{R}}-\int_{0}^{\cdot}\left(\gamma
A{{\bar{\bar{u}}}^{\widetilde{R}}}(s)+B_{\widetilde{R}}({{\bar{\bar{u}}}^{\widetilde{R}}}(s))+F({{\bar{\bar{u}}}^{\widetilde{R}}}(s))\right)\mathrm{d}s\|^{m}_{W^{\alpha,m}(0,T;V^{\prime}+X^{\prime})}$
$\displaystyle\lesssim$ $\displaystyle\leavevmode\nobreak\
|{{\bar{\bar{u}}}_{0}^{\widetilde{R}}}|^{m}+\int_{0}^{T}\gamma|A{{\bar{\bar{u}}}^{\widetilde{R}}}(s)|^{m}_{{V^{\prime}}}\mathrm{d}s+\int_{0}^{T}|B_{\widetilde{R}}({{\bar{\bar{u}}}^{\widetilde{R}}}(s))|_{{V^{\prime}}}^{m}\mathrm{d}s+\int_{0}^{T}|F({{\bar{\bar{u}}}^{\widetilde{R}}}(s))|^{m}_{{X^{\prime}}}\mathrm{d}s,$

where the hidden constants depend on $T,p,m$ but not on $\widetilde{R}$.
Observe that since $m<2$, the expected values of the first two terms above are
treated like in (5.10). Next, using (2.7) and observing that
$\frac{m}{2p}+\frac{3m}{4}+(1-\frac{m(2+3p)}{4p})=1$ we apply the Hölder
inequality with these exponents to obtain the following estimates in place of
(4.26),

$\displaystyle\widetilde{\bf
E}\int_{0}^{T}|B_{\widetilde{R}}({\bar{\bar{u}}^{\widetilde{R}}(s)})|_{V^{\prime}}^{m}\mathrm{d}s$
$\displaystyle\leq\widetilde{\bf
E}\int_{0}^{T}|{\bar{\bar{u}}^{\widetilde{R}}(s)}|^{\frac{m}{2}}\|{\bar{\bar{u}}^{\widetilde{R}}(s)}\|^{\frac{3m}{2}}\mathrm{d}s$
$\displaystyle\lesssim T^{(1-\frac{m(2+3p)}{4p})}\left(\widetilde{\bf
E}\int_{0}^{T}|{\bar{\bar{u}}^{\widetilde{R}}(s)}|^{p}\mathrm{d}s\right)^{\frac{m}{2p}}\left(\widetilde{\bf
E}\int_{0}^{T}\|{\bar{\bar{u}}^{\widetilde{R}}(s)}\|^{2}\mathrm{d}s\right)^{\frac{3m}{4}}.$
Since the embedding $X\subset H$ is continuous we can, thanks to (6.1), bound
this, independently of $\widetilde{R}$ by $\displaystyle
T^{(1-\frac{m(2+3p)}{4p})}\left(\widetilde{\bf
E}\int_{0}^{T}\|{\bar{\bar{u}}^{\widetilde{R}}(s)}\|_{X}^{p}\mathrm{d}s\right)^{\frac{m}{2p}}\left(\widetilde{\bf
E}\int_{0}^{T}\|{\bar{\bar{u}}^{\widetilde{R}}(s)}\|^{2}\mathrm{d}s\right)^{\frac{3m}{4}}$
(6.3) $\displaystyle\leq C(T,m,p,K_{3}).$

Using the condition (2.18) and the fact that $m<\frac{p}{p-1}$, we obtain the
following bounds where $C=C(T,m,p,k_{3})>0$ is independent of $\widetilde{R}$

$\displaystyle\widetilde{\bf
E}\int_{0}^{T}|F(\bar{\bar{u}}^{\widetilde{R}}(s))|^{m}_{X^{\prime}}\mathrm{d}s$
$\displaystyle\leq T^{(1-{\frac{m(p-1)}{p}})}\left(\widetilde{\bf
E}\int_{0}^{T}|F(\bar{\bar{u}}^{\widetilde{R}}(s))|^{\frac{p}{p-1}}_{X^{\prime}}\mathrm{d}s\right)^{\frac{m(p-1)}{p}}$
$\displaystyle\lesssim C(T,m,p)\left(\widetilde{\bf
E}\int_{0}^{T}\|{{\bar{\bar{u}}^{\widetilde{R}}(s)}}\|^{p}_{X}\mathrm{d}s\right)^{\frac{m(p-1)}{p}}$
(6.4) $\displaystyle\leq C.$

Now observe that Proposition 4.4 holds for $\beta=m$ and with
$P_{n}(G(u^{n}))$, $P_{n}(K(u^{n}))$ replaced by
$G(\bar{\bar{u}}^{\widetilde{R}})$, $K(\bar{\bar{u}}^{\widetilde{R}})$
respectively. Combining with (6.3) and (6.4) we obtain (6.2). ∎

Next define,

(6.5) $\displaystyle\mu_{u}^{\widetilde{R}}(\cdot)$
$\displaystyle=\widetilde{{\bf P}}(\bar{\bar{u}}^{\widetilde{R}}\in\cdot)\in
Pr\left(L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime}+X^{\prime})\right),$

where $Pr(\mathcal{S})$ denotes, as before, the set of probability measures on
a metric space $\mathcal{S}$.

######  Proposition 6.2.

Assume that $p>2$ and let $\bar{\bar{u}}^{\widetilde{R}}$ solve the SPDE (5.1)
with respect to the basis
$({\widetilde{\Omega}},\widetilde{\mathcal{F}},{(\bar{\bar{\mathcal{F}}}^{\widetilde{R}}_{t})_{t\geq
0}},\widetilde{\bf
P},{\bar{\bar{W}}^{\widetilde{R}},{\bar{\bar{\pi}}}^{\widetilde{R}}})$. Then
the laws $\\{\mu_{u}^{\widetilde{R}}\\}_{\widetilde{R}>1}$ of
$\\{\bar{\bar{u}}^{\widetilde{R}}\\}_{\widetilde{R}>1}$ are tight in the space
$\mathcal{Z}_{1}$, defined in (5.13).

###### Proof.

A part of the proof is to show that a suitable Aldous condition (4.31) is
satisfied by $\bar{\bar{u}}^{\widetilde{R}}$.
With that in mind, let $(\tau_{\widetilde{R}})_{\widetilde{R}>1}$ be a
sequence of stopping times such that $0\leq\tau_{\widetilde{R}}\leq T$. We
define the terms $\mathcal{M}_{1,2,3,4,5}$ as in (4.34) in Proposition 4.8 and
(5.15) in Proposition 5.4. Thanks to the estimates (6.1), the terms
$\mathcal{M}_{1,2,4,5}$ are treated identically as in the proof of Proposition
4.8. We emphasize here that the bounds found in (4.36) are independent of