Stochastic Chemotaxis-Stokes system]A Meta Theorem for nonlinear stochastic coupled systems: Application
to stochastic  chemotaxis-Stokes  porous media
E. Hausenblas]Erika Hausenblas
Montanuniversität Leoben

Department Mathematik und Informationstechnologie

Franz Josef Straße 18

8700 Leoben

B.  Jidjou Moghomye]Boris Jidjou Moghomye
Montanuniversität Leoben

Department Mathematik und Informationstechnologie

Franz Josef Straße 18

8700 Leoben

The purpose of the paper is twofold.
Firstly, we want to present a Meta Theorem to show the existence of a martingale solution for coupled systems of non-linear stochastic differential equations.
The idea is first to split the system by rewriting the non-linear part in a linear part acting on a given process $\xi$. This is done in such a way that the fixpoint with respect to $\xi$ would be the solution. However, to show the well posedness of the linearized system, one needs a cut-off argument. Under which conditions one can handle  the limits of the cut-off parameter to get in the end a martingale solution of the original system is given in the Meta-Theorem.
This approached was used in [14] to show the existence of a martingale solution of a chemical activator-inhibitor

Secondly, we want to verify the full applicability of the Meta Theorem by  showing the existence of a martingale solution of a highly nonlinear chemotaxis system with underlying fluid dynamic. In particular, we investigate
the coupled system

\begin{align*}
\label{aa}
\begin{cases}
d {n} +\bu\cdot \nabla ndt= \lk[ r_n\Delta   |n|^{q-1}n- \chi \Div( n\nabla c) \rk]\, dt+g_{\gamma_1}(n) \circ dW_1(t),
\\
d{c}+\bu\cdot\nabla cdt  =\lk[r_c \Delta c  -\zeta c+ \beta n \rk]\, dt+g_{\gamma_2}(c) \circ dW_2(t),
\\
d \bu +\nabla Pdt=\left[r_\bu\Delta \bu
+n\star\Phi \right]dt+ \sigma_{\gamma_3}\, dW_3(t), %\mbox{and Navier Stokes (Dirchlet boundary condition}
\\
\nabla \cdot \bu =0,
\end{cases}
\end{align*}

with initial condition $(n_0,c_0,\bu_0)$ on a filtered probability space  $\mathfrak{A}$ and $W_1$, $W_2$, and $W_3$ be three  independent time homogeneous spatial Wiener processes over $\mathfrak{A}$. Here $n$ is the cell density,  $c$ is the concentration of the chemical signal, and $\bu$ is a vector field over $\mathcal{O}$.
The positive  terms $r_n$ and $r_c$  are the diffusivity of the  cells and chemoattractant, respectively,
the positive value
$\chi$ is the chemotactic sensitivity.

[2010]Primary 35K57, 60H15; Secondary 37N25, 47H10, 76S05, 92C15

Boris Jidjou Moghomye was supported by the Austrian Science Foundation (FWF), Project P 32295,

Moreover we would like to thank Paul Razafamandimby, City University Dublin, Ireland, for his inspiring discussions and helpful remarks.

§ INTRODUCTION

Cross-diffusion systems are a type of coupled partial differential equations system that arises in many applications involving multi-species transport. In these systems, the diffusion coefficients of different species are coupled. In particular, the diffusion of one variable may depend on the gradients of another variable. Cross-diffusion models have been used to study a wide range of phenomena, including pattern formation in biology,  see [10],
and chemistry, phase separation in materials science, and flow in porous media.
Sometimes, these systems are even coupled with the Navier-Stokes describing an underlying fluid.
Cross diffusion coupled with fluid dynamics
arises e.g. in fluid dynamics of mixtures, electro-chemistry, cell biology, and biofilm modelling.
For more details, we refer to  [17, 19, 21].

Here are a few references that provide more information on cross-diffusion systems:

Murray, J. D. Mathematical Biology I: An Introduction, 3rd edition. Springer, 2002.

Gurtin, M. E.,     MacCamy, R. C. On the diffusion of biological populations. Mathematical Biosciences, 33(1-2), 35-49, 1977.

Chen, X.,     Liu, C. Global existence and uniqueness of solutions to a cross-diffusion system. Archive for Rational Mechanics and Analysis, 147(4), 299-340, 1999.

Garcke, H.,     Nestler, B. Nonlinear PDE models in image processing and computer vision. Springer, 2014.

Mimura, M.,     Yamaguti, M. Singular limit of a nonlinear parabolic equation for a reversible reaction system. Journal of Differential Equations, 61(3), 336-362, 1986.

Coupled nonlinear systems of partial differential equations, like reaction-diffusion systems, with e.g. cross-diffusion systems, are fundamental modeling tools for
mathematical biology with applications to ecology, population dynamics, pattern formation,
enzymatic reactions and chemotaxis, see [17, 19, 21].
These systems of equations arise in many application areas, such as fluid dynamics of mixtures, electrochemistry, cell biology, and biofilm modeling.
Systems with cross-diffusion occur if the gradient in the concentration of one species induces a flux of another species.
However, often these systems are not monotone or do not satisfy the maximum principle such that
standard techniques do not work, and one has to search for more sophisticated methods, especially if
a stochastic term is involved.
Here, often, one shows first the existence of a local solution. This solution is globalised using Galerkin method or another approximate scheme and showing that the
approximate solution is uniformly bounded by an entropy functional.
This has the disadvantage, that for each equation one has to design an approximate scheme, such that the entropy functional is preserved.
Our methods has the advantage that it does not involve an approximating scheme.
In this work, we first present a  Meta Theorem by which nonlinear stochastic strongly coupled
parabolic systems can be solved. To demonstrate its applicability, we apply the Meta theorem to a chemotaxis system in porous media coupled with the Navier Stokes and show the existence of a martingale solution.

There are several recent works tackling systems of stochastic partial differential equations being coupled by nonlinearity and/or
cross-diffusion. Here, let us mention the work of Dahirwal et al. [7], where the authors study cross-diffusion systems.
In addition, the first named author investigated in cooperation with different systems appearing in physical and biological applications, see [5, 12, 13].

The rest of the paper is organized as follow. In Section 2, we state and prove a meta theorem for an existence result of  nonlinear coupled systems and in Section 3, we prove the existence of a martingale solution to a stochastic porous media chemotaxis-fluid by verifying that the hypothesis of meta theorem obtained in section 2 are satisfied.

§ THE MAIN ABSTRACT RESULT

Let $\Ocal\subset \R^d$ be an open domain, $d\ge 1$,  $J\in \mathbb{N}$ and
$ E,E_1,...,E_\Jott\subset\Dcal^{\prime}(\Ocal)$ be the Hilbert spaces and $ U,U_1,...,U_J\subset\Dcal^{\prime}(\Ocal)$ be the Banach spaces[Here, $\Dcal^\prime(\Ocal)$ denotes the space of Schwartz distributions on $\Ocal$, that is, the topological dual space of smooth functions with compact support $\Dcal(\Ocal)=C_0^\infty(\Ocal)$.], such that

$$  E\hookrightarrow U,\text{ and } E_\jott\hookrightarrow U_j,\qquad j=1,...,J.$$

Let $\mathbb{X}\subset\{ \eta:[0,T]\to U \}$ be a Banach function space.

Let $\Afrak=(\Omega,\Fcal,\mathbb{F},\P)$ be a filtered probability space
with filtration $\mathbb{F}=(\Fcal_{t})_{t\in [0,T]}$ satisfying the usual conditions. Let $H$ be a separable Hilbert space and let $(W_j(t))_{t\in [0,T]}$, $j=0,...,J$ be a cylindrical Wiener process

in $H$,
such that $W_j$ has the representation

W(t)=\sum_{i\in\mathbb{I}}  \psi_i\beta^j_i(t),\quad t\in [0,T],

where $\{\psi_i:i\in \mathbb{I}\}$ is a complete orthonormal system in $H$, $\mathbb{I}$ a suitably chosen countable index set, and $\{\beta^j_i:i\in\mathbb{I}\}$ a family of independent real-valued standard Brownian motions on $[0,T]$ modeled in $\Afrak=(\Omega,\Fcal,\mathbb{F},\P)$. Due to <cit.>, this representation does not pose a restriction.

Here, we are interested in   a solution  $(w,v^1,...,v^J)$ of the following system of  Itô stochastic partial differential equations (SPDEs)

{ dw(t)    =   (_0 w(t)+ F_0(w(t),v^1(t),⋯,v^(t)))  dt +Σ_0(w(t)) dW_0(t),

dv^(t)    =   (_v^(t)+ F_(w(t),v^(t),v^-1(t),⋯,v^(t)))  dt +Σ_(v^(t)) dW_(t),

w(0)   =   w_0,

v^(0)   =   v^_0,  =1,…,,

where the equalities hold respectively in $U_j$, $j=0,1,...,J$.

We shall also assume that $\DeltaA_0:D(\DeltaA_0)\subset E_0\to E_0$ is a possibly nonlinear and measurable (single-valued) operator,
$\DeltaA_\jott:D(\DeltaA_\jott)\subset E_\jott\to E_\jott$, $\jott=1,\ldots,\Jott$, are  linear  (single-valued) operators. The mappings $\Sigma_\jott$, $\jott=0,\ldots,\Jott$, are densely defined
mappings from $E_\jott$ into $L_{HS}(H;E_\jott)$ specified later on[For a separable Hilbert space $H$ and a any Hilbert space $E_j$ defined $L_{HS}(H;E_j)$ the Hilbert-Schmidt operators from $H$ to $E_j$.] and

$$F_0:E\times E_1 \times \cdots \times E_\Jott\longrightarrow U, \ \text{ and } \ F_\jott:E\times E_j \times \cdots \times E_\Jott\longrightarrow U_j,\quad j=1,...J,$$
are nonlinear mapping. 	Later on we will show in our examples that cross diffusions systems or reaction-diffusion equations will be covered.

For clarity, we define the notion of martingale solution to the system (<ref>).

Fix $T>0$ and initial conditions $\mathfrak{w}_0=(w_0,v^1_0,\ldots,v^\Jott _0)$.	A martingale solution or weak solution to the problem
is given by the tuple

\left(\Omega ,{{\mathcal{F}}},\mathbb{P},{\mathbb{F}},
\mathbb{W} %=(W_0,W_1,\ldots, W_\Jott )
, \mathfrak{w}%=(w,v^1,\ldots,v^\Jott )
\right)

such that

* $\mathfrak{A}:=(\Omega ,{{\mathcal{F}}},{\mathbb{F}},\mathbb{P})$ is a complete filtered
probability space with a filtration ${\mathbb{F}}=\{{{\mathcal{F}}}_t:t\in [0,T]\}$ satisfying the usual conditions,

* the vector-valued Wiener process
$\mathbb{W}$ is given by $\mathbb{W} =(W_0,W_1,\ldots, W_\Jott )$, where $W_0,W_1,$ $\ldots, W_\Jott $  are mutually independent  cylindrical Wiener processes on $H$ over the probability space
$\mathfrak{A}$, and

* the tuple $\mathfrak{w}=(w,v^1,\ldots,v^\Jott )$ is such that  $w:[0,T] \times \Omega \rightarrow U$, and $v^\jott:[0,T] \times \Omega \rightarrow
U_\jott$, $\jott=1,\ldots,\Jott $
are $\BF$–progressively  measurable processes such that $\PP$-a.s.,
$$(w,v^1,\ldots,v^\Jott )\in C([0,T];U)\times C([0,T];U_1)\times\cdot\cdot\cdot\times C([0,T];U_J), $$

and satisfy  for all $t\in [0,T]$ the following  stochastic integral equations $\PP$–a.s.

\begin{equation}\label{eq:nStratonovich}
w(t)=w_0+\int_0^t (A_0w(s)+F(w(s),v^1(s),\ldots,v^\Jott (s)\,  )\,ds+\int_0^t \Sigma_0 % g_{\gamma_1}
(w(s)) dW_1(s),\ \text{in } U,
\end{equation}

and for  $\jott =1,\ldots, \Jott $,

\begin{equation}\label{eq:nStratonovich2}
v^\jott(t)=v^\jott_0+\int_0^t (A_\jott  v^\jott(s)+F_\jott (w(s),v^\jott(s),v^{\jott+1}(s),\ldots,v^\Jott (s)\,  )\,ds+\int_0^t \Sigma_j %g_{\gamma_\jott }
(v^\jott(s)) dW_\jott (s),\ \text{in } U_j.
\end{equation}

The main objective of this section is to prove an existence result for a solution of the abstract stochastic coupled system (<ref>) in the sense of Definition <ref>. A way to find a solution to this nonlinear system  is using a Schauder fix point argument. In this way,  for a certain fixed $m\ge1$, we define the collection of processes

\begin{equation}\label{eq:MMdef}
\begin{aligned}  \Mcal_{\Afrak}^{m}(\X)
:= & \Big\{ \xi:\Omega\times[0,T]\times \CO\to \mathbb{R}\;\colon\;\\
&\qquad\text{\ensuremath{\xi} is \ensuremath{\mathbb{F}}-progressively measurable}\;\text{and}\;\Eb|\xi|_{\X}^{m}<\infty\Big\}
\end{aligned}
\end{equation}

equipped with the norm

$$\X:=L^m(0,T;X)\subset\{ \eta: [0,T]\to U \}$$

with $X$ be a given UMD Banach space.

In order to overcome some difficulties caused by the non linearities of $F_j$, we first  introduce a family of the cut-off functions. Let $\phi \in C^{\infty}(\RR)$
be a smooth cut-off function that satisfies

\phi(x) \bcase =0, &\mbox{ if } |x|\ge 2,
\\ \in [0,1], &\mbox{ if } 1<|x|<2,
\\=1, &\mbox{ if } |x|\le 1,
\ecase

and let

\begin{equation}
\phi_\kk(x):= \phi(x/\kk), \ x\in\RR,\,\kk \in \mathbb{N}. \label{2.5}
\end{equation}

For  $\jott=1,\ldots, \Jott $, let  $\YY_\jott$ be a function space, e.g. $\YY_\jott=L^{m_\jott}(0,T;Y_\jott)$   or $\YY _\jott=C(0,T;Y_\jott)$,  with $1\leq m_j\leq \infty$ and  $Y_\jott$ being some given  UMD Banach spaces of type $2$. Let $h_\jott  :\YY _\jott\to\RR_0^+$ be given by
$$h_\jott (\eta_j,t)=\|\eta_j \mathds{1}_{[0,t)}\|_{\YY _\jott},\ \
\eta_j\in \YY _\jott.$$

\Theta^\jott_\kappa(\eta_j,t):= \phi_\kappa(h_\jott  (\eta_j,t  ))\ \text{ and } \ \Xi^\jott_\kappa(t):=\prod_{i=j}^\Jott \Theta^i_\kappa(\eta_j,t),\quad t\ge 0.

For any $\xi\in \Mcal_{\Afrak}^{m}(\X)$ and $\kappa\in \mathbb{N}$, we consider on the  fixed filtered  probability space $\mathfrak{A}=(\Omega,\CF,(\CF_t)_{t\in[0,T]},\PP)$ given above, the following system

\begin{align}
\label{spdeskappa} \tag{{\arabic{equation}$_\kappa$}} %\tag*{[foo]}_\kappa
dw_\kappa(t) &=&\lk(\DeltaA_0 w_\kappa(t)+   \Xi^1_\kappa(t) \cdot F_0(\xi(t),v^1(t),\cdots,v^\Jott (t)) %F_0(\xi(t),w_\kappa(t))
\,\rk)\, dt +\Sigma_0(w_\kappa(t))\,dW_0(t),\ \text{in } U,
\\
dv^\jott_\kk(t) &=&\lk(\DeltaA_\jott   v^\jott_\kk(t)+  \Xi^\jott_\kappa(t)\cdot F_\jott  (\xi(t),v_\kk^{\jott}(t),v_\kk^{\jott+1}(t),\cdots,v_\kk^\Jott (t))\rk)\, dt +\Sigma_j(v_\kk^\jott(t))\,dW_\jott(t),\ \text{in } U_j,
\\
&&\quad \jott=1,\ldots ,\Jott ,
\\
\\
v^\jott(0)&=&v^\jott_0,\quad \jott=1,\ldots ,\Jott .
\earray\rk.
\end{align}

Before we present the statement of the main abstract result, we  outline shortly the idea.  We implicitly assume that the equation (<ref>) is well posed. Let us assume that the cut-off functions are chosen in such a way that for any cut-off parameter $\kappa\in\NN$,
there exists a bounded subset $\mathcal{X}_\MA^\kappa$ of $\Mcal_{\MA}^m(\X)$
such that for all $\xi\in\mathcal{X}_\MA^\kappa$
there exists in the stochastic sense  a unique strong solution  $(w_\kappa, v^1_\kappa,...,v^J_\kappa)$ to the system (<ref>) such that $w_\kappa\in \mathcal{X}_\MA^\kappa$.
If we want to underline the dependence of $w_\kappa$ or $v^\jott_\kappa$ on $\xi$, we will write $w_\kappa(\xi)$ or $v^\jott_\kappa(\xi)$, respectively.
we have to provide a bounded convex subset of $\Mcal_{\Afrak}^{m}(\X)$, such that the operator $\Vcal_{\Afrak,W,\eta }$ will map this set into itself.

Here, it is essential to characterize the set in such a way that  one can
transfer the definition to the set of probability measures.
This can be done in the following they. For each $\kappa\in\NN$,
we have to find two measurable functions $\Phi_\kappa:\DD(0,T;U)\to\RR$ and $\Psi_\kappa:\DD(0,T;U)\to\RR\cup\{\infty\}$ such that for any bounded closed interval $I$ the set $\Psi_\kappa^{\leftarrow}(I)$[For a measurable function $f:X\to\RR$, $f^{\leftarrow}$ denotes the preimage

given by $\{ x\in X: f(x)\in A\}$ for all $A\in\CB(I)$.] is closed in $\mathbb{X}$.
Let us now define for any   $R>0$
a subset
$\mathcal{X}^\kappa_{\MA}(R)$ of $\mathcal{M}_\MA^m(\mathbb{X})$ by

𝒳^κ_(R)   :=   { ξ∈ℳ_^m(𝕏):Φ_κ(ξ)≤R^m  (Ψ_κ(ξ)<∞)=1}.
The set $\mathcal{X}^\kappa_{\MA}(R)$ has to be chosen in such a way that for all $R>0$  the system (<ref>) has  a unique strong solution (in the stochastic sense) $w\in\Mcal_{\MA}^m(\X)$ for all $\xi\in\Mcal_{\MA}^m(\X)\cap \mathcal{X}^\kappa_{\MA}(R)$.

In the next, on the fixed filtered  probability space $\mathfrak{A}=(\Omega,\CF,(\CF_t)_{t\in[0,T]},\PP)$ given above, for a vector valued Wiener process $\mathbb{W}=(W_0,W_1,\ldots,W_\Jott)$, we then define the operator
^k_,𝕎:𝒳_^κ(R)   →   ℳ_^m()

ξ   ↦   ^k_,𝕎(ξ):=w_κ(ξ),

where $w_\kappa(\xi)$ should be the first component of the solution $(w_\kappa(\xi), v^1_\kappa(\xi),...,v^J_\kappa(\xi))$   of  system (<ref>).
Secondly, we have to find a sequence $\{R_\kappa:\kappa\in\NN\}$, such that we have $\mathcal{X}^{\kappa-1}_{\MA}(R_{\kappa-1})\subset \mathcal{X}^{\kappa}_{\MA}(R_\kappa)$ and
$\mathcal{X}^{\kappa}_{\MA}(R_\kappa)$ is invariant for $\Vcal^k_{\Afrak,\mathbb{W}}$ for any $\kappa\ge 1$.
For simplicity let us put $ \mathcal{X}^{\kappa}_{\MA}:=\mathcal{X}^{\kappa}_{\MA}(R_\kappa)$.

Let us assume that $w_\kappa$ is a fixed point of the system (<ref>) on a new filtered  probability space $\mathfrak{A}_\kappa=(\Omega_\kappa,\CF_\kappa,(\CF^\kappa_t)_{t\in[0,T]},\PP_\kappa)$ with respect to the new Wiener process  $\mathbb{W}_\kappa=(W^\kappa_0,W^\kappa_1,\ldots,W^\kappa_\Jott)$,. Then, on the time interval
$[0,\tau^\ast_\kappa)$, where

\begin{align}\label{defintau}
&\tau^\ast_\kappa:=\min_{1\le \jott \le \Jott }\tau^\jott_\kappa\quad\mbox{with}\quad
\tau^\jott_\kappa=\tau^\jott_\kappa(v^\jott_\kappa)
:=\inf_{t>0} \{ h_\jott(v^\jott_\kappa,t)\ge \kappa\},\,\, \jott=1,\ldots,\Jott,
\end{align}

the tuple $\mathfrak{w}_\kappa=(w_\kappa,v_\kappa^1(w_\kappa),\ldots,v^\Jott_\kappa(w_\kappa))$ solves the original system (<ref>). This means that, in addition to the new probability space and the new Wiener process, $\{(\mathfrak{w}_\kappa,\tau_\kappa^\ast)\}_{\kappa\geq 0}$ is a family of local martingale solution of problem  (<ref>).  The next step is to extend the local solution  to a global solution. This will be done by  finding some Lyapunov functional

\Lambda:\YY _1\times\cdots\times \YY _\Jott\longrightarrow \RR^+_0

such that there exists some $p>0$ and $C>0$ with

\EE \lk|\Lambda(v^1(w_\kappa),v^2(w_\kappa),\ldots, v^\Jott (w_\kappa))\rk|^p\le C,\quad \forall\,\kappa\in\NN.

Here, it is important, that the inner part of the cut-off function is bounded by the functional. In particular, we need that there exists some $\delta>0$ such that we have

h_(v^_κ,T)^δ≤C  Λ(w_κ,v^1(w_κ),v^2(w_κ),…, v^(w_κ)),  ∀ κ∈,     and   =1,….
If  (<ref>) is satisfied, then one can show by the Chebyscheff inequality that
the probability of the set

$$ \Omega_\kappa:=\{\omega\in \Omega:\tau^\ast_\kappa\le  T\}
converges to zero as $\kappa$ tends to infinity. This implies that

$$\PP(\liminf_{\kappa\to \infty }\Omega_\kappa)\leq \lim_{\kappa\to \infty}\P(\Omega_\kappa)=0,$$

\begin{equation}
\liminf_{\kappa\to\infty}\Omega_\kappa=\cup_{\kappa=0}^\infty\cap_{j=\kappa}^\infty\Omega_j.\notag
\end{equation}

Hence,  we know that the set  $\Omega_0:=\Omega\setminus\liminf_{k}\Omega_\kappa$
has probability one  and
the system with cut-off coincides on $\Omega_0$ with the original equation.
In this way, the existence of a global martingale solution  is proven.

We are  now ready to formulate our main theorem.

Given an

* underlying space $\mathbb{X}=L^p(0,T;E)$,

* possible cut–off $(\phi_\kappa)_{\kappa\in\NN}$ being continuous w.r.t. $\mathbb{X}$,
i.e. given a sequence $\{\xi_n:n\in\NN\}$ such that $\xi_n\to \xi$ in  $\mathbb{X}$, then we have
$\phi_\kappa(\xi_n)\to \phi_\kappa(\xi)$.

* a subspace $\mathcal{X}_\kappa$ of $\CM^p( \xi:[0,T]\times \Omega\to \mathcal{D}, \quad \mbox{$\xi$ is progressively measurabel
and $\xi\in\mathbb{X}$} \}$

* and an operator $\operator _\kappa$ with

Let $m>1$ be given. Let us  assume that
for  any $R>0$ and $\kappa\in \mathbb{N}$, there exists a
bounded and convex  set

^κ_(R):={ ξ∈^m_(): Φ^κ(ξ)≤R,  (Ψ_κ(ξ)<∞)=1}

in $\CM^m_\MA(\BX)$, such that
for all  $\xi \in \CM^m_\MA(\BX)$ the system (<ref>) has a unique probabilistic strong solution
$$\mathfrak{w}_\kappa(\xi)=({w}_\kappa(\xi),{v}^1_\kappa(\xi),\dots,{v}_\kappa^\Jott(\xi))\in \X\times\X_1\times...\times \X_J, \ \mathbb{P}\text{-a.s},$$

where $\X_j\subset\{ \eta: [0,T]\to U_j \}$, $j=1,...,J$,  are given Banach spaces. In addition we assume that for any $R>0$ we have
$\Xcal^{\kappa-1}_{\Afrak}(R)\subset \Xcal^\kappa_{\Afrak}(R)$.

Finally, we assume that there exists some initial condition given as follows:
there exists some Banach spaces $E_0$ and $E_\jott$, $\jott=1,\ldots, \Jott$ and measurable functions $g_0:E_0\to \RR$,  $g_\jott:E_\jott\to \RR$, $\jott=1,\ldots,\Jott$, such that

\begin{align}\label{initcond}
\EE g_0(w_0),\EE g_1(v^1_0),\ldots,\EE g(v^\Jott_0)<\infty.
\end{align}

We then consider the following  operator
^κ_,𝕎:^m_()   ⟶   _^m()

ξ   ⟼   ^κ_,𝕎(ξ):=w_κ(ξ),
where $w_\kappa(\xi)$ is  the  first component of $({w}_\kappa(\xi),{v}^1_\kappa(\xi),\dots,{v}_\kappa^\Jott(\xi))$ solution of  system (<ref>).

If for any $\kappa\in \mathbb{N}$,
there exists
two measurable functions $\Phi_\kappa: \X\to \mathbb{R}$ and $\Psi_\kappa:\X\to\RR\cup\{\infty\}$ such that

for any $R>0$ 	the operator $\Vcal^\kappa_{\Afrak,\mathbb{W}}$ is well posed on  $\Xcal^\kappa_{\Afrak}(R)$;  in particular,
for all $\xi \in \Xcal^\kappa_{\Afrak}(R_\kk)$, there exists a unique strong solution $\mathfrak{w}_\kappa=(w_\kappa(\xi), v^1_\kappa(\xi),...,v^J_\kappa(\xi))$ of system (<ref>) exists, with $\Vcal^\kappa_{\Afrak,\mathbb{W}}(\xi)=w_\kappa$,  and	$w_\kappa\in \Mcal_{\Afrak}^{m}(\X)$.
for any $\xi\in \CX_\kappa$, there exists a tuple of processes
\mathfrak{w}_\kappa(\xi)=({w}_\kappa(\xi),({v}^1_\kappa)(\xi),\dots,({v}_\kappa^\Jott)(\xi))
solving the system

for any $\kappa\in\NN$, there exists a number $R_\kappa>0$ such that the operator $\Vcal^\kappa_{\Afrak,\mathbb{W}}$   maps $\Xcal^\kappa_{\Afrak}(R)$ into itself for any $R\ge R_\kappa$;

* let us define the operator
the operator $\Vcal^\kappa_{\Afrak,\mathbb{W}}$ by where $\tilde w$ is a solution to
(<ref>), where we replaced $\xi$ by $\xi\mathds{1}_{[0,\tau^0_\kappa(\xi))}$,
and $v^k$ by  $v^k\mathds{1}_{[0,\tau^l_\kappa(v^k))}$.
Then, if $\{\xi_n:n\in\NN\}$ is a sequence converging to $\xi$ in $\Mcal_{\Afrak}^{m}(\X)$, then
the sequence $\{\osperator(\xi_n):n\in\NN\}$ converges to $\osperator(\xi)$ in $\Mcal_{\Afrak}^{m}(\X)$.
In addition we assume that there exists $p_1,\ldots,p_\Jott\ge 1$ such that if $\{\xi_n:n\in\NN\}$ is a sequence converging to $\xi$ in $\Mcal_{\Afrak}^{m}(\X)$, then
$v^\jott_\kappa(\xi_n)$ converges to $v^\jott_\kappa(\xi)$ on $L^{p_\jott}(\Omega;\YY _\jott)$.

for any $R\ge R_\kappa$, 	the operator $\Vcal^\kappa_{\Afrak,\mathbb{W}}$ restricted to $\Xcal^\kappa_{\Afrak}(R)$ is
continuous on $\Mcal_{\Afrak}^{m}(\X)$.

In addition,
there exists Banach spaces $\YY _1,\ldots, \YY _K$ such that if a sequence $\{\xi_n:n\in\NN\}\subset \mathcal{X}_\kappa$ converges to $\xi$ then for all uniformly in $v^n_\jott(\xi_n)  \to v_\jott (\xi) $ in $\mathcal{M}^{m_\jott  }(\YY _\jott  )$.

for any $R\ge R_\kappa$, 	there exists a number $m_0>0$ and a space $\BX'$ such that $\BX'\hookrightarrow\BX$ compactly  and  there exists a $C_0>0$ with
$\EE\|w_\kappa(\xi)\|_{\BX'}^{m_0}\le C_0$ for all $\xi\in\Xcal^\kappa_{\Afrak}(R)$;

for any $R\ge R_\kappa$, 	there exists a number $m_1>m$    and   a constant $C_1>0$ such that
$\EE\|w_\kappa(\xi)\|_{\BX}^{m_1}\le C_1$ for all $\xi\in\Xcal^\kappa_{\Afrak}(R)$;

for any $R\ge R_\kappa$,  we have $\mathbb{P}\text{-a.s}$, $({w}_\kappa(\xi),{v}^1_\kappa(\xi),\dots,{v}_\kappa^\Jott(\xi))\in C([0,T];U)\times C([0,T];U_1)\times\dots\times  C([0,T];U_J), $ for all $\xi\in \Xcal^\kappa_{\Afrak}(R)$.

Let us assume that for any $\kappa\in\NN$, there exists
a probability space $\mathfrak{A}_\kappa$,  a Wiener process $\mathbb{W}_\kappa$ defined on $\mathfrak{A}_\kappa$, and an element $w_\kappa\in \Xcal^\kappa_{\Afrak_\kappa}(R_\kk)$,

such that for all initial conditions being $\CF_0$-measurable we have $\Vcal^\kappa_{\Afrak_\kappa}(w_\kappa)=w_\kappa$.
Let us assume that the condition on the initial data are preserved, i.e. at time  $\tau^\ast_\kappa$ (defined in (<ref>)) the random variable $w_\kappa(\tau^\ast_\kappa),v^1_\kappa(\tau^\ast_\kappa),\ldots ,v^1_\kappa(\tau^\ast_\kappa)$ satisfy condition (<ref>).

If for   the  corresponding solution

\mathfrak{w}_\kappa=({w}_\kappa,{v}^1_\kappa,\dots,{v}_\kappa^\Jott),

of the system (<ref>),
there exists $m_j>1$, $j=1,\ldots,J$, and a constant $C(T)>0$ (both independent of $\kappa$)
such that

max_=1,…,J[ h^m__(v^_κ,t )]≤C(T),  t∈[0,T], κ∈,

then there exists a global martingale solution to the problem
(<ref>) in the sense of Definition <ref>.

let $w_\kappa\in\Xcal^\kappa_{\Afrak}(R_\kk)$ be a fix point of $\Vcal^\kappa_{\tilde{\Afrak},\tilde{\mathbb{W}}}$ and let
\mathfrak{w}_\kappa=({w}_\kappa,{v}^1_\kappa,\dots,{v}_\kappa^\Jott)

be the corresponding tuple  solving the system
There exists $m_j>1$, $j=1,\ldots,J$, and a constant $C(T)>0$ such that

\max_{\jott =1,\ldots,J}\tilde{\EE}\left[ h^{m_\jott }_\jott(v^\jott_\kappa,t )\right]\le C(T),\quad t\in[0,T],\;\kappa\in\NN;

The inner part of the cut-off function  satisfy the following condition:
given a sequence $\{\xi_n:n\in\NN\}$ such that $\xi_n\to \xi$ in  $\mathbb{X}$,  we have
$h_\jott(v^\jott_\kappa(\xi_n),t)\to h(v^\jott_\kappa(\xi),t)$.

Showing the uniform bound for the cut-off functions,
it is essential that
to find a Lyapunov type functional $\Lambda$, such that $h_\jott(v^\jott_\kappa,T)$ are bounded by the Lyapunov functional.
In particular, one has to show that

\EE h^{m_j}_\jott(v^\jott_\kappa(\xi),T)\le \EE \Lambda( \mathfrak{w}_\kappa(\xi)),\quad j=1,\ldots, J,  %\le C.

\EE \Lambda( \mathfrak{w}_\kappa(\xi))  \le C.

Now, we give a proof of Theorem <ref>.

As outlined before, the proof of the Theorem <ref> consists of several steps.	Part of the steps we have already introduced in the first paragraphs, as well as part of the ingredients, like the cut-off function $\phi$, the mappings $h$ and $\Theta$, the system (<ref>),  the operator $\mathcal{V}^\kappa_{\MA,W}$, and the corresponding bounded and convex sets   $\mathcal{X}_\kappa$ on which the operator $\mathcal{V}^\kappa_{\MA,W}$ acting on.

* In this step we will show that for any $\kappa\in\NN$ there exists a fix point and that the fix point solves the system given in (<ref>) on a new probability space.
In particular, one can get for each $\kappa\in\NN$ a local martingale solution.

For any $\kappa\in\NN$, there exists a probability space
$\tilde{\mathfrak{A}}_\kk =(\tilde \Omega_\kk ,\tilde{\CF}_\kk ,\tilde{\mathbb{F}}_\kk ,\tilde{\PP}_\kk ),
a vector valued Wiener process
$\tilde{\mathbb{W}}^\kk=(\tilde W_0^\kappa,\tilde W_1^\kappa,\ldots,\tilde W^\kappa_\Jott)
being defined on  $\tilde {\mathfrak{A}}_\kk $, and a tuple of random variable

$$\tilde{\mathfrak{w}}_\kappa:=(\tilde w_{\kappa},\tilde v^\jott_\kappa,\ldots ,\tilde v^\Jott_\kappa)\in C([0,T];U)\times C([0,T];U_1)\times \cdots\times C([0,T];U_J), \ \tilde{\PP}_\kk-a.s.

solving the system (<ref>) $\tilde{\PP}_\kk$-a.s.

The proof is closed to the prove of the stochastic Schauder Tychonoff Theorem given in [14].
In [11] a refined version with a Lévy process is given. However, our main difference is that we have only to proof the assumption for the existing fixpoint for one process, i.e. $w_\kappa$. Here, it is essential that only a dependence from equation for $v_\kappa^\jott$ to $v_\kappa^{\jott-1}$ exists.

we have a $\tilde \PP$-a.s. the fixed point

\mathcal{V}_{\kk ,\tilde{\MA}_\kappa}(\tilde u_{\kappa}(t))=(\tilde u_{\kappa}(t))

for every $t\in[0,T]$.
Due to the construction of
$\mathcal{V}_{\scriptstyle \tilde{\MA}}$, the process $\tilde u_{\kappa}$
solves the system (<ref>) refs eq:cutoffuu/eq:cutoffvv gibts nicht
over the stochastic basis $\tilde{\mathfrak{A}}_\kk $ with the Wiener noise $\tilde W^\kk $.

*  In this step we will approximate the operator  $	\Vcal^\kappa_{\Afrak,\mathbb{W}}$ via a time discretisation. To this aim, in a similar way as in [4], for $\kl\in \mathbb{N}$ we introduce a dyadic time grid
$$\pi:=\lk\{\frac{iT}{2^{\kl}}; \ \ i=0,1,...,2^\kl-1\rk\},$$

and define  the shifted  Haar projection $\Pro_\kl $ of order $\ell$ by  for $f\in  \X$ such that $f(0)$ exists,

\begin{equation*}\label{hatdefined} (\Pro_\kl f)(s):=\bcase f(0), \ \text{if } \ s\in [0,\frac{T}{2^\ell})\\
\frac {2^\ell}{T} \int_{\frac{iT}{2^{\kl}}}^{\frac{(i+1)T}{2^{\kl}}} f(r-\frac{T}{2^{\ell}})\: dr, \text{ if } \ s\in [\frac{iT}{2^{\kl}},\frac{(i+1)T}{2^{\kl}}),  \ i=1,2,3..., 2^\kl-1.\ecase
\end{equation*}

Note, that if $f\in \CM_\MA^{m}(\X)$ is such that $f(0)$ exists, then $\Pro_\kl f$ is a
progressively measurable, piecewise  constant,  $X$-valued

We recall  in  <cit.>, that for  all $\ell\in \mathbb{N}$,

* $\Pro_
\kl$ is a bounded linear operator from $\X$ into $\X$. In fact
\kl f\|_{\X}\le \|f\|_{\X},\quad \forall f\in \X.

* if $K$ is a bounded compact subset of $\X$,
then for all $\ep>0$ there exists a $\kl_\eps\in\NN$ such that

\kl  f - f\|_{\X}\le \ep,\quad f\in K,\quad \forall \kl\ge\kl_\eps.

* $\Pro_
\kl$ is convergence in  $\X$.
$$	\Pro_
\kl f \longrightarrow f \quad \text{as}\quad  \kl \longrightarrow  \infty\quad  \text{ in }\quad \X.$$

It follows that for all $\ep>0$ there exists a $\kl_\eps\in\NN$ such that

\kl\Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi) - \Vcal^\kappa_{\Afrak,\mathbb{W}}(\xi)\|_{\X}\le \ep,\quad \forall \, \xi\in \Xcal^\kappa_{\Afrak}(R_\kk)\quad\forall \kl\ge\kl_\eps.

Indeed let us fix $\eps>0$. Due to the assumption (<ref>), we know that there exists $C_0>0$ such that
$$\EE\| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X}^{m_0}\leq C_0,\quad \forall \xi\in \Xcal^\kappa_{\Afrak}(R_\kk).$$

Let us consider  $m'_0 >0$ is such that $\frac{1}{m_0}+\frac{1}{m'_0}=1$ and $R_\eps>0$ such that

$$ 2C\lk(\frac{C_0 }{R^{m_0}_\ep}\rk)^\frac 1{m'_0}\,
\lk(C_0\rk)^\frac 1{m_0}\leq \frac{\eps}{2},$$

where $C>0$ is a constant coming from the continue embedding of $\X'$ into $\X$. We then consider the set

$$K_\eps=\left\{\xi \in \X: \|\xi\|_{\X'}\leq R_\ep\right\}.$$

Since $\BX'\hookrightarrow\hookrightarrow\BX$ compactly, the set $K_\eps$ is a compact set of $\X$. Hence, invoking the  Chebycheff inequality we derive that

\begin{align*}
\lk(\PP\lk( \Vcal^\kappa_{\Afrak,\mathbb{W}}  (\xi) \not\in K_\ep\rk) \rk)^\frac 1{m'_0}\,
\lk(\EE\| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X}^{m_0}\rk)^\frac 1{m_0}&\leq C\lk(\PP\lk( \| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X'}> R_\ep\rk) \rk)^\frac 1{m'_0}\,
\lk(\EE\| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X'}^{m_0}\rk)^\frac 1{m_0}\\
&\leq C\lk(\frac{\EE \| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|^{m_0}_{\X'} }{R^{m_0}_\ep}\rk)^\frac 1{m'_0}\,
\lk(C_0\rk)^\frac 1{m_0}.\\
&\leq C\lk(\frac{C_0 }{R^{m_0}_\ep}\rk)^\frac 1{m'_0}\,
\lk(C_0\rk)^\frac 1{m_0}.
\end{align*}

Let $\ell_\eps\in\NN$ large such that

\|\Pro_
\kl  f- f\|_{\X} \le \frac \ep2,\quad \forall f\in K_{\ep} ,\text{and for all }  \kl\ge \ell_\eps.

Using the first item of Remark <ref>, we note that for $\ell\geq \ell_\eps$

\begin{align*}
\EE\| \Pro_
\kl \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi) - \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X}  &\leq \EE1_{\Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\in K_\ep}\| \Pro_
\kl \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi) - \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X}\\
&\qquad +\EE 1_{\Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\not \in K_\ep}\lk(\|\Pro_
\kl \Vcal^\kappa_{\Afrak,\mathbb{W}}  (\xi) \|_{\X}+\| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X}\rk)\\
&\leq \frac \ep2+2\EE 1_{\Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\not \in K_\ep}\| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X}\\
&\leq \frac \ep2+2C\lk(\EE 1_{\Vcal^\kappa_{\Afrak,W}(\xi)\not \in K_\ep}^{m'_0}\rk)^\frac 1{m'_0}\,
\lk(\EE\| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X'}^{m_0}\rk)^\frac 1{m_0}\\
&\leq \frac \ep2+2C\lk(\PP\lk( \Vcal^\kappa_{\Afrak,\mathbb{W}}  (\xi) \not\in K_\ep\rk) \rk)^\frac 1{m'_0}\,
\lk(\EE\| \Vcal^\kappa_{\Afrak,\mathbb{W}} (\xi)\|_{\X}^{m_0}\rk)^\frac 1{m_0}\\
&\leq \frac \ep2+2C\lk(\frac{C_0 }{R^{m_0}_\ep}\rk)^\frac 1{m'_0}\,
\lk(C_0\rk)^\frac 1{m_0}\\
&\leq \ep,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall\xi\in \Xcal^\kappa_{\Afrak}(R_\kk).
\end{align*}

Finally, let us define the operators

\Vcal^{\kappa,\ell}_{\Afrak,\mathbb{W}}(\xi):=   \Pro_\ell\Vcal^\kappa_{\Afrak,\mathbb{W}}(\xi),\quad \xi\in \Xcal^\kappa_{\Afrak}(R_\kk).

For any $\ell  \in\NN$ the family of operators $\{\Vcal^{\kappa,\ell}_{\Afrak,\mathbb{W}}:\kl\in\NN\}$
induces  a family of operators $\{\mathscr{V}^{\kappa,\kl}:\kl\in\NN\}$ on the set of Borel probability measures on $\X$ denoted by $\mathscr{M}_1(\X)$[

Let $(E,d)$ be a metric space.  Let $\mathscr{M}_1(E)$ be the set of Borel probability measures on the metric space $(X,d)$. Note, that the KyFun metric is defined  in the following (see <cit.>): For $\nu,\mu\in\mathscr{M}_1(X)$,

$$d_{K}(\mu,\nu):=\inf\{\alpha>0: \mu(A)\le \nu(A_\alpha)+\alpha \mbox{ and }
\nu(A)\le \mu(A_\alpha)+\alpha \mbox{ for all } A\in\CB(X)\}.

Here $A_\alpha:=\{ x\in X: d(x,A)<\alpha\}$. The Ky Fan and Prohorov metric and the Prohorov metric characterizes weak convergence (see <cit.>).
Note that if $(S,d)$ is a separable complete metric space, then the set of all laws on $S$ defined on $\CB(S)$ is complete for the Prohorov or KyFan metric $d_K$, see <cit.>.]. The construction of $\mathscr{V}^{\kappa,\kl}$ is done in the following way. Firstly, let us observe that due to assumption  (<ref>), for all $\xi\in\Xcal^\kappa_{\Afrak}(R_\kk)$ the process $\Vcal^{\kappa,\ell}_{\Afrak,\mathbb{W}}(\xi)$ belongs $
\PP$-a.s. to  $\DD(0,T,U)$. In fact, due to the point (<ref>),  $\Vcal^{\kappa,\ell}_{\Afrak,\mathbb{W}}(\cdot)$ maps $\xi$ into $C(0,T;U)$ and the projection $\Pro_\kl$ projects $C(0,T;U)$ onto $\DD(0,T,U)$. Hence,  let us define the subset of probability measures $\mathscr{X}_\kk$ given by

𝒳_   :=   { 𝒬∈ℳ_1(∩(0,T;U)): ∫_Ψ(x)d𝒬≤R_}.
Let $\mathscr{Q}$ be a  probability measure belonging to $\mathscr{X}_\kk$.

Then, by the Skorohod theorem,  there exists a
probability space  $\mathfrak{A}_0=(\Omega_0,\CF^0,\PP_0)$ and a random variable
$\xi:\Omega_0\to \X\cap \DD(0,T;U)$ such that the law of $\xi$ coincides with  $\mathscr{Q}$.
In particular,
the probability measure $\mathscr{P}_\xi:\CB(\X)\to[0,1]$ induced by $\xi$ and given by

\mathscr{P}_\xi:\CB(\X)\ni A\mapsto \PP_0\lk( \lk\{ \omega:\xi(\omega)\in A\rk\}\rk)

coincides with the probability measure $\mathscr{Q}$.

Since $\PP$-a.s., $\xi\in\DD(0,T;U)$ we know, that for any $t\in[0,T]$, $\xi(t)$ is a $U$–valued random variable over $\mathfrak{A}_0$.
In particular, the mapping $\xi:\Omega_0\times [0,T]\to U$ is well defined.

\CF^0_t:=\sigma \lk( \lk\{\,\xi (s),\;\colon\; 0\le s\le t \, \rk\}\cup\tilde \CN_0\rk),\quad t\in [0,T],

where $\CN_0$ denotes the zero sets of ${\mathfrak{A}}_0$.
Let us put $\mathfrak{A}_0:=(\Omega_0,\CF^0,(\CF_t^0)_{t\in[0,T]},\PP_0)$.

Next, we have to construct the $\Jott+1$-tuple of Wiener processes and extend the probability space $\mathfrak{A}_0$.
Then, we consider the filtered probability space   $\mathfrak{A}_1=(\Omega_1,\CF^1,(\CF_t)_{t\in[0,T]},\PP_1)$ where on the  cylindrical Wiener processes $\mathbb{W}=(W_0,\ldots,W_\Jott)$  is defined, and let $\mathfrak{A}_{\mathscr{Q}}$ the product probability space of $\mathfrak{A}_0$ and $\mathfrak{A}_1$.
In particular, we put

Ω_𝒬 =  Ω_0×Ω_1,

_𝒬 = ^0⊗^1,

^𝒬_t = ^0_t⊗^1_t,

_𝒬 = _0×_1.

Here, we know that $\xi$ at time $t$ is independent of the increments of the Wiener process after time t. In particular, the
integral w.r.t. the Wiener process is an Ito integral.
By the choice of $\mathscr{X}_\kk$, we know that  $\xi\in\Xcal^\kappa_{\mathfrak{A}_\mathscr{Q}}(R_\kk)$ and therefore,  $\Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)$ is well defined with
$$\mathfrak{A}_\mathscr{Q}=(\Omega_\mathscr{Q},\CF_\mathscr{Q}, (\CF^\mathscr{Q}_t )_{t\in [0,T]}, \PP_\mathscr{Q} ).$$
Let us define a   family of   operators
\{\ST^{\kk,\kl} :\kl\in\NN\}$ acting on $\mathscr{M}_1(\X)$ and
having the same law as  $\Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)$.
To be more precise,  the mapping
$$\ST^{\kk,\kl}: \mathscr{M}_1(\X)\longrightarrow \mathscr{M}_1(\X)$$
the probability measure $\mathscr{P}_\xi:\CB(\X)\to[0,1]$ induced by $\xi$ (the random variable $\xi$ is obtained via a given probability measure $\mathscr{Q}\in \mathscr{M}_1(\X)$) to the probability measure $\mathscr{P}_{\Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)}:\CB(\X)\to[0,1]$ given by

\mathscr{P}_{\Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)}(A):=\PP_\mathscr{Q}\lk( \lk\{ \omega\in\Omega_\mathscr{Q}: \Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi(\omega))\in A\rk\}\rk), \ \forall A\in\CB(\X).

Note, since $\X$ is a separable Banach space,
the metric space of probability measures over $\X$ equipped with the
Prohorov metric   is separable and complete.

The set $\mathcal{C}_R $ corresponds to[We denote the set of all probability measure on a Banach space $E$ by $\mathscr{M}_1(E)$ topology]

{ P∈ℳ_1(∩𝕌): P(|).

∫_ ξ^2_ dP≤R)=1

The following points are valid:

* For a fixed $\kk\in \mathbb{N}$,  due to the fact that $\Vcal^\kappa_{\Afrak_{\mathscr{Q}},\mathbb{W}}$ restricted to $\Xcal^\kappa_{\mathfrak{A}_\mathscr{Q}}(R_\kk)$ is continuous (see point (<ref>)) and the fact that the family $\{\mathscr{V}^{\kk,\kl}, \kl\in \mathbb{N}\}$ is uniformly continuous,  the family of operator $\{\text{Proj}_\ell, \kl\in \mathbb{N}\}$  restricted to $\mathscr{X}_\kappa$ is uniformly continuous on $\mathscr{M}_1(\X) $ in the Prohorov metric. This point follows from <cit.>.

* $\mathscr{V}^{\kk,\kl}$ restricted to $\mathscr{X}_\kappa $ is compact on $\mathscr{M}_1(\X) $.
In particular, it maps bounded sets into compact sets.
In fact, we have to show that for all $\kappa\in\NN$ and $\ep>0$ there exists a compact subset in $\CB(\X)$ such that

\mathscr{P}_ {\Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)}\lk (\mathbb{X}\setminus K_\ep\rk)\le \ep, \quad \forall \mathscr{Q}= \mathscr{P}_\xi\in \mathscr{K}_\kappa\quad \mbox{and}\quad \PP_ {\Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)}:=\mathscr{V}^{\kk,\kl}(\mathscr{P}_\xi).

Indeed, from assumption  (<ref>) we know that there exists a constant  $C>0$ such that

\EE\|\Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)\|_{\mathbb{X}'}^{m_0}\le C,\quad \forall \xi\in\Xcal^\kappa_{\mathfrak{A}_\mathscr{Q}}(R_\kk).

Let $\tilde R>0$ so large that

\frac {C}{\tilde R^{m_0}}\le \ep

and let $K_\ep:=\{f\in\mathbb{X}:\|x\|_{\mathbb{X}'}\le \tilde{R}\}$.
Note, we know by the Chebyscheff inequality that

$$\mathscr{P}_{ \Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)}\lk (\mathbb{X}\setminus K_\ep\rk)
\mathscr{P}_{ \Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)}\lk (\overline{ \lk\{
f\in\mathbb{X}:\|x\|_{\mathbb{X}'} \ge \tilde{R}\rk\}}\rk)
\le \epsilon$$

for all $\mathscr{P}_\xi\in \mathscr{K}_\kappa$ where $\mathscr{P}_{ \Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}(\xi)}:=\mathscr{V}^{\kk,\kl}(\mathscr{P}_\xi)$.

Since $\mathbb{X}'\hookrightarrow \mathbb{X}$ compactly,
we have proven the tightness and the Prohorov theorem gives compactness on $\mathscr{M}_1(\X)$.

* $\mathscr{X}_\kappa$  is a convex subset of $\mathscr{M}_1(\X)$ and is invariant under $\mathscr{V}^{\kk,\kl}$. In fact the convexity follows by direct calculations and the invariance follows from the assumption (<ref>). Let  us consider the probability measure $ \mathscr{Q}\in \mathscr{M}_1(\X)$. Then there exists a filtered probability space $\mathfrak{A}_\mathscr{Q}$, the Wiener process $\mathbb{W}$ and we can construct  a bounded set $\Xcal^\kappa_{\mathfrak{A}_\mathscr{Q}}(R_\kk)$ such that  $\Vcal_{\mathfrak{A}_\mathscr{Q},\mathbb{W}}^{\kk,\kl}$  maps  $\Xcal^\kappa_{\mathfrak{A}_\mathscr{Q}}(R_\kk)$  into itself.  Since $\|\Pro_
\kl \xi\|_\X\le \|\xi\|_\X,$ we then know that $\mathscr{X}_\kappa$  is invariant under  $\mathscr{V}^{\kk,\kl}$.

In particular, the mapping $\mathscr{V}^{\kk,\kl}$ restricted to $\mathscr{X}_\kappa$
satisfies the assumption of the Schauder-Tychonov Theorem (see <cit.>).
Hence, for any $\kl\in\NN$ there exists a probability measure $\mathscr{P}^\ast_{\kk,\kl}$  such that $\mathscr{V}^{\kk,\kl}(\mathscr{P}^\ast_{\kk,\kl})=\mathscr{P}^\ast_{\kk,\kl}$.

Note, for a fixed $\kk\in\NN$,  since the estimate on $\X'$ is uniform, the set

\lk\{ \mathscr{P}^\ast_{\kk,\kl}\ : \  \kl\in\NN\rk\}

is tight. Hence,  there exists a subsequence

$\{\kl_j :j\in\NN\}$ and a Borel probability measure $\mathscr{P}^\ast_\kk$ such that
$ \mathscr{P}_{\kk,\kl_j}\to 	 \mathscr{P}_\kk^\ast$, as $j\to\infty$.	 By the Skorokhod lemma <cit.>,
there exists a probability space $\mathfrak{A}^\ast_0=(\Omega^\ast_0,\CF^{\ast}_0,\PP^\ast_0)$ and a  sequence of $\X$-valued random variables $\{ {w}^\ast_{\kappa,\kl_j }:j\in\NN\}$
and ${w}^\ast_\kk$ where
the random variable $w^\ast_\kk:\Omega^\ast_0\to \X$ has the  law $\mathscr{P}^\ast_\kk$  and

𝒫_κ,_j^∗,  j∈,

and such that

\begin{equation}
{w}^\ast_{\kappa,\kl_j }\to{w}^\ast_\kk\quad \mbox{ as $j\to \infty$ } \,\,{\P}^\ast_0\text{-a.s.}
\text{ on } \X.\label{2.11}
\end{equation}

Next, we have  to construct  the Wiener process.

Let us first consider the  filtered probability space

\mathfrak{A}_1=\lk(\Omega_1,\PP_1,\CF_1,(\CG^1_t)_{t\in[0,T]}\rk),

where on the  cylindrical Wiener processes $\mathbb{W}=(W_0,\ldots,W_\Jott)$  is defined. 			In particular, we put
Ω^∗=  Ω_0^∗×Ω^∗_1,
^∗=   _0^∗⊗_1^∗,
^∗=   ^∗_0⊗_1.
and extend  the random variables $\mathbb{W}$, ${w}^\ast_{\kappa ,\kl_j}$  on $(\Omega^\ast, \mathcal{F}^\ast,\mathbb{P}^\ast)$ as follows for all $t\in [0.T]$.

\begin{align*}
&\mathbb{W}(t,\omega_0,\omega_1)=\mathbb{W}(t,\omega_1), \qquad\forall\  (\omega_0,\omega_1)\in \Omega^\ast,\\
&{w}^\ast_{\kappa ,\kl_j}(\omega_0,\omega_1)={w}^\ast_{\kappa ,\kl_j}(\omega_0), \qquad\forall\  (\omega_0,\omega_1)\in \Omega^\ast,\\
&{w}^\ast_{\kappa}(\omega_0,\omega_1)={w}^\ast_{\kappa}(\omega_0), \qquad\forall\  (\omega_0,\omega_1)\in \Omega^\ast.
\end{align*}

Moreover, let us introduce
the filtration $\CG^\ast =(\mathcal{G}_t^{\ast})_{t\in[0,T]}$  given by

\CG^{\ast}_t:=\sigma \lk( \lk\{\,(w _{\kk,\kl_j}^\ast(s), w^\ast_\kk(s), \mathbb{W}(s)) \;\colon\; 0\le s\le t \ , \ j\in\NN
\rk\}\cup\tilde \CN_0\rk),\quad t\in [0,T],

where $\CN_0$ denotes the zero sets of $(\Omega^\ast, \mathcal{F}^\ast,\mathbb{P}^\ast)$.  Let us set  $\mathfrak{A}^\ast:=(\Omega^\ast, \mathcal{F}^\ast,\CG^\ast , \mathbb{P}^\ast)$ and recall that by the definition of the filtration $\CG^\ast $, the  process $ \mathbb{W}$ is Wiener process over $\mathfrak{A}^\ast$.

In addition, $\Xcal^\kappa_{\mathfrak{A}^\ast}(R_\kk)$ can be defined, and also the operators ${\Vcal^\kappa_{\mathfrak{A}^\ast,\mathbb{W}}}$ and its  projection $\Vcal^{\kappa,\ell}_{\mathfrak{A}^\ast,\mathbb{W}}$.

We start by remarking that although  $\mathscr{V}^{\kk,\kl_j}(\mathscr{P}^\ast_{\kk,\kl_j})=\mathscr{P}^\ast_{\kk,\kl_j}$, the process $w^\ast_{\kappa,\ell_j }$, does not have any guaranties  to satisfy

{\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}} ( w_{\kk,\kl_j}^\ast )(s) =  w^\ast_{\kappa ,\kl_j}(s)\rk\}\rk)=1 \quad \mbox{for} \quad 0\le s\le T.$$

In this step, we  will construct a fixed point denoted by $\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}$  to the operator $\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}$  and that the probability measure will not change. We will construct this new process by induction. To start with let us note that since  $\mathscr{V}^{\kk,\kl_j}(\mathscr{P}^\ast_{\kk,\kl_j})=\mathscr{P}^\ast_{\kk,\kl_j}$, then  by the equality (<ref>) and the definition of $\mathscr{X}_\kk$ and $\Xcal^\kappa_{\mathfrak{A}^\ast}(R_\kk)$, the process $w^\ast_{\kappa,\ell_j }\in\Xcal^\kappa_{\mathfrak{A}^\ast}(R_\kk)$ and hence,
$\Vcal^{\kappa}_{\mathfrak{A}^\ast,\mathbb{W}}(w^\ast_{\kappa,\ell_j })$
are well defined. Indeed for $\xi=w^\ast_{\kappa,\ell_j }$, by the hypothesis of Meta theorem, considering the filtered probability space $\mathfrak{A}^\ast$, we can solve the system (<ref>) to obtain the $J+1$-couples $(\Vcal^{\kappa}_{\mathfrak{A}^\ast,\mathbb{W}}(w^\ast_{\kappa,\ell_j }),{v}^1_\kappa(w^\ast_{\kappa,\ell_j }),\dots,{v}_\kappa^\Jott(w^\ast_{\kappa,\ell_j }))$ such that by assumption  (<ref>), $\mathbb{P}^\ast\text{-a.s},$
$$(\Vcal^{\kappa}_{\mathfrak{A}^\ast,\mathbb{W}}(w^\ast_{\kappa,\ell_j }),{v}^1_\kappa(w^\ast_{\kappa,\ell_j }),\dots,{v}_\kappa^\Jott(w^\ast_{\kappa,\ell_j }))\in C(0,T;U)\times C(0,T;U_1)\times\dots\times  C(0,T;U_J).$$

We also note that since $\ST^{\kk,\kl_j}(\mathscr{P}^\ast_{\kappa ,\ell_j})=\mathscr{P}^\ast_{\kappa,\ell_j }$, we have
$$\Law(\Vcal^{\kappa,\ell_j}_{\mathfrak{A}^\ast,\mathbb{W}}( w^\ast_{\kappa,\ell_j }))=\Law(w^\ast_{\kappa,\ell_j}).$$

Now, we  construct here a fixed point  to the operator $
\Vcal^{\kappa,\ell_j}_{\mathfrak{A}^\ast,\mathbb{W}}$. To start with, by induction  let

$$\tilde w^\ast_{\kappa,\kl_j ,0}(s) =w^\ast_{\kappa ,\kl_j}(s),\qquad s\in [0,T],$$

w̃^∗_κ,_j ,1(s)    :=
w^∗_κ,_j(s)     0≤s<t^_j_1,

(^κ,_j_𝔄^∗,𝕎(w^∗_κ,_j))(s)     t^_j_1≤s≤T.

Clearly, in the time interval $[0,t_1^{\kl_j})$, the law is the same and by the fact that  the operator  $\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}$ is invariant with respect to the measure $\PP^\ast$, we will not change the law on the time interval $[t_1^{\kl_j},T]$.

Next, let w̃^∗_κ,_j ,2(s)    :=
w̃^∗_κ,_j,1(s)     0≤s<t^_j_2,

(^κ,_j_𝔄^∗,𝕎(w̃^∗_κ,_j,1))(s)     t^_j_2≤s≤T.

Similar like before, by the fact that  in the time interval $[0,t_2^{\kl_j})$, the law is the same and since  the operator  $\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}$ is invariant with respect to the measure $\PP^\ast$, we will not change the law on the time interval $[t_2^{\kl_j},T]$.

Now, having defined $\tilde w^\ast_{\kappa ,\kl_j,i}$,  $i=1,2,3,...,2^{\ell_j}-1$, let
w̃^∗_κ,_j ,i+1(s)    :=
w̃^∗_κ,_j,i(s)     0≤s<t^_j_i+1,

(^κ,_j_𝔄^∗,𝕎(w̃^∗_κ,_j,i))(s)     t^_j_i+1≤s≤T.

Our claim is now, that the process $\tilde w_{\kk,\kl_j,2^{\ell_j}}^\ast$ satisfies

^κ,_j_𝔄^∗,𝕎 (w̃_,_j,2^ℓ_j^∗)(s) =w̃^∗_κ,_j,2^ℓ_j(s)})   =   1      0≤s≤T.
In fact, we have $\PP^\ast$-a.s. by definition we have,

{\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}\rk)(s)=
{\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kappa,\kl_j ,0}\rk)(s), \quad 0\le s<t^{\ell_j}_1.

By the definition of $\Pro_{\kl_j}$ on $[0,t_1^{\kl_j})$
the process on $[0,t_{1}^{\kl_j})$ is defined by the initial data.

In particular, we have  $\PP^\ast$-a.s.

{\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}\rk)(s)
=%\tilde w^\ast_{l_1}(s)=w^\ast_{\kappa }(s)=
w^\ast_0, \quad \mbox{for} \quad 0\le s<t^{\kl_j}_1.

On the other side, we have
$$\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}(s)= w^\ast_{\kk,\kl_j}(s)=w^\ast_0\ \text{ for } \ 0\le s<t^{\kl_j}_1.$$

At time $t^{\kl_j}_1$, we have by (<ref>)

{\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}\rk)(t_1^{\ell_j})
%{\mathcal{T}}_{\kappa }\lk(\tilde  w^\ast_{\kappa ,2^{l_j}},%\tilde r^\ast_{l_j,n,2^{\iota_j}}\rk)(t_1)
={\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j}\rk)(t_1^{\ell_j})=\tilde w^\ast_{\kappa,\ell_j ,1}(t^{\ell_j}_1).

However, by definition we have
$$\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}(t_1^{\ell_j})= \tilde w^\ast_{\kappa,\ell_j ,1}(t^{\ell_j}_1).$$

Now, we  analyse what is happening at the next time interval $[t^{\ell_j}_1,t^{\ell_j}_2)$. Note that, here  the process is constant and equals   $\PP^\ast$-a.s. the value at $t^{\ell_j}_1$, i.e.

{\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}\rk)(s)%{\mathcal{T}}_{\kappa }\lk(w^\ast_{\kappa ,2^{\kappa }}\rk)(s)
=\tilde w^\ast_{\kappa,\ell_j ,1}(s),  \quad \mbox{for} \quad t^{\ell_j}_1\le s<t^{\ell_j}_2.

Note, also that   $\PP^\ast$-a.s. we have $\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}(s)= \tilde w^\ast_{\kappa,\ell_j ,1}(s)$ for $ t^{\ell_j}_1\le s<t^{\ell_j}_2$, and, hence

\PP^\ast\lk( {\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}} (\tilde w_{\kk,\kl_j,2^{\ell_j}}^\ast )(s) =\tilde  w^\ast_{\kappa ,\kl_j,2^{\ell_j}}(s)\rk) =1,  \quad \mbox{for} \quad t^{\ell_j}_1\le s<t^{\ell_j}_2.

Next we analyse what happens in $t^{\ell_j}_2$. We note that by  definition  we have

{\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}\rk)(t_2^{\ell_j})
%{\mathcal{T}}_{\kappa }\lk(\tilde  w^\ast_{\kappa ,2^{l_j}},%\tilde r^\ast_{l_j,n,2^{\iota_j}}\rk)(t_1)
={\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j,1}\rk)(t_2^{\ell_j})=\tilde w^\ast_{\kappa,\ell_j ,2}(t^{\ell_j}_2),

and $\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}(t_2^{\ell_j})= \tilde w^\ast_{\kappa,\ell_j ,2}(t^{\ell_j}_2)$.

Now, by induction let us assume that in $[0,t^{\ell_j}_i)$ we have shown that

^∗( ^κ,_j_𝔄^∗,𝕎 (w̃_,_j,2^ℓ_j^∗)(s) =w̃^∗_κ,_j,2^ℓ_j(s)) =1      0≤s≤t^ℓ_j_i
Then,  since by definition, on $t_i^{\ell_j}\le s<t_{i+1}^{\ell_j}$

{\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}\rk)(t_i^{\ell_j})
%{\mathcal{T}}_{\kappa }\lk(\tilde  w^\ast_{\kappa ,2^{l_j}},%\tilde r^\ast_{l_j,n,2^{\iota_j}}\rk)(t_1)
={\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}}\lk(\tilde  w^\ast_{\kk,\kl_j,i-1}\rk)(t_i^{\ell_j})=\tilde w^\ast_{\kappa,\ell_j ,i}(t^{\ell_j}_i),

the conclusion follows by the fact that

$\tilde  w^\ast_{\kk,\kl_j, 2^{\ell_j}}(t_i^{\ell_j})= \tilde w^\ast_{\kappa,\ell_j ,i}(t^{\ell_j}_i)$.

Next, we verify the following statements with the goal to pass to the limit. We know due to the construction and the properties of the projection that

$$\text{Law}(\tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}})=\text{Law}({w}^\ast_{\kappa ,\kl_j}).$$

Then, by assumption (<ref>) we note that

* there exists a constant $C>0$ such that $ \sup_{j\in{\mathbb{N}}}  \EE^\ast \left[ \| \tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}}
\|^{ m_1}_{\mathbb{X}}\right]\le C$ and

* for any $r\in (m,m_1)$ we have

$$\lim_{j\to \infty}{{\mathbb{E}^\ast }}\left[   \| \tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}}- w_\kk^{\ast} \|_{\mathbb{X}}^r\right] = 0. $$

Indeed, since $\{  {w}^\ast_{\kappa,\kl_j }\}_{j\in\NN}\subset\Xcal^\kappa_{\mathfrak{A}^\ast}(R_\kk)$ and $\Xcal^\kappa_{\mathfrak{A}^\ast}(R_\kk)$ is bounded in $\X$,
we can conclude due to the equality of law that

\[
\EE^\ast \lk\|\tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}}\rk\|_{\X}^{r}={\Eb}^\ast \|{w}^\ast_{\kappa,\kl_j }\|_{\X}^{r},
\]

for any $r\in (m, m_1]$,
so that we get by assumption (<ref>) that

\[
\sup_{j}{\Eb}^\ast \|\tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}}\|_{\X}^{ r}=	\sup_{j}{\Eb}^\ast \|{w}^\ast_{\kappa,\kl_j }\|_{\X}^{ r}\le CR,\qquad \forall r\in [1,m_0],
\]

Hence, we know that $\{\| w^\ast_{\kk,\kl_j}\|_{\X}^{r}\}$
is uniformly integrable for any $r\in (m,m_1]$ w.r.t. the probability measure $\P^\ast$.
Moreover, by the convergence (<ref>), 	we get by the Vitali convergence theorem that

\begin{equation*}
\lim_{j\to\infty}{\Eb}^\ast\left\| {w}^\ast_{\kappa,\kl_j }-{w}_\kk^\ast\right\|_{\X}^{r}=0\label{eq:strong-v}
\end{equation*}

for any $r\in(m,m_1]$. By the equality of the laws, we then infer that

\begin{equation*}
\lim_{j\to\infty}{\Eb}^\ast\left\| \tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}}-{w}_\kk^\ast\right\|_{\X}^{r}=0.
\end{equation*}

* In this step we show that $w_\kk^\ast=\Vcal^\kappa_{\Afrak^\ast,\mathbb{W}}(w_\kk^\ast)$ and derive that over $\MA^\ast$,  the random variable  $(w_\kk^\ast, v_\kk^1(w_\kk^\ast),...,v^J(w_\kk^\ast))$ together with the Wiener process $\mathbb{W}$  is indeed a martingale solution to (<ref>) where $v_\kk^1(w_\kk^\ast),...,v_\kk^J(w_\kk^\ast)$ is obtained via $\Vcal^\kappa_{\Afrak^\ast,\mathbb{W}}(w_\kk^\ast)$.
We have the following decomposition for any $j\in \mathbb{N}$.

=   w_^∗-w̃^∗_,_j, 2^ℓ_j_:=I
w̃^∗_,_j, 2^ℓ_j-^κ,_j_𝔄^∗,𝕎( w̃^∗_,_j, 2^ℓ_j)_=:II+
^κ,_j_𝔄^∗,𝕎(w̃^∗_,_j, 2^ℓ_j)-^κ,_j_𝔄^∗,𝕎(w_^∗)_=:III
Now, we analyse $I$, $II$, $III$, and $IV$.

Note, due to the convergence we know

$$\EE\| w_\kk^\ast-\tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}}\|_{\mathbb{X}}^{m_1}\le \frac \ep 3.

Next, to tackle II, we know due to the well posedness and we have by the step before

\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}( \tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}})=\tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}}.

To tackle III, due to the continuity of the operator
$\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}$ we know that there exists  a function $\phi$ with $\lim_{x\to 0}\phi(x)=0$, such that

\EE \lk\| \Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}( \tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}})-\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}(w_\kk^\ast)\rk\|_{\mathbb{X}}^m
\le C\phi \lk(\EE\lk\| \tilde w^\ast_{\kk,\kl_j, 2^{\ell_j}}-w_\kk^\ast\rk\|_{\mathbb{X}}^{m_1}\rk)^{\frac 1m_1}

Finally, since $\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}= \Pro_{\kl_j}{\Vcal^{\kappa}_{\mathfrak{A}^\ast,\mathbb{W}}}$ and the  convergence of the projection $\Pro_{\kl_j}$, the difference

\EE  \lk\|\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}(w_\kk^\ast)-\Vcal^{\kappa,\kl_j}_{\mathfrak{A}^\ast,\mathbb{W}}(w_\kk^\ast)\rk\|_{\mathbb{X}}^m

tends to zero (see remark <ref>).
Finally, IV tends to zero, due to the continuity of the operator $\Vcal^{\kappa}_{\mathfrak{A}^\ast,\mathbb{W}}$.

As a consequence, we have

\[\Vcal^{\kappa}_{\mathfrak{A}^\ast,\mathbb{W}}(w_\kk^\ast)=w_\kk^\ast,\quad {\P}^\ast \text{-a.s.}\]

As seen above, $w_\kk^\ast\in\Xcal^\kappa_{\mathfrak{A}^\ast}(R_\kk)$,
so that by the assumption (<ref>),

$$(\Vcal^{\kappa}_{\mathfrak{A}^\ast,\mathbb{W}} (w_\kk^\ast),v^1(w_\kk^\ast),\ldots,v^\Jott(w_\kk^\ast) )\in C([0,T];U)\times C([0,T];U_1)\cdot\cdot\cdot\times C([0,T];U_J), $$

Hence for
all $t\in[0,T]$, ${\P^\ast}$-a.s.

$$(w_\kk^\ast,v^1(w_\kk^\ast),\ldots,v^\Jott(w_\kk^\ast) )(t)=(\Vcal^{\kappa}_{\mathfrak{A}^\ast,\mathbb{W}}(w_\kk^\ast),v^1(w_\kk^\ast),\ldots,v^\Jott(w_\kk^\ast) )(t),$$

and by construction, we see that $(w_\kk^\ast,v^1(w_\kk^\ast),\ldots,v^\Jott(w_\kk^\ast) )$ solves
the system (<ref>)
on $\Afrak^\ast$ and the proof is complete.

Independent gluing together of the local solutions and Extension to a global  solution:
In this step,  we will show that
there exists a martingale solution of system (<ref>).
That is, there exists a filtered probability space
$\bar{\mathfrak{A}} := (\bar\Omega, \bar{\mathcal{F}}, (\bar{\mathcal{F}}_t)_{t\geq 0}, \bar{\mathbb{P}})$,

a vector–valued Wiener process $\bar{\mathbb{W}}=(\bar{W}_0, \bar{W}_1,\ldots,\bar{W}_\Jott)$ over $\bar{\mathfrak{A}}$,
and a tuple  of processes $\bar{\mathfrak{w}}=(\bar{w},\bar{v}_1,\ldots,\bar{v}_\Jott)$ being a strong solution
of the system (<ref>)  over $\bar{\mathfrak{A}}$.

To do so, we construct a  family of solutions  $\{\bar{ \mathfrak{w}}_{\kappa} :\kappa\in\NN\}$   following the solution to the  original problem until a certain stopping time ${\bar \tau}_\kappa(\bar{ \mathfrak{w}}_\kappa)$ (being defined later on in (<ref>)). In particular,  we will introduce for each $\kappa\in\NN$ a new tuple  of processes $\bar{ \mathfrak{w}}_{\kappa} $ following the original  system up to the stopping time ${\bar \tau}_\kappa(\bar{ \mathfrak{w}}_\kappa)$. Besides, we will have

\bar{ \mathfrak{w}}_{\kappa}\big|_{\big[0,{\bar \tau}_\kappa(\bar{ \mathfrak{w}}_\kappa)\big)}=\bar{ \mathfrak{w}}_{\kappa +1}\big|_{\big[0,{\bar \tau}_\kappa(\bar{ \mathfrak{w}}_\kappa)\big )}.

Let us start with $\kappa =1$. From the steps before, we know there exists a martingale solution consisting of a probability space $\mathfrak{A}_\zahl =(\Omega_\zahl ,\CF_\zahl ,\mathbb{F}_\zahl ,\PP_\zahl )$, an  independent vector valued  Wiener process $\mathbb{W}_\zahl=(W^\zahl_0, W^\zahl_1 ,\ldots,W^\zahl_\Jott)$ defined over  $\mathfrak{A}_\zahl $, and a process $\mathfrak{w}_\zahl =(w_1,v_1^1,...,v_1^J)$ solving $\PP_\zahl $–a.s. the system

{dw_(t)    =   (_0 w_(t)+  F_0(w_(t),v^1_1(t),⋯,v_1^(t))
·Ξ^0_1(t)  )  dt +Σ_0(w_(t)) dW_0(t),

dv^_(t)    =   (_v^_(t)+ F_(w_(t),v_^(t),v_^+1(t),⋯,v_^(t))·Ξ^_1(t))  dt +Σ_(v_^(t)) dW_(t),

w(0)   =   w_0,

v^(0)   =   v^_0,  =1,…,.

{ du_1(t)    =   A u_1(t) + F(u_1(t))Θ_(u_1(t),u_1(t),t)+ σu_1(t) d W^1(t),

u_1(0)   =   u_0.
Let us define now the stopping times

\tau_\zahl^\ast(\mathfrak{w}_\zahl):=\inf \{s\ge 0\;\colon\; \max_{1\le j\le J}h_j(v^j_\zahl,s) \ge \zahl \}\wedge T,

on $\mathfrak{A}_1$.
Observe, on the time
interval $[0,\tau_\zahl^\ast(\mathfrak{w}_\zahl)))$, the process ${ \mathfrak{w}}_\zahl$ solves the  original system given in
Now, we define a new  vector valued process $\bar{ \mathfrak{w}}_\zahl$  following $\mathfrak{w}_\zahl$  on $[0,\tau_\zahl^\ast(\mathfrak{w}_\zahl) )$ and extend this
to the whole interval $[0,T]$ in the following way.

First, we put
$$\overline{\mathfrak{A}}_1:=\mathfrak{A}_1\ \text{ and } \ \bar {W}_\jott^1:={W}_\jott^1,\ \jott=0,1,2,\ldots,\Jott,$$

and let us introduce the processes $y_1$, $z^1_\zahl,\ldots, z^\Jott_\zahl$
being a strong solution over $\overline{\mathfrak{A}}_\zahl$ to

{dy_(t)    =   _0 y_(t) +Σ_0(y_(t)) d(θ_τ^∗_1( 𝔴_1)W̅^1_0)(t),  t≥0

dz^_(t)    =   _z^_(t)  dt +Σ(z_^(t)) d(θ_τ^∗_1( 𝔴_1)W̅^1_)(t),
=1,…,,  t≥0,

y_(0)   =   w( τ^∗_( 𝔴_)),
z^_(0)=v_^(τ^∗_1( 𝔴_)),  =1,…,.

Here, $\theta_\sigma$ is the shift operator which maps ${\bar{W}}_\jott(t)$ to ${\bar{W}}_\jott(t+\sigma)-\bar{W}_\jott(\sigma)$.
Following the idea of  standard Theorems (see e.g. <cit.> or Theorem 4.2.4 <cit.>) the existence of a unique solution $y_\zahl$ over $\overline{\mathfrak{A}}_1$ to the first equation is obtained. Also following   <cit.>,
the solutions to $(z^1_\zahl,z^2_\zahl,\ldots,z^\Jott_\zahl)$ are obtain by the fact that the operators $A_j$ are linear.
Let us denote by $\mathfrak{y}_1$ the tuple of processes $(y_\zahl,z_\zahl^1,\ldots,z_\zahl^\Jott)$.
Now, let us define a  process $\bar{ \mathfrak{w}}_\zahl $
being identical to ${ \mathfrak{w}}_\zahl$  on the time interval $[0,\tau^\ast_\zahl({ \mathfrak{w}_\zahl}))$ and
following the  linear part with noise, i.e.,
$$\mathfrak{y}_\zahl(\tau^\ast_1( \mathfrak{w}_\zahl)):=(y_1(\tau^\ast_1( \mathfrak{w}_\zahl)), z_1^1(\tau^\ast_1( \mathfrak{w}_\zahl)),\cdots,z_1^J( \mathfrak{w}_\zahl)),$$
In particular, let

\bar{ \mathfrak{w}}_\zahl  (t) = \bcase { \mathfrak{w}}_\zahl(t) & \mbox{ for } 0\le t< \tau^\ast_\zahl(\mathfrak{w}_1),\\
\mathfrak{y}_\zahl(\tau^\ast_1( \mathfrak{w}_\zahl))& \mbox{ for } \tau^\ast_\zahl(\mathfrak{w}_\zahl)\le  t \le T.
\ecase

We also define

Let us now construct the probability space and the processes for the next time interval. First, let $\bar{\mathfrak{w}}_\zahl(\tau^\ast_\zahl(\mathfrak{w}_\zahl))$ have probability law $\mu_\zahl$ on $U_0\times U_1\times\cdots\times U_\Jott$.

Again, from the first part, we know
there is a martingale solution consisting of a probability space $\mathfrak{A}_\zahl=(\Omega_\zahl ,\CF_\zahl ,\mathbb{F}_\zahl ,\PP_\zahl )$, an independent Wiener processes $\mathbb{W}_\zahl=(W^\zahl_0, W^\zahl_1 ,\ldots,W^\zahl_\Jott)$,
a  process $\mathfrak{w}_\zahl $ solving $\PP_\zahl $-a.s. the system

{dw_(t)    =   (_0w_(t)+  F_0(w_(t),v^1_(t),⋯,v^_(t))
·Ξ^0_(t)  )  dt +Σ_0(w_(t)) dW^2_0(t),

dv^_(t)    =   (_v^_(t)+ F_(w_(t),v_^(t),v_^+1(t),⋯,v_^(t))·Ξ^_(t))  dt +Σ_(v_^(t)) dW^2_(t),

w(0)   =   w_0,

v^(0)   =   v^_0,  =1,…,.

with initial condition

$$\mathfrak{w}_\zahl(0)=(w_0(\tau_1^\ast(\mathfrak{w}_1 ),v_1(\tau_1^\ast(\mathfrak{w}_1)),\ldots, v_1(\tau_1^\ast(\mathfrak{w}_1) )

having law $\mu_1$.

Let us define now the stopping times

$$\tau_\zahl^\ast(\mathfrak{w}_\zahl):=\inf \{s\ge 0\;\colon\; \max_{1\le j\le J}h_j(v^j_\zahl,s) \ge \zahl \}\wedge T,

on $\mathfrak{A}_\zahl$.
Let $\overline{\mathfrak{A}}_1:=(\overline{\Omega}_1,\overline{\CF}_1,\overline{\mathbb{F}}_1,\overline{\PP}_1):=\mathfrak{A}_1$, with $\overline{\mathbb{F}}_1:=(\overline{\CF}^1_t)_{t\in [0,T]}$.
Let $\overline{\Omega}_2:=\overline{\Omega}_1\times\Omega_2$, $\overline{\CF}_2:=\overline{\CF}_1\otimes \CF_2$, $\overline{\PP}_2:=\overline{\PP}_1\otimes\PP_2$ and let
$\overline{\mathbb{F}}_2:=(\overline{\CF}^2_t)_{t\in [0,T]}$, where

$$\overline{\CF}^2_t:=\bcase \overline{\CF}^1_t, & \mbox{if} \quad t<\bar{\tau}_1(\bar{ \mathfrak{w}}_1),
\\ \CF^2_{t-\bar{\tau}_1(\bar{ \mathfrak{w}}_1)}, & \mbox{if}\quad  t\ge \bar{\tau}_1(\bar{ \mathfrak{w}}_1).
\ecase

Let $\overline{\mathfrak{A}}_2:=(\overline{\Omega}_2,\overline{\CF}_2,\overline{\mathbb{F}}_2,\overline{\PP}_2)$.
Finally, let us set for $j=0, 1, 2,\cdots J,$

$$\bar{W}_j^2(t):=\bcase \bar{W}_j^1(t), & \mbox{if} \quad t<\bar{\tau}_1(\bar{ \mathfrak{w}}_1),
\\   W_j^2({t-\bar{\tau}_1(\bar{ \mathfrak{w}}_1)})+\bar{W}_j^1(\bar{\tau}_1(\bar{ \mathfrak{w}}_1)), & \mbox{if}\quad  t\ge \bar{\tau}_1(\bar{ \mathfrak{w}}_1).
\ecase

which gives a Wiener process  w.r.t. the filtration $\overline{\mathbb{F}}_2$.

Now, let us define two  processes $\bar{ \mathfrak{w}}_2 $
being identical to $\bar{ \mathfrak{w}}_1$  on the time interval $[0,\bar{\tau}_1(\bar{ \mathfrak{w}}_1))$, being identical to $\mathfrak{w}_2$ on the time interval $[ \bar{\tau}_1(\bar{ \mathfrak{w}}_1), \bar{\tau}_1(\bar{ \mathfrak{w}}_1)+ \tau_2^\ast(\bar{ \mathfrak{w}}_2))$ and
following the tuple  of processes
$$	\mathfrak{y}_2(t)=(y_2(t),z_2^1(t),\ldots,z_2^\Jott(t)), \ t\geq 0,$$

solving the following system

{dy_(t)    =   _0y_(t) +Σ_0(y_(t)) d(θ_τ̅_2(𝔴̅_2)  W̅^2_0)(t),  t≥0

dz^_(t)    =   _z^_(t)  dt +Σ_j(z_^(t)) d(θ_τ̅_2(𝔴̅_2)W̅^2_)(t),
=1,…,,  t≥0,

y_(0)   =   w( τ^∗_( 𝔴_)),
z^_(0)=v_^(τ^∗_2( 𝔴_)),  =1,…,.
on $\overline{\mathfrak{A}}_2$ with
$$\bar{\tau}_2(\bar{ \mathfrak{w}}_2)= \bar{\tau}_1(\bar{ \mathfrak{w}}_1)+\tau^\ast_2( \mathfrak{w}_2).$$

Let for $t\in[0,T]$
In particular, we define

\bar{ \mathfrak{w}}_2  (t) = \bcase { \bar{\mathfrak{w}}}_1(t) & \mbox{ for } 0\le t< \bar{\tau}_1(\bar{\mathfrak{w}}_1),
\\
{ \mathfrak{w}}_2(t-\bar{\tau}_1(\bar{\mathfrak{w}}_1)) & \mbox{ for } \bar{\tau}_1(\bar{\mathfrak{w}}_1)\le t<\bar{\tau}_2(\bar{\mathfrak{w}}_2),
\\
\mathfrak{y}_2(\bar{\tau}_2(\bar{\mathfrak{w}}_2)) & \mbox{ for } \bar{\tau}_2(\bar{\mathfrak{w}}_2)\le  t \le T.\ecase

By construction, we observe that
\bar{ \mathfrak{w}}_{1}\big|_{\big[0,{\bar \tau}_1(\bar{ \mathfrak{w}}_1)\big)}=\bar{ \mathfrak{w}}_{2}\big|_{\big[0,{\bar \tau}_1(\bar{ \mathfrak{w}}_1)\big )},

and $\bar{ \mathfrak{w}}_{1}$, $\bar{ \mathfrak{w}}_{2}$ solve the original problem (<ref>) until stopping times ${\bar \tau}_1(\bar{ \mathfrak{w}}_{1})$ and ${\bar \tau}_2(\bar{ \mathfrak{w}}_{2})$ in the filtered  probability space $\overline{\mathfrak{A}}_1$ and $\overline{\mathfrak{A}}_2$ respectively with respective associated Wiener processes $\bar{W}_j^1$ and $\bar{W}_j^2$, $j=0,1,\cdots,J$.

In the same way we will construct for any $\zahl \in\NN$ a
probability space $\mathfrak{A}_\zahl=(\Omega_\zahl ,\CF_\zahl ,\mathbb{F}_\zahl ,\PP_\zahl )$, an independent Wiener processes $\mathbb{W}_\zahl=(W^\zahl_0, W^\zahl_1 ,\ldots,W^\zahl_\Jott)$,
a  process $\mathfrak{w}_\zahl $ solving $\PP_\zahl $-a.s.

{dw_(t)    =   (_0 w_(t)+  F_0(w_(t),v^1(t)_,⋯,v^_(t))
·Ξ^0_(t)  )  dt +Σ_0(w_(t)) dW^_0(t),

dv^_(t)    =   (_v^_(t)+ F_(w_(t),v_^(t),v_^+1(t),⋯,v_^(t))·Ξ^_(t))  dt +Σ_(v_^(t)) dW^_(t),

w(0)   =   w_0,

v^(0)   =   v^_0,  =1,…,.

with initial condition

$$\mathfrak{w}_\zahl(0)=(w_0(\tau_\zahlbefore^\ast(\mathfrak{w}_\zahlbefore ),v_\zahlbefore(\tau_\zahlbefore^\ast(\mathfrak{w}_\zahlbefore)),\ldots, v_\zahlbefore(\tau_\zahlbefore^\ast(\mathfrak{w}_\zahlbefore) )

having law $\mu_\zahlbefore$, with

$$\tau_{\kk-1}^\ast(\mathfrak{w}_{\kk-1}):=\inf \{s\ge 0\;\colon\; \max_{1\le j\le J}h_j(v^j_{\kk-1},s) \ge \zahl -1\}\wedge T.

By induction, similarly, as done for $\kappa=2$ we extend  the filtered probability space and the Wienner processes as follows: We set
$\overline{\Omega}_\kk:=\overline{\Omega}_{\kk-1}\times\Omega_\kk$, $\overline{\CF}_\kk:=\overline{\CF}_{\kk-1}\otimes \CF_\kk$, $\overline{\PP}_\kk:=\overline{\PP}_{\kk-1}\otimes\PP_\kk$,
$\overline{\mathbb{F}}_\kk:=(\overline{\CF}^\kk_t)_{t\in [0,T]}$, where

$$\overline{\CF}^\kk_t:=\bcase \overline{\CF}^{\kk-1}_t, & \mbox{if} \quad t<\bar{\tau}_1(\bar{ \mathfrak{w}}_{\kk-1}),
\\ \CF^\kk_{t-\bar{\tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1})}, & \mbox{if}\quad  t\ge \bar{\tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1}).
\ecase

and let
$\overline{\mathfrak{A}}_\kk:=(\overline{\Omega}_\kk,\overline{\CF}_\kk,\overline{\mathbb{F}}_\kk,\overline{\PP}_\kk)$ and for $j=0, 1, 2,\cdots J,$

$$\bar{W}_j^\kk(t):=\bcase \bar{W}_j^{\kk-1}(t), & \mbox{if} \quad t<\bar{\tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1}),
\\   W_j^\kk({t-\bar{\tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1})})+\bar{W}_j^{\kk-1}(\bar{\tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1})), & \mbox{if}\quad  t\ge \bar{\tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1}).
\ecase

which defines a Wiener process  w.r.t. the filtration $\overline{\mathbb{F}}_\kk$.

{dy_(t,y(0),σ_)    =   _0 y_(t) +Σ_0(y_(t)) dθ_σ_W_0(t),  t≥0

dz^_(t, y(0),σ_)    =   _z^_(t)  dt +Σ(z_^(t)) dθ_σ_dW_(t),
=1,…,,  t≥0,

y_(0)   =   w( τ^∗_( 𝔴_)),
z^_(0)=v_^(τ^∗_1( 𝔴_)),  =1,…,.

We also consider the following  tuple  of processes
$$	\mathfrak{y}_\kk(t)=(y_\kk(t),z_\kk^1(t),\ldots,z_\kk^\Jott(t)), \ t\geq 0,$$

solving the following system
and following  the linearized system

{dy_(t)    =   y_(t) +Σ_0(y_(t)) d(θ_τ̅_( 𝔴_)W̅^_0)(t),  t≥0

dz^_(t)    =   _z^_(t)  dt +Σ(z_^(t)) d(θ_τ̅_(𝔴̅_)W̅^_)(t),
=1,…,,  t≥0,

y_(0)   =   w( τ^∗_( 𝔴_)),
z^_(0)=v_^(τ^∗_( 𝔴_)),  =1,…,.

afterwards on $\overline{\mathfrak{A}}_\kk$ where

\begin{equation}
\bar{\tau}_\kk(\bar{ \mathfrak{w}}_\kk)= \bar{\tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1})+\tau^\ast_\kk( \mathfrak{w}_\kk).\label{stopp_time}
\end{equation}

Finally, we construct

\bar{ \mathfrak{w}}_\kk  (t) := \bcase { \bar{\mathfrak{w}}}_{\kk-1}(t) & \mbox{ for } 0\le t< \bar{\tau}_{\kk-1}(\bar{\mathfrak{w}}_{\kk-1}),
\\
{ \mathfrak{w}}_\kk(t-\bar{\tau}_{\kk-1}(\bar{\mathfrak{w}}_{\kk-1})) & \mbox{ for } \bar{\tau}_{\kk-1}(\bar{\mathfrak{w}}_{\kk-1})\le t<\bar{\tau}_\kk(\bar{\mathfrak{w}}_\kk),
\\
\mathfrak{y}_\kk(\bar{\tau}_\kk(\bar{\mathfrak{w}}_\kk)) & \mbox{ for } \bar{\tau}_\kk(\bar{\mathfrak{w}}_\kk)\le  t \le T.\ecase

Besides, we know  that
\bar{ \mathfrak{w}}_{\kk-1}\big|_{\big[0,{\bar \tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1})\big)}=\bar{ \mathfrak{w}}_{\kk}\big|_{\big[0,{\bar \tau}_{\kk-1}(\bar{ \mathfrak{w}}_{\kk-1})\big )},

Note that, for any $\kk\in \mathbb{N}$, the processes $\buk_\zahl$  solves system (<ref>) on the time interval $[0,\bar \tau_\zahl ^\ast(\bar{\mathfrak{w}}_\zahl)]$.
By the uniform bound, we will show that for all $T>0$ with probability one we have

$$\lim\limits_{\zahl \to\infty}\bar \tau_\zahl ^\ast(\bar{\mathfrak{w}}_\zahl)\ge T.

Now, due to assumption (<ref>), we know that there exists a constant $C(T)>0$ such that all cut-off functions are uniformly bounded
independent of $\kappa\geq1$. Due to this property we will be able to extend our local solution to a global  solution:
Let us define

\bar \Omega := \prod_{\kk\in\mathbb{N}} \bar \Omega_\zahl , \quad 	\bar {\mathcal{F}} := \bigotimes_{\kk\in\mathbb{N}} \bar {\mathcal{F}}_\zahl, \quad \bar{\mathcal{F}}_t := \bigcap_{\zahl \in \mathbb{N}} \bar{\mathcal{F}}^\zahl _{t}, \ t\in [0, T], \quad  \overline{\mathbb{F}}:=(\overline{\CF}_t)_{t\in [0,T]}, \quad \bar\PP := \bigotimes_{\kk\in\mathbb{N}}\bar \PP_\zahl, \quad \bar{\mathfrak{A}} = (\bar{\Omega}, \bar{\mathcal{F}}, \bar{\BF}, \bar\PP),

$$  \bar{\mathfrak{w}}(t):=(\bar w, \bar v_1, \bar v_2,\cdots,\bar v_J):=\bcase \bar{\mathfrak{w}}_1(t), & \mbox{if} \quad 0\leq t<\bar{\tau}_1(\bar{ \mathfrak{w}}_1),
\\   \bar{\mathfrak{w}}_\kk(t), & \mbox{if}\quad  \bar{\tau}_{\kk}(\bar{ \mathfrak{w}}_\kk)\leq t< \bar{\tau}_{\kk+1}(\bar{ \mathfrak{w}}_{\kk+1}),\quad \kk\in \mathbb{N}.
\ecase

$$   \bar{W}_j(t):=\bcase \bar{W}_j^1(t), & \mbox{if} \quad 0\leq t<\bar{\tau}_1(\bar{ \mathfrak{w}}_1),
\\   \bar{W}_j^\kk(t), & \mbox{if}\quad  \bar{\tau}_{\kk}(\bar{ \mathfrak{w}}_\kk)\leq t< \bar{\tau}_{\kk+1}(\bar{ \mathfrak{w}}_{\kk+1}),\quad \kk\in \mathbb{N}.
\ecase
\qquad\text{for } j=0,1,\cdots,J,

and observe that
$$(\bar{\mathbb{W}}(t):=(\bar{W}_0(t),\bar{W}_1(t),\cdots,\bar{W}_J(t)))_{t\in [0, T]},$$
is a  Wiener processes with
respect to the filtration $\bar{\BF}$
over $\bar{\mathfrak{A}} $.

Next we will prove that the tuple  $	\left(\bar{\mathfrak{A}},
\bar{\mathbb{W}} %=(W_0,W_1,\ldots, W_\Jott )
, \bar{\mathfrak{w}}%=(w,v^1,\ldots,v^\Jott )
\right)$  is in fact a global solution of the original system (<ref>). For this aims, for any $\zahl \geq 1$, we define the set

$$\bar{\Omega}_\zahl  :=
\lk\{ \omega\in \bar{\Omega}\;\colon\;\bar\tau _\kappa(\bar{\mathfrak{w}} _\zahl)\ge T\rk\}.

By construction, we know that the  progressively measurable process $\buk$,  over $\bar{\mathfrak{A}}$  solves  the system (<ref>) up to time $T$ on the set $\bar{\Omega}_\zahl $. In particular, we have for the conditional probability

\begin{equation}\label{e:conditionalsol}
\bar{	\PP}\lk( \{ \omega\in \bar{\Omega}:  \mbox{ $\buk(\omega)$ solves systems  \eqref{spdes1} up to times $T$} \} \mid \bar{\Omega}_\zahl \rk)=1.
\end{equation}

let us set
$$\bar \Omega_0=\bigcup_{\zahl =1}^{\infty} \bar{\Omega}_\zahl.$$

Then, since by construction we have  $\bar{\Omega}_\zahl  \subset \bar{\Omega}_{\zahl +1}$, we infer that

( {

=    lim_→∞ (  { } ∩Ω̅_)

=    lim_→∞ (  {  } |Ω̅_)·( Ω̅_).

Due to (<ref>)

it remains to show that
${\lim\limits_{\zahl \to\infty}}$ $\bar{\PP}( \bar{\Omega}_\zahl)=1$.
However, we note that by construction of the positive stopping  times $\bar\tau _\kappa(\bar{\mathfrak{w}} _\zahl)$ in (<ref>), for any $\kk\geq 1$, the following injection holds

$$\left\{\omega\in \bar{\Omega}:\bar\tau _\kappa(\bar{\mathfrak{w}} _\zahl)< T \right\}\subset \left\{\omega\in \bar{\Omega}:\tau^\ast _\kappa(\mathfrak{w} _\zahl)< T \right\},$$

and therefore by definition of stopping times $\tau^\ast _\kappa(\mathfrak{w} _\zahl)$, for any $\kk\geq 1$,  we derive that

\begin{align*}
\PP\lk\{\omega\in\bar \Omega\setminus  \bar \Omega_\zahl \rk\}&=\PP\left\{\omega\in \bar{\Omega}:\bar\tau _\kappa(\bar{\mathfrak{w}} _\zahl)< T \right\}\\
&\leq \PP \left\{\omega\in \bar{\Omega}:\tau^\ast _\kappa(\mathfrak{w} _\zahl)< T \right\}\\
&\leq \PP \left\{\omega\in \bar{\Omega}:h_\jott(v^\jott_\kappa,\tau^\ast _\kappa(\mathfrak{w} _\zahl ))\geq \kk \right\}.
\end{align*}

From assumption (<ref>) we know there exists a ($\zahl $-independent) constant $C(T)>0$ and ${m_j}>0$, $j=1,2,\cdots,J$ such that we have

\max_{\jott =1,\ldots,J}\bar{\EE}\left[h^{m_\jott }_\jott(v^\jott_\kappa,t )\right]\le C(T),\quad t\in[0,T],\quad \zahl \in\NN.

Thus we have by the Markov inequality

$$\PP\lk\{\omega\in\bar \Omega\setminus  \bar \Omega_\zahl \rk\}\le \frac {C(T)}{\zahl ^{m_j} }\to 0,\qquad \mbox{ as }\zahl  \to \infty.

Hence the solution process is well defined on $\bar \Omega_0=\bigcup_{\zahl =1}^{\infty} \bar \Omega_\zahl $ with $\PP(\bar \Omega_0)=1$.
This finishes the proof. $\square$

§ APPLICATION TO STOCHASTIC CHEMOTAXIS ON FLUID MODEL

Chemotaxis is defined as the oriented movement of cells (or an organism) in response to a chemical gradient. Many different kinds of motile cells undergo chemotaxis. For example, bacteria and many
amoeboid cells move toward a food source; immune cells like macrophages and neutrophils move towards invading cells in our bodies. Other cells, connected with the immune response and wound healing, are attracted to areas of inflammation by chemical signals.
In particular, there are many processes in nature governed by chemotaxis. Besides Pfeffer, who described already chemotaxis $1884$ by sperm offern and mosses [27],
Raper studied in $1935$ chemotaxis at the cellular slime mold Dictyostelium discoideuum, a simple model of morphogenesis [29].
During its life cycle, a population of `Dicto' grows by cell division as long as there is sufficient nourishment. If food is rare and the environment living
conditions are getting poor; then the individual amoeba starts to  emit CAM
(Cyclic Adenosine Monophosphate).
Amoeba having sufficient nutriment do not strongly react on CAM. While suffering from hunger, the reaction sensitivity
to CAM increases, and with a small delay, the CAM concentration
is amplified by famished cells that, additionally by being hungry, scent CAM. By this stimulus, the amoeba enters the aggregation phase, which is characterised by all hungry amoeba coming
together, form a slug, and  ends with building a fruiting body
which finally releases the spores. Theoretical and mathematical modeling of chemotaxis
dates to the works of Patlak [24] in the $1950$s and Keller and Segel [20] in the 1970s.
The Keller-Segel model describes the aggregation phase
which models the  directed movement of microorganisms and cells stimulated by
a chemical produced by themselves.
For more details, we refer to the
surveys about chemotaxis by the compelling articles of Horstmann [16], the  summary of  Hillen et al. [15] and  the more recently
summaries of Bellomo et  al. [2] and of Perthame [26].

A fourth
work is the book of Triebel <cit.>.
Here, he treats the underlying Keller–Segel equations using general function spaces and applying similar methods as he has applied to the nonlinear heat equation, see <cit.>.
Finally, he considers the Keller-Segel model combined with the  Navier Stokes equations.
In <cit.> hyperbolic models of chemotaxis are introduced.
Chemotaxis is a typical example of cross-diffusion. Here, it is important to note that these systems are usually not of monotone type
or do not satisfy any maximum principle. On the contrary, in certain situations, a blow-up may occur.

Aerobic bacteria often live in thin fluid layers near solid-air-water contact lines. In the absence of fluid, cells can move in a straight line or exhibit random motion, while in the presence of fluid, cells may experience hydrodynamic interactions with the surrounding fluid and may undergo fluid-mediated transport, such as diffusion, advection, or convective mixing. Fluid flow can also affect the formation and stability of chemical gradients, which are essential for chemotaxis. Here, the chemo–attractant diffuses linearly, is transported in the fluid flow direction and is consumed by the cells. On the other hand, the cell's motion is
influenced by linear or nonlinear diffusion,  transported along the fluid velocity, and partially directed in the direction of the chemoattractant
The chemotaxis system describes a biological movement in which cells (e.g., bacteria) accumulate by preferentially moving towards higher concentrations of chemotactic substances. Experiments have shown that the mechanism is a chemotactic movement of bacteria, often towards a higher concentration of oxygen, which they consume, a gravitational effect on the motion of the fluid by the heavier bacteria, and a convective transport of both cells and oxygen through the water. We refer to [22, 31] for more details about its physical background.

We consider an initial–boundary value problem for the incompressible stochastic chemotaxis–Stokes equations with the porous-media-type diffusion in  a   complete probability space $\mathfrak{A}=(\Omega, \CF,\BF,\PP)$
with filtration $\BF=(\CF_t)_{t\ge 0}$  satisfying the usual conditions i.e.,
$\mathbb{P}$ is complete on $(\Omega, \CF)$,
for each $t\geq 0$, $\CF_t$ contains all $(\CF,\mathbb{P})$-null sets, and
the filtration $(\CF_t)_{t\ge 0}$ is right-continuous.
Here, the underlying spatial domain $\CO$ is a bounded domain of $\mathbb{R}^2$. Let  $W_1$, $W_2$ and $W_3$  be  three  independent cylindrical Wiener processes on $L^2(\CO)$ defined over the probability space
$\mathfrak{A}$  by
W_j(t)    =   ∑_k=1^∞φ_k 𝐰^j_k(t),  t∈[0,T],   j=1,2,3,
where $\{\mathbf{w}^j_k:k\in\NN\}$, $j=1,2,3$, are two mutually independent families of i.i.d. standard scalar Wiener processes and  $\{ \varphi_k:k\in\NN\}$ the eigenfunction of the Laplace operator $(-\Delta)$ on $L^2(\CO)$ corresponding to the associated eigenvalues  $\{\lambda_k:k\in\NN\}$. We investigate the  following coupled system

\begin{align}\label{1.1*}
\lk\{
\barray d {n} +\delta _n \bu\cdot \nabla ndt& =& \lk[ r_n\Delta   |n|^{q-1}n+\theta n- \chi \Div( n\nabla c) \rk]\, dt+g_{\gamma_1}(n) \circ dW_1,\vspace{0.2cm}
\\
d{c}+\delta _c \bu\cdot\nabla cdt & =&\lk[r_c \Delta c  -\zeta c+ \beta n \rk]\, dt+g_{\gamma_2}(c) \circ dW_2,\vspace{0.2cm}
\\
d \bu +\nabla Pdt&=&\left[r_\bu\Delta \bu
+n\star\Phi \right]dt+ \sigma_{\gamma_3}\, dW_3\vspace{0.2cm} %\mbox{and Navier Stokes (Dirchlet boundary condition}
\\
\nabla \cdot \bu& =&0
\earray \rk.
\end{align}

with boundary and initial condition conditions

\begin{equation}\label{init}
\begin{split}
&\nabla c\cdot \nu=0,\qquad \nabla n\cdot \nu=0, \qquad\bu=0, \quad\text{ on } \partial\bo,\\
&c(0)=c_0,\qquad n(0)=n_0,\qquad \bu(0)=\bu_0,
\end{split}
\end{equation}

on a filtered probability space  $\mathfrak{A}$ where the noise intensities $\gamma_j$, $j=1,2,3,$ are given positive real numbers and  the linear diffusion coefficients $g_{\gamma_1}$, $g_{\gamma_2}$ and $\sigma_{\gamma_3}$ are defined respectively   by

g_γ_j(ψ) h    :=     ψ·(-Δ)^-γ_j/2 h=∑_k=1^∞λ_k^-γ_j/2 u·φ_k h,φ_k,   ψ∈L^2(), h∈H ,  j=1,2,

σ_γ_3  h    :=     (-Δ)^-γ_3/2 h=∑_k=1^∞λ_k^-γ_3/2 φ_k h,φ_k,  h∈H.

In system (<ref>),  the unknowns are  the cell density $n$,  the concentration of the chemical signal $c$, the pressure $P$ and velocity  field  $\bu$ a  over the domain $\mathcal{O}$.
The positive  terms $r_u$ and $r_v$  are the diffusivity of the  cells and chemo-attractant respectively, the positive value   $\chi$ is the chemotactic sensitivity,
and  the positive constant $\zeta$ is the so-called damping constant.  The given function $\Phi=(\phi_1,\phi_2)$  represents the kernel of  gravitational potential.  We  then  recall that the system (1.1) is equivalent to the following Itô system

\begin{align}\label{1.1}
\lk\{
\barray d {n} +\delta _n \bu\cdot \nabla ndt& =& \lk[ r_n\Delta   |n|^{q-1}n+\theta n- \chi \Div( n\nabla c) \rk]\, dt+g_{\gamma_1}(n)  dW_1,\vspace{0.2cm}
\\
d{c}+\delta _c \bu\cdot\nabla cdt & =&\lk[r_c \Delta c  -\alpha c+ \beta n \rk]\, dt+g_{\gamma_2}(c) dW_2,\vspace{0.2cm}
\\
d \bu +\nabla Pdt&=&\left[r_\bu\Delta \bu
+n\star\Phi \right]dt+ \sigma_{\gamma_3}\, dW_3 ,\vspace{0.2cm}%\mbox{and Navier Stokes (Dirchlet boundary condition}
\\
\nabla \cdot \bu& =&0,
\earray \rk.
\end{align}

where $\alpha=\zeta-\frac{1}{2}\gamma^2_2$ and  $\theta=\frac{1}{2}\sum_{k=1}^{\infty}\lambda_k^{-\gamma_1}$.

We now introduce the following spaces

\begin{align*}
&\mathcal{V} =\left\{ \bu\in \mathcal{C}_{c}^{\infty }(\bo,\mathbb{R}^2)\,\,\text{such that}%
\,\,\nabla\cdot \bu=0\right\}, \\
&\mathbf{L}^2(\bo)=\,\,\text{closure of $\mathcal{V}$ in }\,\,(L^{2}(\bo))^2 ,\\
&	\mathbf{H} =\,\,\text{closure of $\mathcal{V}$ in }\,\,(H_0^{1}(\bo))^2 ,\qquad\text{ and }\qquad\mathbf{V}=	\mathbf{H}\cap (H^{2}(\bo))^2.
\end{align*}

We endow $\mathbf{L}^2(\bo)$ with the scalar product and norm of $(L^2(\bo))^2$. As
usual we equip the space $\mathbf{H}$ with the the scalar product
$(\nabla \bu, \nabla \bv)_{\mathbf{L}^2}$ which, owing to the Poincaré inequality, is equivalent to the
$(H^1(\bo))^2$-scalar product. Let $\Pi: (L^2(\bo))^2 \rightarrow \mathbf{L}^2(\bo)$ be the Helmholtz-Leray projection. We denote by
Stokes operator with domain $D(A)=\mathbf{V}$.

In case of our setting, we define the notion of martingale solution to our problem as follows.

Fix $T>0$. A martingale solution to the problem
(<ref>)  is given by the tuple

\left(\Omega ,{{\mathcal{F}}},\mathbb{P},{\mathbb{F}},

such that

* $\mathfrak{A}:=(\Omega ,{{\mathcal{F}}},{\mathbb{F}},\mathbb{P})$ is a complete filtered
probability space with a filtration ${\mathbb{F}}=\{{{\mathcal{F}}}_t:t\in [0,T]\}$ satisfying the usual conditions,

* $W_1$, $W_2$ and $W_3$ are three cylindrical Wiener processes on $H$ over the probability space
$\mathfrak{A}$, and

* $n:[0,T] \times \Omega \rightarrow H^{-1}_2(\CO)$, and $c:[0,T] \times \Omega \rightarrow
L^2(\CO)$, and $\bu:[0,T] \times \Omega \rightarrow\nsH$
are $\BF$–progressively  measurable processes  such that $\PP$-a.s.
$$(n,c,\bu)\in C([0,T];H^{-1}_{2}(\CO))\times  C([0,T];L^2(\CO))\times  C([0,T];\mathbf{L}^2(\CO)),$$

and satisfy for all $t\in [0,T]$ the following stochastic integral equations

\begin{align}\label{eq:nStratonovich:app}
n(t)+\int_0^ t\bu(s)\cdot\nabla n(s)\, ds&=n_0+\int_0^t (r_u\Delta  \abs{n(s)}^{q-1}n(s) +\theta n(s)-\chi\mbox{div}( n(s) \nabla c(s) )\,ds\notag\\
&\qquad+\int_0^t g_{\gamma_1}(n(s)) dW_1(s),
\end{align}

\begin{equation}\label{eq:cStratonovich:app}
c(t)+\int_0^ t\bu(s)\cdot\nabla c(s)\, ds=c_0+\int_0^t (r_c\Delta c(s)-\alpha c(s)+\beta n(s))\,ds+
\int_0^t g_{\gamma_2}(c(s))dW_2(s),
\end{equation}

\begin{equation} \label{eq:buStratonovich:app}
\bu(t) -r_\bu\int_0^ t A \bu(s)ds=\int_0^ t n(s)\star\Phi ds+  \int_0^ t\sigma_{\gamma_3} %\int_0^ t\circ
\end{equation}

where the equations (<ref>), (<ref>) and (<ref>) hold in $H^{-1}_{2}(\bo)$, $L^2(\CO)$ and $\mathbf{L}^2(\bo)$ respectively.

In what follows, we gather   assumptions that we will need for the proof of the main result of this section.

We will assume that

(A1) The noise  intensity $\gamma_j$, $j=1,2,3$, satisfy $\gamma_1>2$, $\gamma_2>2$ and $\gamma_3>1$.

(A2) $q>4$ and the kernel of gravitational potential $\Phi=(\phi_1,\phi_2)$ where $\phi_i=e^{-\delta_i\Delta}$, with some fixed  real numbers $\delta_i>0$, $i=1,2$.

(A3) The initial condition $(n_0,c_0,\bu_0)$ is  a $H^{-1}_2(\bo)\times H^1_2(\bo)\times\nsH$-valued   $\CF_0$-measurable random variable   satisfying

\begin{equation}
\EE|n_0|_{H^{-1}_2}^\frac{4m^*}{r^*}<\infty,\  \EE\abs{c_0}^\frac{8m^*}{r^*}_{H_2^1}<\infty,\  \EE\abs{\bu_0}^{8\frac{m^*}{r^*}}_\nsH<\infty,\label{3.7}
\end{equation}

where $r^\ast$ and $m^\ast$ are real numbers  such that

\begin{align*}
&q\leq r^*<q+1\quad \mbox{and}\quad   m^*\geq 2q+2.
\end{align*}

Assumption (A1) is needed to insure that  $|\sigma_{\gamma_3}|_{L_{HS}(L^2,\nsH)}<\infty$ and also the existence of  a constant $C>0$ such that

\begin{align}
|g_{\gamma_1}(\psi)|_{L_{HS}(L^2,H^{-1}_2)}^2\le  C|\psi|_{H^{-1}_2} ^2 , \quad \psi\in L^2(\CO), \label{eq:H-1-2q}\\
&	|g_{\gamma_2 }(\psi)|_{L_{HS}(L^2,L^2)}^2\le C |\psi|^2_{L^2}, \quad \psi\in L^2(\CO),	\label{eq:Hil-Schi2}
\end{align}

where $L_{HS}(L^2,E)$ denote the space of Hilbert-Schmit operators defined from $L^2(\bo)$ into $E$, with $E$ either $\nsH$, $L^2(\bo)$ or  $H^{-1}_2(\bo)$.
We refer to  Appendix <ref> for more explanation.

The main result of this section is stated in the following theorem.

Let  Assumption <ref> holds. Then
the system (<ref>)-(<ref>) has a martingale solution in the sense of Definition <ref>.
In addition, for all $p\in [1,\frac{2m^*}{r^*}]$ there exists a constant $C>0$ such that

\begin{align}\label{3.8}
&	\EE	\sup_{0\le s\le  {T} }	\abs{\ukk(s)}^{2p}_\nsH+\EE\sup_{0\le s\le T}	\abs{c_k(s)}^{2p}_{L^2}+\EE\sup_{0\le s\le  {T}  }|\nk(s)|_{H^{-1}_2}^{2p}\notag\\
&+\EE\left(\int_0^{T} \abs{\nk(s)}^{q+1}_{L^{q+1}} ds\right)^p+\EE\left(\int_0^T\abs{ \ckk(s)}^2_{H^1_2}ds\right)^p+\EE\left(\int_0^T\abs{\ukk(s)}^2_\nsV ds\right)^p\notag
\\
&	\leq e^{CT}\left(1+ \EE\abs{\bu_0}^{2p}_\nsH+\EE\abs{c_0}^{2p}_{H_2^1}+\EE |n_0|_{H^{-1}_2}^{2p}\right).
\end{align}

The proof will use the Meta Theorem <ref>. Here, we will only prove that the assumptions  (<ref>)
to (<ref>) and the bound in (<ref>) are satisfied.

The proof consists of three steps. In the first step, we will define the underlying spaces and the operator.
In the second step, we will
show that the assumptions  (<ref>)
to (<ref>) and the bound in <ref>  of Theorem <ref> are satisfied.
In the third step, we will apply the Theorem <ref>.
The estimates regarded in (<ref>) will give the regularity property of the solution.

* In the first step, we will define the underlying spaces, the argument of the cut-off function, the operator, and the subspace on which the operator acts.
Let us denote by $\X$ the  space
$$\X= L^{m^\ast}(0,T;H^{-s^{**}}_{r^*}(\CO)), \text{ where }  s^{**}=\frac{2}{q+1},$$
and by $\CM^{m^\ast}_\MA(\X)$ the space  being defined by
^m^∗_()   :=   { ξ:×[0,T]×Ω→,

Let us define the arguments  of the cut-off  function by

The cut-off function is given by the functions $\phi_\kk$, defined by the relation (<ref>)
Θ_κ(,t)   =   ϕ_κ(h(,t)),

Define for some real number $R>0$,  the set $\Xcal_\kappa^{\MA}=\Xcal_\kappa^\MA(R)$, where

\begin{equation*}
\Xcal^\kappa_{\Afrak}(R):=\lk\{\xi\in \CM^{m^\ast}_\MA(\X):\
\EE\sup_{0\le s\le T}|\xi(s)|_{H^{-1}_2}^{4\frac{m^*}{r^*}}+ \EE\left(\int_0^T |\xi(s)|_{L^{q+1}}^{q+1}\, ds \right)^{2\frac{m^*}{r^*}}\le R
%sup_{0\le s\le T}|\xi(t)|_{H^{-1}_2}^2\le R_2
\rk\}.
\end{equation*}

We recall that  since $m^\ast>q+1$, $2<r^\ast<q+1$ and $s^{\ast\ast}\in (0,1)$, we have $\frac{m^\ast}{r^\ast}>1$. By Proposition <ref>,  we can derive that for all $R>0$, the set $\Xcal^\kappa_{\Afrak}(R)$ is a bounded convex set in $\CM^{m^\ast}_\MA(\X)$. In fact, for $\xi\in \Xcal^\kappa_{\Afrak}(R)$ it follows by  Proposition <ref>, that we have

\begin{align}
\EE\|\xi\|_{\BX}^{m^\ast}&\leq C \EE\sup_{0\le s\le T}|\xi(s)|_{H^{-1}_2}^{2\frac{m^*}{r^*}}+ C\EE\left(\int_0^T |\xi(s)|_{L^{q+1}}^{q+1}\, ds \right)^{\frac{m^*}{r^*}}\notag\\
&\leq C\left(\EE\sup_{0\le s\le T}|\xi(s)|_{H^{-1}_2}^{4\frac{m^*}{r^*}}\right)^\frac{1}{2}+ C\left(\EE\left(\int_0^T |\xi(s)|_{L^{q+1}}^{q+1}\, ds \right)^{2\frac{m^*}{r^*}}\right)^\frac{1}{2}\notag\\
&\leq CR^\frac{1}{2}\notag.
\end{align}

* We will show that the framework satisfies the assumptions of Theorem <ref>.  For this aim,  let us consider for any integer $\kk\geq 1$, the mapping
$$\CV_{\MA,\mathbb{W}}^\kappa:\CM^{m^\ast}_\MA(\X)\to \CM^{m^\ast}_\MA(\X),$$
defined by for $\xi \in \CM^{m^\ast}_\MA(\X)$,  $\CV_{\MA,\mathbb{W}}^\kappa(\xi):=n_\kappa$, where the triple $(n_\kappa,\ckk,\bu_\kappa)$ solves the following system.

{dn_κ(t)     =    (r_nΔ|n_κ(t)|^q-1n_κ(t)+θξ(t)- χ( ξ(t) ∇(t)))  dt

_κ(t) ·∇ξ(t)  dt+g_γ_1(n_κ(t))   dW_1(t),|

d(t)    =   ( r_c Δ(t)  -α(t)+ β
ξ(t) )  dt

+ Θ_κ(_κ,t)_κ(t) ·∇(t)  dt+ g_γ_2((t))  dW_2(t),|

d _κ(t)   =   r_ A _κ(t)
dt+ξ(t)⋆Φdt+ σ_γ_3 dW_3(t),.

with initial condition $n_\kappa(0)=n_0$, $\ckk(0)=c_0$, and $\bu_\kappa(0)=\bu_0$.

Note, that $\ckk$ and $\bu_\kappa$ does not depend on $\nk$ but on $\xi$. In particular, the system (<ref>) is split.

Verification of conditions (<ref>)  and (<ref>) of Theorem <ref>: Here, we will show that the mapping $\CV_{\MA,\mathbb{W}}^\kappa$ is well defined and maps element of $ \CX_\MA(R)$ into itself for a certain $R>0$.
In particular, to verify assumption (<ref>),  we will show that for any $R>0$, any $\kappa\in\NN$, any initial conditions $(n_0,c_0,\bu_0)$ satisfying estimates (<ref>)
and any $\xi\in  \CM^{m^\ast}_\MA(\X)\cap\CX_\kappa(\MA)$, there exists a triple of processes $(n_\kappa,\ckk,\bu_\kappa)$  solving system
By the a priori estimates, which we will prove in the following claim,  condition (<ref>)  of Theorem <ref> follows.

Let Assumption <ref> be satisfied and let  $R>0$ be arbitrary. For
any $\xi\in \CM^{m^\ast}_{\mathfrak{A}}(\mathbb{X})\cap\CX(R)$, there exists a unique triple of processes $(n_\kappa,\ckk,\bu_\kappa)$  solving system (<ref>) such that $n_\kappa\in \CM^{m^\ast}_{\mathfrak{A}}(\mathbb{X})$.
In addition, for any $\kappa\in\NN$, there exists some constant $C=C(\kappa,m^*,r^*,T,q)$ such that the following estimates are satisfied:

\begin{equation}
\EE	\sup_{0\le s\le  {T} }	\abs{\ukk(s)}^{8\frac{m^*}{r^*}}_\nsH+r_u^{4\frac{m^*}{r^*}}\EE\left(\int_0^T\abs{\ukk(s)}^2_\nsV ds\right)^{4\frac{m^*}{r^*}}\leq C\EE\abs{\bu_0}^{8\frac{m^*}{r^*}}_\nsH+ CR^\frac{4}{q+1}+C,\label{3.14*}
\end{equation}

\begin{equation}
\EE\sup_{0\le t\le T}\abs{c_k(t)}^\frac{8m^*}{r^*}_{H_2^1}+C^\frac{4m^*}{r^*}_1\EE\left(\int_0^T\abs{ \ck(s)}^2_{H^2_2}ds\right)^\frac{4m^*}{r^*}\leq C\EE\abs{c_0}^\frac{8m^*}{r^*}_{H_2^1}+CR^\frac{4}{q+1},\label{3.12}
\end{equation}

\begin{align}
&\frac{1}{2}\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{4m^*}{r^*}+(2r_n)^\frac{2m^*}{r^*}\EE\left(\int_0^{T} \abs{\nk(s)}^{q+1}_{L^{q+1}} ds\right)^\frac{2m^*}{r^*}\label{3.12*}\\
&\leq C\EE|n_0|_{H^{-1}_2}^\frac{4m^*}{r^*}+C\EE\abs{\bu_0}^{8\frac{m^*}{r^*}}_\nsH+C\EE\abs{c_0}^\frac{8m^*}{r^*}_{H_2^1}+C+CR^\frac{2}{q+1}+CR^\frac{4}{q+1}.\notag
\end{align}

_C([0,T];H^2 q/q+1 _q+1)^q+1,∇_L^q+1(0,T;H^1_q+1)^q+1≤C_6(|c_0|_H^2 q/q+1_2
sup_0≤s≤T _L^2^2≤C_7(|c_0|_H^-1_2

where $C_1$ is a constant to be given below.

From the estimates in Claim <ref> follows assumption (<ref>) of Theorem <ref>.
To see this, let us for the time being  assume that Claim <ref> is true. Then, from the estimate (<ref>), for all $\kappa\in\NN$ and  for all $R>0$
such that
R   ≥

we know that for any $\xi\in \CX_\MA(R)$ we have  $\CV_\MA^\kappa(\xi)\in  \CX_\MA(R)$ (since $\frac{2}{q+1}<1$ and $\frac{4}{q+1}<1$ ).
In particular, the operator $\CV_\MA^\kappa$ maps element of $  \CX_\MA(R)$ into itself.

Let us fix $R>0$ and $\xi\in \CX_\MA(R)$.
We will start with the last estimate and will end with the first estimate.
In the first step we consider the system

\begin{align}
\label{a3}
& d \bu _\kappa(t) - r_{\bu}A \bu_\kappa(t)dt
=\xi(t)\star\Phi dt+  \sigma_{\gamma_3} \, dW_3(t),
\end{align}

with initial condition $ \bu(0)=\bu_0$.
Here, the existence and uniqueness of a solution are given by standard methods; see, e.g., <cit.>.

By the Itô formula to the process $t\mapsto\abs{\ukk}^2_\nsH$, we infer that

\begin{equation*}
\abs{\ukk(t)}^2_\nsH+2r_u\int_0^t\abs{\ukk(s)}^2_\nsV ds=\abs{\bu_0}^2_\nsH+2\int_0^t(\nabla(\xi(s)\star\Phi) ,\nabla \ukk(s))ds+t\abs{\sigma_{\gamma_3}}^2_{L_{HS}(H,\nsH)}+2\int_0^t(\nabla\ukk(s),\nabla\sigma_{\gamma_3} ) dW_3(s).
\end{equation*}

We note that

\begin{align}
(\nabla(\xi\star\Phi) ,\nabla \ukk)&\leq \abs{\xi \star\Phi}_{\mathbf{L}^2}\abs{\nabla \ukk}_{\mathbf{L}^2}\notag
\leq C\abs{ \ukk}^2_{\nsH}+\abs{\nabla(\xi\star\Phi)}^2_{\mathbf{L}^2}\notag\\
&= C\abs{ \ukk}^2_{\nsH}+\sum_{i=1}^2	|\nabla(\xi\star \phi_i)|^2_{L^2}.\notag
\end{align}

This implies that

\begin{align*}
\sup_{0\le s\le  {T} }	\abs{\ukk(s)}^2_\nsH+r_u\int_0^T\abs{\ukk(s)}^2_\nsV ds&\leq\abs{\bu_0}^2_\nsH+ 2\int_0^T|\nabla(\xi(s)\star \Phi)|^2_{\mathbf{L}^2}ds+C\int_0^T\abs{\ukk(s)}_\nsH^2ds\\
&\qquad+T\abs{\sigma_{\gamma_3}}^2_{L_{HS}(H,\nsH)}+2\sup_{0\le t\le T}\abs{\int_0^t(\nabla\ukk(s),\nabla\sigma_{\gamma_3} ) dW_3(s)}.
\end{align*}

From the point (A1) of Assumption <ref> and some properties of  semigroup theory (see, e.g. <cit.>), we know that

\begin{align}
|\nabla(\xi\star \Phi)|^2_{\mathbf{L}^2}=\sum_{i=1}^2	|\nabla(\xi\star \phi_i)|^2_{L^2}&\leq \sum_{i=1}^2	|\xi\star \phi_i|^2_{H^1_2}= \sum_{i=1}^2	\abs{e^{-\delta_i\Delta}\xi}_{H^1_2}^2\notag\\
&\leq C \sum_{i=1}^2\abs{e^{-\delta_i\Delta}}_{L(H^{-1}_2;H^1_2)}^2\abs{\xi}^2_{H^{-1}_2}\notag\\
&\leq  C\abs{\xi}^2_{H^{-1}_2}\sum_{i=1}^2\abs{-\Delta e^{-\delta_i\Delta}}^2_{L(H^{-1}_2;H^{-1}_2)}\notag\\
&\leq C\abs{\xi}^2_{H^{-1}_2}\sum_{i=1}^2\frac{C}{\delta_i^{1/2}} \notag.
\end{align}

We then infer that

\begin{align*}
&\EE	\sup_{0\le s\le  {T} }	\abs{\ukk(s)}^{8\frac{m^*}{r^*}}_\nsH+r_u^{4\frac{m^*}{r^*}}\EE\left(\int_0^T\abs{\ukk(s)}^2_\nsV ds\right)^{4\frac{m^*}{r^*}}\\
&\leq\EE\abs{\bu_0}^{8\frac{m^*}{r^*}}_\nsH+ C(\delta_1,\delta_2)\EE\left(\int_0^T\abs{\xi(s)}^2_{H^{-1}_2}ds\right)^{4\frac{m^*}{r^*}}+C\\
&\qquad+ C\EE\left(\int_0^T\abs{\ukk(s)}^2_{\nsH}ds\right)^{4\frac{m^*}{r^*}}+C\EE\sup_{0\le t\le T}\abs{\int_0^t(\nabla\ukk(s),\nabla\sigma_{\gamma_3} ) \, dW_3(s)}^{4\frac{m^*}{r^*}}.
\end{align*}

Since $q>3$, we know that  $\frac{4}{q+1}<1$. Therefore, by the Hölder inequality  and the embedding  of $L^{q+1}(\CO)$ into $H^{-1}_2(\CO)$,  we derive that

\begin{align}
C\EE\left(\int_0^T\abs{\xi(s)}^2_{H^{-1}_2}ds\right)^{4\frac{m^*}{r^*}}&\leq C(T)C\EE\left(\int_0^T\abs{\xi(s)}^{q+1}_{L^{q+1}}ds\right)^{\frac{8m^*}{(q+1)r^*}}\notag\\
&\leq C(T)\left(\EE\left(\int_0^T\abs{\xi(s)}^{q+1}_{L^{q+1}}ds\right)^{2\frac{m^*}{r^*}}\right)^{\frac{4}{q+1}}\notag\\
&\leq C(T)R^\frac{4}{q+1}.\notag
\end{align}

By the Burkholder-Gundy-Davis inequality and Young inequality, we obtain

sup_0≤t≤T∫_0^t(∇(s),∇σ_γ_3 ) dW_3(s)^4m^*/r^*

(∫_0^T(s)^2_σ_γ_3^2_L_HS(H,) ds)^2m^*/r^*
σ_γ_3^4m^*/r^*_L_HS(H,) (∫_0^T(s)^2_ds)^2m^*/r^*

≤    C∫_0^T(s)^4m^*/r^*_ds≤C∫_0^T(s)^8m^*/r^*_ds+C.In addition, the Hölder inequality gives

$$\EE\left(\int_0^T\abs{\ukk(s)}^2_\nsH ds\right)^{4\frac{m^*}{r^*}}\leq \EE\int_0^T\abs{\ukk(s)}^{8{\frac{m^*}{r^*}}}_\nsH ds.$$

Combining these four  last inequalities we arrive at

\begin{equation*}
\EE	\sup_{0\le s\le  {T} }	\abs{\ukk(s)}^{8\frac{m^*}{r^*}}_\nsH+r_u^{4\frac{m^*}{r^*}}\EE\left(\int_0^T\abs{\ukk(s)}^2_\nsV ds\right)^{4\frac{m^*}{r^*}}\leq\EE\abs{\bu_0}^{8\frac{m^*}{r^*}}_\nsH+ CR^\frac{4}{q+1}+C+C\EE \int_0^T\abs{\ukk(s)}^{8\frac{m^*}{r^*}}_\nsH ds.
\end{equation*}

Applying the Gronwall Lemma, we obtain the inequality (<ref>).

As next, we consider the equation

\begin{align}
\label{sysv}
d{\ck }(t)& =\lk( r_c  \Delta \ck (t)  -\alpha \ck (t)+ \beta %\Theta^1_\kappa(\ckk,t)
\xi(t)\rk)\, dt
\\\notag
&{} + \Theta_\kappa (\bu,t)\bu_\kappa(t)\cdot \nabla c_\kappa(t) \, dt +\ g_{\gamma_2}(\ck (t)) dW_2(t),\quad \ck (0)=c_0.
\end{align}

To show existence and uniqueness of a solution, we will verify that   the assumptions (H1), (H2$^{\prime}$), (H3), and  (H4$^{\prime}$) of Theorem 5.1.3 in <cit.> are satisfied. For this aim, let us consider the Gelfand triple

H^1_2(\CO)\subset  L^2(\CO)\subset H^{-1}_2(\CO),

and recall that we have

\begin{equation}
_{H^{-1}_2}\langle c,w \rangle_{H^1_2}=(c,w), \ \ \text{ for all } \  \ c\in   H^{-1}_2(\CO) \text{ and } \ w\in H^{1}_2(\CO).\notag
\end{equation}

For $t\in [0,T]$, we define an operator $A_1(t,\cdot):H^{1}_2(\CO)\to H^{-1}_2(\CO)$ by
=r_c  Δc  -αc+ βξ(t) + Θ_κ(,t)_κ(t)·∇c,  c∈H^1_2(),

where $\bu_\kappa$ solves (<ref>).

Since $A_1(t,\cdot)$ is affine, then for all $c_1$, $c_2$ and  $w$ in $H^1_2(\CO)$,  the application

$$\lambda\longmapsto(A_1(t,c_1+\lambda c_2),w)$$

is continue on $\mathbb{R}$. Hence the assumption (H1) is satisfy.

Now we prove the assumption (H2$^{\prime}$). We fix $c_1$ and $c_2$ in $H^1_2(\CO)$ and note that

\begin{align}
&\leq C\abs{c_1-c_2}_{L_2}^2\notag,
\end{align}

and the local monotonicity assumption  (H2$^{\prime}$) follows. We note that we have used the fact that after an integration-by-part
$$\Theta_\kappa (\bu,t)(\bu_\kappa(t)\cdot \nabla (c_1-c_2),c_1-c_2)=\frac{1}{2}\Theta_\kappa (\bu,t)(\bu_\kappa(t)\cdot \nabla (c_1-c_2)^2,1)=0.$$

To prove assumption (H3), we fix $c\in H^1(\CO)$ and use the fact that $\Theta_\kappa (\bu,t)(\bu_\kappa(t)\cdot \nabla c,c)=0$ to  note that

\begin{align}
&\leq C\abs{c}_{L_2}^2-(r_c+\alpha)\abs{c}_{H^1_2}^2+C(\beta)\abs{\xi(t)}^2_{L^2}\notag\\
&\leq C\abs{c}_{L_2}^2-\theta_1\abs{c}_{H^1_2}^{\alpha_1}+f_1(t),\notag
\end{align}

with $\theta_1=r_c+\alpha$ and  $f_1(t)=C(\beta)\abs{\xi(t)}^2_{L^2}.$ Since $f_1\in L^1(\Omega\times[0,T];\mathbb{R})$, the coercivity assumption (H3) follow with $\alpha_1=2$.

For the assumption (H4$^{\prime}$),  we fix  $c\in H^1(\CO)$ and note that for all $w\in  H^1(\CO)$, we have

\begin{align}
(A_1(t,c),w)&=(r_c  \Delta c  -\alpha c+ \beta
\xi(t) + \Theta_\kappa (\ukk,t) \bu_\kappa(t)\cdot \nabla c,
&\leq -\theta_1\abs{c}_{H^1_2}\abs{w}_{H^1_2}+\beta\abs{\xi}_{L^2}\abs{w}_{L^2}+\Theta_\kappa (\ukk,t)\abs{\ukk(t)}_{\mathbf{L}^4}\abs{\nabla c}_{L^2}\abs{w}_{L^4}\notag\\
&\leq -\theta_1\abs{c}_{H^1_2}\abs{w}_{H^1_2}+\beta\abs{\xi}_{L^2}\abs{w}_{L^2}+\Theta_\kappa (\ukk,t)\abs{\ukk(t)}_{\mathbf{L}^4}\abs{\nabla c}_{L^2}\abs{w}_{L^4}\notag\\
&\leq \beta\abs{\xi}_{L^2}\abs{w}_{L^2}+\Theta_\kappa (\ukk,t)\abs{\ukk(t)}_{\nsH}\abs{\nabla c}_{L^2}\abs{w}_{H^1_2}\notag\\
&\leq (\beta\abs{\xi}_{L^2}+C(\kappa)\abs{\nabla c}_{L^2})\abs{w}_{H^1_2}\notag.
\end{align}

This implies that  for $\alpha_1=2$,

\begin{align}
\abs{A_1(t,c)}^{\frac{\alpha_1}{\alpha_1-1}}_{H^{-1}_2}&\leq C(\beta)\abs{\xi}^2_{L^2}+C(\kappa)\abs{c}^2_{H^1_2}\notag\\
&\leq f_1(t)+C(\kappa)\abs{c}^2_{H^1_2}\notag\\
&\leq (f_1(t)+C(\kappa)\abs{c}^2_{H^1_2})(1+\abs{c}_{L^2}^2)\notag,
\end{align}

and the growth condition  (H4$^{\prime}$) follows.
Since the above conditions are satisfy, we invoke Theorem 5.1.3 in  [23] to derive that  equation (<ref>) has a unique solution such that  for all $p\geq 2$ we have

$$\EE \sup_{0\le s\le  {T}  }\abs{c_\kappa(s)}_{L^2}^p<\infty.$$

Now we prove the a'priori estimate for $c_\kappa$. For this purpose,   we apply the Itô formula to the function $\phi(c_k):=\abs{c_k(t)}^2_{H_2^1}$ to derive an estimate for
$\abs{c_k(t)}^2_{H_2^1}$. Doing so, we obtain

\begin{align}
\abs{c_k(t)}^2_{H_2^1}&=\abs{c_0}^2_{H_2^1}+2\int_0^t( r_c \Delta \ckk(s)  -\alpha \ckk(s), \ckk(s))_{H_2^1}ds+ 2\beta
\int_0^t( \xi(s) ,c_k(s))_{H_2^1}ds\notag\\
&\quad+2\delta_c\int_0^t( \Theta_\kappa(\bu_\kappa,s)\bu_\kappa (s) \cdot\nabla \ckk(s),c_k(s))_{H_2^1}ds\label{3.14}\\
&\quad+\int_0^t\abs{g_{\gamma_2}(\ckk(s)) }^2_{L_{HS}(H_1;H_2^1)}ds+2\int_0^t(c_k(s),g_{\gamma_2}(\ckk(s)) dW_2(s))_{H_2^1}\notag
\end{align}

Using  integration-by-part and the Young inequality we derive that

\begin{align}
2( r_c \Delta \ckk  -\alpha \ckk, \ckk)_{H_2^1}&=-2(r_c+\alpha)\abs{\nabla \ck}^2_{L^2}-2\alpha\abs{ \ck}^2_{L^2}-2r_c\abs{\Delta \ck}^2_{L^2}\notag\\
&\leq-2\alpha\abs{ \ck}^2_{H^1_2}-2r_c\abs{\Delta \ck}^2_{L^2},\notag
\end{align}

\begin{align}
2\beta( \xi ,c_k)_{H_2^1}&=2\beta( \xi ,c_k)_{L^2}-2\beta( \xi ,\Delta c_k)_{L^2}\notag\\
&\leq C\abs{ \xi}^2_{L^2}+\eps_1\abs{ \ck}^2_{H^1_2}+\eps_1\abs{ \Delta\ck}^2_{L^2}\notag\\
&\leq C\abs{ \xi}^2_{L^2}+\eps_1\abs{ \ck}^2_{H^2_2}.\notag
\end{align}

In a similar way, using in addition the fact that $( \Theta_\kappa(\bu_\kappa)\bu_\kappa  \cdot\nabla \ckk,c_k)_{L^2}=0$ and the embedding of $\nsH$ into $\mathbf{L}^4(\CO)$,  we have

\begin{align}
2( \Theta_\kappa(\bu_\kappa)\bu_\kappa  \cdot\nabla \ckk,c_k)_{H_2^1}&=2\delta_c( \Theta_\kappa(\bu_\kappa)\bu_\kappa  \cdot\nabla \ckk,c_k)_{L^2}-2\delta_c( \Theta_\kappa(\bu_\kappa)\bu_\kappa  \cdot\nabla \ckk,\Delta c_k)_{L^2}\notag\\
&\leq 2 \Theta_\kappa(\bu_\kappa)\abs{ \bu_\kappa}_{\mathbf{L}^4}\abs{ \nabla \ck}_{L^4}\abs{ \Delta\ck}_{L^2}\notag\\
&\leq 2 \Theta_\kappa(\bu_\kappa)\abs{ \bu_\kappa}_{\nsH}\abs{ \ck}^\frac{1}{2}_{H_2^1}\abs{ \ck}^\frac{1}{2}_{H_2^2}\abs{ \Delta\ck}_{L^2}\notag\\
&\leq C(\kappa)\abs{ \ck}^\frac{1}{2}_{H_2^1}\abs{ \ck}^\frac{3}{2}_{H_2^2}\notag\\
&\leq C(\kappa)\abs{ \ck}_{H_2^1}^2+\eps_1\abs{ \ck}^2_{H_2^2}\notag.
\end{align}

Since $\gamma_2>1$,  by a simple calculation, we derive that

\begin{align}
\abs{g_{\gamma_2}(\ckk) }^2_{L_{HS}(H_1;H_2^1)}&\leq C(\abs{ \ck}_{L^4}+\abs{\nabla \ck}_{L^4}^2)\notag\\
&\leq C(\abs{ \ck}^2_{H^1_2}+\abs{ \ck}^\frac{1}{2}_{H_2^1}\abs{ \ck}^\frac{1}{2}_{H_2^2})
\leq C\abs{ \ck}^2_{H^1_2}+\eps_1\abs{ \ck}^2_{H_2^2}.\notag
\end{align}

By using these estimates we infer from the equality (<ref>) that

\begin{align}
&\abs{c_k(t)}^2_{H_2^1}+2\int_0^t(\alpha\abs{ \ck(s)}^2_{H^1_2}+r_c\abs{\Delta \ck(s)}^2_{L^2})ds\notag\\
&\leq\abs{c_0}^2_{H_2^1}+3\eps_1\int_0^t\abs{ \ck(s)}^2_{H^2_2}ds+C\int_0^t\abs{ \xi(s)}^2_{L^2}ds+C(\kappa)\int_0^t\abs{ \ck(s)}^2_{H^1_2}ds\notag\\
&\quad+2\int_0^t(c_k(s),g_{\gamma_2}(\ckk(s)) dW_2(s))_{H_2^1}\notag.
\end{align}

There exists a constant $C_0>0$ such that $ \abs{ \ck}^2_{H^2_2}\leq C_0(\abs{ \ck(s)}^2_{H^1_2}+\abs{ \Delta\ck(s)}^2_{L^2})$. Therefore taking $\eps_1=C_1:=\frac{\min(\alpha,r_c)}{3}$ in the previous inequality, we derive that

\begin{align}
\abs{c_k(t)}^2_{H_2^1}+C_1\int_0^t\abs{ \ck(s)}^2_{H^2_2}ds
&\leq\abs{c_0}^2_{H_2^1}+C\int_0^t\abs{ \xi(s)}^2_{L^2}ds\notag\\
&\quad+C(\kappa)\int_0^t\abs{ \ck(s)}^2_{H^1_2}ds+2\int_0^t(c_k(s),g_{\gamma_2}(\ckk(s)) dW_2(s))_{H_2^1}\notag.
\end{align}

Taking the supremum over $[0,T]$, the  exponent $\frac{4m^*}{r^*}$ and the expectation, we arrive at

\begin{align}
&	\EE\sup_{0\le t\le T}\abs{c_k(t)}^\frac{8m^*}{r^*}_{H_2^1}+C^\frac{4m^*}{r^*}_1\EE\left(\int_0^T\abs{ \ck(s)}^2_{H^2_2}ds\right)^\frac{4m^*}{r^*}\notag\\
&\leq\EE\abs{c_0}^\frac{8m^*}{r^*}_{H_2^1}+C(m^*,r^*)\EE\left(\int_0^T\abs{ \xi(s)}^2_{L^2}ds\right)^\frac{4m^*}{r^*}+C(\kappa,m^*,r^*)\EE\left(\int_0^T\abs{ \ck(s)}^2_{H^1_2}ds\right)^\frac{4m^*}{r^*}\label{3.15}\\
&\quad+C(m^*,r^*)\EE\sup_{0\le t\le T}\abs{\int_0^t(c_k(s),g_{\gamma_2}(\ckk(s)) dW_2(s))_{H_2^1}}^\frac{4m^*}{r^*}\notag.
\end{align}

We note that since $q>3$ then $\frac{4}{q+1}<1$ and therefore we obtain by  the Hölder inequality,

\begin{align}
\EE\left(\int_0^T\abs{ \xi(s)}^2_{L^2}ds\right)^\frac{4m^*}{r^*}%&\leq \EE\left(\int_0^T\abs{ \xi(s)}^2_{L^{q+1}}ds\right)^\frac{2m^*}{r^*}\notag\\
&\leq C(T)\EE\left(\int_0^T\abs{ \xi(s)}^{q+1}_{L^{q+1}}ds\right)^\frac{8m^*}{r^*(q+1)}\notag\\
&\leq C(T)\left(\EE\left(\int_0^T\abs{ \xi(s)}^{q+1}_{L^{q+1}}ds\right)^{2\frac{m^*}{r^*}}\right)^\frac{4}{q+1}\notag.
\end{align}

Since $\xi\in \CX_\MA^\kappa$, we get

\EE\left(\int_0^T\abs{ \xi(s)}^2_{L^2}ds\right)^\frac{4m^*}{r^*}
\le C(T)R^\frac{4}{q+1}.

By the  Hölder inequality we obtain

\begin{equation}
\EE\left(\int_0^T\abs{ \ck(s)}^2_{H^1_2}ds\right)^\frac{4m^*}{r^*}\leq C(T)\EE\int_0^T\abs{ \ck(s)}^\frac{8m^*}{r^*}_{H^1_2}ds.
\end{equation}

By the Burkholder-Gundy-Davis inequality, we have

\begin{align}
&C(m^*,r^*)\EE\sup_{0\le t\le T}\abs{\int_0^t(c_k(s),g_{\gamma_2}(\ckk(s)) dW_2(s))_{H_2^1}}^\frac{4m^*}{r^*}\notag\\
&\leq C\EE\left(\int_0^T\abs{\ck(s)}^2_{H^1_2}\abs{g_{\gamma_2}(\ckk(s))}^2_{L_{HS}(H_1;H_2^1)}ds\right)^\frac{2m^*}{r^*}\notag\\
&\leq C\EE\sup_{0\le t\le T}\abs{\ck(s)}^\frac{4m^*}{r^*}_{H^1_2}\left(\int_0^T\abs{g_{\gamma_2}(\ckk(s))}^2_{L_{HS}(H_1;H_2^1)}ds\right)^\frac{2m^*}{r^*}\notag\\
&\leq\frac{1}{2}\EE\sup_{0\le t\le T}\abs{\ck(s)}^\frac{8m^*}{r^*}_{H^1_2}+C\EE\left(\int_0^T\abs{g_{\gamma_2}(\ckk(s))}^2_{L_{HS}(H_1;H_2^1)}ds\right)^\frac{4m^*}{r^*}\notag.
\end{align}

By using the definition of $g_{\gamma_2}$, we note that

\begin{align}
&\leq C\EE\left(\int_0^T(\abs{\ckk(s)}^2_{L^4}+\abs{\nabla\ckk(s)}^2_{L^4})ds\right)^\frac{4m^*}{r^*}\notag\\
&\leq C\EE\left(\int_0^T\abs{\ckk(s)}^2_{H^1_2}ds\right)^\frac{2m^*}{r^*}+C\EE\left(\int_0^T\abs{\ckk(s)}_{H^1_2}\abs{\ckk(s)}_{H^2_2}ds\right)^\frac{4m^*}{r^*}\notag\\
&\leq C\EE\left(\int_0^T\abs{\ckk(s)}^2_{H^1_2}ds\right)^\frac{2m^*}{r^*}+C\EE\left(\int_0^T(C\abs{\ckk(s)}^2_{H^1_2}+\frac{C_1}{(2C)^{\frac{r^*}{2m^*}}}\abs{\ckk(s)}^2_{H^2_2})ds\right)^\frac{4m^*}{r^*}\notag\\
&\leq C\EE\left(\int_0^T\abs{\ckk(s)}^2_{H^1_2}ds\right)^\frac{4m^*}{r^*}+\frac{C_1^\frac{2m^*}{r^*}}{2}\EE\left(\int_0^T\abs{\ckk(s)}^2_{H^2_2}ds\right)^\frac{4m^*}{r^*}\notag\\
&\leq C\EE\int_0^T\abs{ \ck(s)}^\frac{8m^*}{r^*}_{H^1_2}ds+\frac{C_1^\frac{2m^*}{r^*}}{2}\EE\left(\int_0^T\abs{\ckk(s)}^2_{H^2_2}ds\right)^\frac{4m^*}{r^*}\notag,
\end{align}

and therefore

\begin{align}
&C(m^*,r^*)\EE\sup_{0\le t\le T}\abs{\int_0^t(c_k(s),g_{\gamma_2}(\ckk(s)) dW_2(s))_{H_2^1}}^\frac{2m^*}{r^*}\notag\\
&\leq\frac{1}{2}\EE\sup_{0\le t\le T}\abs{\ck(s)}^\frac{8m^*}{r^*}_{H^1_2}+\frac{C_1^\frac{2m^*}{r^*}}{2}\EE\left(\int_0^T\abs{\ckk(s)}^2_{H^2_2}ds\right)^\frac{4m^*}{r^*}+ C\EE\int_0^T\abs{ \ck(s)}^\frac{8m^*}{r^*}_{H^1_2}ds.\notag
\end{align}

Combining these last  inequalities we infer from the inequality (<ref>) that

\begin{equation}
\EE\sup_{0\le t\le T}\abs{c_k(t)}^\frac{8m^*}{r^*}_{H_2^1}+C^\frac{4m^*}{r^*}_1\EE\left(\int_0^T\abs{ \ck(s)}^2_{H^2_2}ds\right)^\frac{4m^*}{r^*}\leq\EE\abs{c_0}^\frac{8m^*}{r^*}_{H_2^1}+CR^\frac{4}{q+1}+C\EE\int_0^T\abs{ \ck(s)}^\frac{4m^*}{r^*}_{H^1_2}ds,\notag
\end{equation}

and the Gronwall lemma implies (<ref>).

Finally, we consider the system

\begin{align}
\label{sysn}
d{n}_\kappa(t)  = (r_n\Delta   |n_\kappa(t)|^{q-1}n_\kappa(t)+\theta\xi(t)
- \chi \Div( \xi(t) \nabla \ckk(t))) dt +\ukk(t)\cdot\nabla \xi(t)+g_{\gamma_3}(\nk(t))  dW_1(t).
\end{align}

We now show that there exists a
unique solution $\nk$ to the system (<ref>) by
verifying  the assumptions (H1), (H2$^{\prime}$), (H3), and  (H4$^{\prime}$) of Theorem 5.1.3 in <cit.>. Let us consider the Gelfand triple

\begin{equation}
V\subset \CH\cong \CH^{\ast}\subset V^{\ast},\label{3.1}
\end{equation}

with $\CH:=H_{2}^{-1}(\CO)$,  $V:=L^{q+1}(\CO)$ and $V^{\ast}=L^\frac{{q+1}}{q}(\CO)$.
The duality $_{V^{\ast}}\langle \cdot, \cdot \rangle_V$ is defined as

\begin{equation}
_{V^{\ast}}\langle n,w \rangle_V
=\int_{\CO} ((-\Delta)^{-1}n)(x)\, w(x)\, dx
\end{equation}

We set for $t\in [0,T]$,
=r_nΔn^[q]+θξ(t)-χ(ξ(t) ∇c_κ(t))+(t)·∇ξ(t),

where $c_\kappa$ solves (<ref>) and $\bu_\kappa$ solves (<ref>).

First let us consider (H1).
Let $\lambda\in\RR$. We need to show that

\lambda\mapsto {_{V^{\ast}}\langle  A(t,n_{1}+\lambda n_{2}),w\rangle_{V}}

is continuous on $\RR$, for any $n_{1},n_{2},w\in V$ and  $t\in[0,T]$.
As the map $\lambda \mapsto (n_{1}+\lambda n_{2})^{[q]}$ is continuous, it follows by the dominated convergence Theorem
that the mapping $\RR_0^+\ni
\lambda\mapsto \int_{\CO}(n_{1}+\lambda n_{2})^{[q]}w dx $
is continuous. Hence, the mapping $
\lambda\mapsto {_{V^{\ast}}\langle  A(t,n_{1}+\lambda n_{2}),w\rangle_{V}}$ is also continuous.

To show the local monotonicity, i.e. (H2$^{\prime}$),
let us fix $n_1,n_2\in V$ and  $t\in[0,T]$. Then,  using the fact that (see <cit.>),

$$ -\int_{\CO}\left[r_n(n_1^{[q]}-n_2^{[q]})(n_1-n_2)\right]\,dx
\leq0,$$

we have

\begin{align}\label{wasist}
&  \mathbin{_{V^{\ast}}\langle  A(t,n_1)- A(t, n_2),n_1-n_2\rangle_{V}}+|g_{\gamma_1}(n_1)-g_{\gamma_1}(n_2)|^2_{L_{HS}(H_1,\CH)}
\notag\\
&\le |g_{\gamma_1}(n_1)-g_{\gamma_1}(n_2)|^2_{L_{HS}(H_1,\CH)} \\
&\le  C
\end{align}

where $f(t)$ will be given later.

To verify the coercivity, i.e.  (H3)
let us fix $n\in V$ and  $t\in[0,T]$. Then,

\begin{align*}
2	\mathbin{_{V^{\ast}}\langle A(t,n),n \rangle_{V}}+|g_{\gamma_1}(n)|^2_{L_{HS}(H_1,\CH)}
&= - 2r_n |n|_{L^{q+1}}^{q+1}
\di(\xi(t) \nabla c_\kappa(t)),n \rangle_V+2\mathbin{_{V^{\ast}}}\langle\nabla \xi(t)\cdot\bu_\kappa(t),n \rangle_V \notag\\
&\quad+2\mathbin{_{V^{\ast}}}\langle\theta\xi(t),n \rangle_V
\\
& \leq -2r_n |n|_V^{q+1}+C(\theta) |n|_{\CH}^{2}+|\xi|_{\CH}^{2}+2\mathbin{_{V^{\ast}}}\langle\nabla \xi(t)\cdot\bu_\kappa(t),n \rangle_V\\
&\quad-2 \chi\mathbin{_{V^{\ast}}}\langle
\di(\xi(t) \nabla c_\kappa(t)),n \rangle_V.
% \ep_1 \lk|\xi(t,\omega)\rk|_{L^{q+1}}^{q+1} +
\end{align*}

Since $q>3$ then by the embedding of $V$ into $\CH$ and the Young inequality we have

\begin{align}
C(\theta)\abs{n}^2_\CH&\leq C(\theta)\abs{n}^{2}_V\leq r_n\abs{n}^{q+1}_V+C(\theta,q),\label{3.18*}
\end{align}

\begin{align}
\abs{\xi(t)}^2_\CH&\leq C\abs{\xi(t)}^{2}_V\leq \frac{1}{3}\abs{\xi(t)}^{q+1}_V+C(q).\label{3.18**}
\end{align}

Similarly, since $q>3$, by the Hölder inequality and Sobolev embedding,   we note that

\begin{align}\label{3.18}
-2\mathbin{_{V^{\ast}}}\langle \chi
\di(\xi(t) \nabla c_\kappa(t)),n \rangle_V&\leq 2\abs{\di(\xi(t) \nabla c_\kappa(t))}_{H^{-1}_2}\abs{n}_{H^{-1}_2}\notag\\
&\leq 2\abs{\xi(t)}_{L^4}\abs{\nabla c_\kappa(t)}_{L^4}\abs{n}_{H^{-1}_2}\notag\\
&\leq C \abs{\xi(t)}_{L^{q+1}}^2\abs{\nabla c_\kappa(t)}^2_{L^{q+1}}+\frac{1}{2}\abs{n}_{H^{-1}_2}^2\\
&\leq  \abs{\xi(t)}^{4}_{L^{4}}+C\abs{\nabla c_\kappa(t)}^{4}_{L^{4}}+\frac{1}{2}\abs{n}_{H^{-1}_2}^2\notag\\
&\leq \frac{1}{3} \abs{\xi(t)}^{q+1}_{L^{q+1}}+C\abs{\nabla c_\kappa(t)}^{4}_{L^{4}}+\frac{1}{2}\abs{n}_{H^{-1}_2}^2+C(q).\notag
\end{align}

By an integration by parts, we note that for all $w\in H^1_2(\bo)$,

$$\mathbin{_{H^{-1}_2}}\langle \xi(t)\cdot\nabla \bu_\kappa(t) ,w \rangle_{H^1_2}=\int_\bo\nabla\xi(t)\cdot \bu_\kappa (t) wdx=-\int_\bo\xi(t) \bu_\kappa (t) \cdot\nabla wdx\leq \abs{\xi(t) }_{L^4}\abs{\bu_\kappa(t) }_{\mathbf{L}^4} \abs{\nabla w}_{L^2}.$$

This implies that $\abs{\xi(t)\cdot\nabla \bu_\kappa(t)}_{H^{-1}_2}\leq \abs{\xi(t) }_{L^4}\abs{\bu_\kappa(t) }_{\nsH}$  and therefore,

\begin{align}
2 \mathbin{_{V^{\ast}}}\langle\nabla \xi(t)\cdot\bu_\kappa(t),n \rangle_V&\leq 2\abs{\nabla \xi(t)\cdot\bu_\kappa(t)}_{H^{-1}_2}\abs{n}_{H^{-1}_2}\notag\\
&\leq 2\abs{\bu_\kappa(t) }_{\nsH}\abs{\xi(t) }_{L^4}\abs{n}_{H^{-1}_2}\notag\\
&\leq C\abs{\bu_\kappa(t) }_{\nsH}^2\abs{\xi(t) }^2_{L^{q+1}}+\frac{1}{2}\abs{n}_{H^{-1}_2}^2\label{3.19}\\
&\leq C\abs{\xi(t) }^4_{L^{q+1}}+\frac{1}{2}\abs{n}_{H^{-1}_2}^2+C\abs{\bu_\kappa(t) }_{\nsH}^4\notag\\
&\leq \frac{1}{3}\abs{\xi(t) }^{q+1}_{L^{q+1}}+\frac{1}{2}\abs{n}_{H^{-1}_2}^2+C(q)+C\abs{\bu_\kappa(t) }_{\nsH}^4\notag.
\end{align}

By using (<ref>) and (<ref>) we derive that

\begin{equation}
2\mathbin{_{V^{\ast}}\langle A(t,n),n \rangle_{V}}+|g_{\gamma_1}(n)|^2_{L_{HS}(H_1,\CH)}
\leq C_0 |n|_{\CH}^{2}-r_n |n|_V^{\alpha^*}+f(t),
\end{equation}

where $\alpha^*=q+1$, $C_0=\max(1,C(q)r_n^{\frac{q+1}{q}})$ and
$$f(t)=\abs{\xi(t)}^{4}_{L^{q+1}}+C\abs{\nabla c_\kappa(t)}^{4}_{L^{4}}+C\abs{\bu_\kappa(t) }_{\nsH}^4+C(\theta,q).$$

Note that, since $\frac{m^*}{r^*}>1$, due to the estimates (<ref>) and (<ref>), there exists a constant $C(R)>0$ such that

\begin{align}
\EE\int_0^T\abs{\nabla c_\kappa(s)}^{4}_{L^{4}}ds&\leq C\EE \int_0^T \abs{\nabla c_\kappa(s)}^{2}_{L^{2}}\abs{ \nabla c_\kappa(s)}^{2}_{H^1_{2}}ds\notag\\
&\leq C\EE \sup_{0\le s\le  {T}  } \abs{c_\kappa(s)}^{2}_{H^1_{2}}\int_0^T\abs{ c_\kappa(s)}^{2}_{H^2_{2}}ds\notag\\
&\leq C\left(\EE \sup_{0\le s\le  {T}  } \abs{ c_\kappa(s)}^{4}_{H^1_{2}}\right)^\frac{1}{2}\left(\EE\left(\int_0^T\abs{ c_\kappa(s)}^{2}_{H^2_{2}}ds\right)^2\right)^\frac{1}{2}\notag\\
&\leq C\left(\EE \sup_{0\le s\le  {T}  } \abs{ c_\kappa(s)}^{8\frac{m^*}{r^*}}_{H^1_{2}}\right)^\frac{r^*}{4m^*}\left(\EE\left(\int_0^T\abs{ c_\kappa(s)}^{2}_{H^2_{2}}ds\right)^{8\frac{m^*}{r^*}}\right)^\frac{r^*}{4m^*}\notag\\
&\leq C(R)\notag.
\end{align}

By the  Hölder inequality, we obtain

\begin{equation}
\EE\int_0^T\abs{\ukk(s)}^{4}_{\nsH}ds\leq T\EE \sup_{0\le s\le  {T}  }\abs{\ukk(s)}^{4}_{\nsH}\leq T\left(\EE \sup_{0\le s\le  {T}  }\abs{\ukk(s)}^{8^\frac{m^*}{r^*}}_{\nsH}\right)^\frac{r^*}{2m^*}\leq C(R).\notag
\end{equation}

Observe, due to the fact that   $\xi\in\mathcal{X}_\MA^\kappa$, it follows that $f$ belongs to  $L^1([0,T]\times \Omega;dt\otimes\PP)$.
This proves (H3).

The growth condition, i.e.  (H4$^{\prime}$), can be verified similarly to the boundedness.
Let $n\in V$ and  $t\in[0,T]$. Then for all $w\in V$, by the Hölder inequality we have

\begin{align*}
\mathbin{_{V^{\ast}}\langle A(t,n),w\rangle_{V}}& \leq r_n\abs{n}^q_V\abs{w}_V +\mathbin{_{V^{\ast}}}\langle\theta\xi(t),w \rangle_V
-\mathbin{_{V^{\ast}}}\langle \chi
\di(\xi(t) \nabla c_\kappa(t)),w \rangle_V\\
&\quad+\mathbin{_{V^{\ast}}}\langle\nabla \xi(t)\cdot\bu_\kappa(t),w \rangle_V \notag\\
&\leq r_n\abs{n}^q_V\abs{w}_V+\theta\abs{\xi(t)}_{H^{-1}_2}\abs{w}_{H^{-1}_2}+\abs{\di(\xi(t) \nabla c_\kappa(t))}_{H^{-1}_2}\abs{w}_{H^{-1}_2}\notag\\
&\quad+\abs{\nabla \xi(t)\cdot\bu_\kappa(t)}_{H^{-1}_2}\abs{w}_{H^{-1}_2}.
\end{align*}

Using the continuous embedding of $V$ into $H^{-1}_2(\CO)$ we infer that

\begin{align*}
\abs{ A(t,n)}_{V^\star}&\leq r_n\abs{n}^q_V+C\theta\abs{\xi(t)}_{H^{-1}_2}+C\abs{\di(\xi(t) \nabla c_\kappa(t))}_{H^{-1}_2}+C\abs{\nabla \xi(t)\cdot\bu_\kappa(t)}_{H^{-1}_2}\\
&\leq r_n\abs{n}^q_V+C\theta\abs{\xi(t)}_{H^{-1}_2}+C\abs{\xi(t)}_{L^4}\abs{\nabla c_\kappa(t)}_{L^4}+C\abs{\bu_\kappa(t) }_{\nsH}\abs{\xi(t) }_{L^4}\\
&\leq r_n\abs{n}^q_V+C\theta\abs{\xi(t)}_{V}+C\abs{\xi(t)}_{L^{q+1}}\abs{\nabla c_\kappa(t)}_{L^{4}}+C\abs{\bu_\kappa(t) }_{\nsH}\abs{\xi(t) }_{L^{q+1}}.
\end{align*}

Since $\frac{q+1}{q-1}<4$, we deduce  that

\begin{align*}
\abs{ A(t,n)}^{\frac{\alpha^*}{\alpha^*-1}}_{V^\star}&\leq C(q)r^{\frac{q+1}{q}}_n\abs{n}^{q+1}_V+C(\theta,q)\abs{\xi(t)}^{\frac{q+1}{q}}_{V}+C(q)\abs{\xi(t)}^{\frac{q+1}{q}}_{L^{q+1}}\abs{\nabla c_\kappa(t)}^{\frac{q+1}{q}}_{L^{4}}+C(q)\abs{\bu_\kappa(t) }_{\nsH}^{\frac{q+1}{q}}\abs{\xi(t) }^{\frac{q+1}{q}}_{L^{q+1}}\\
&\leq C(q)r^{\frac{q+1}{q}}_n\abs{n}^{q+1}_V+\abs{\xi(t)}^{q+1}_{L^{q+1}}+C(q)\abs{\nabla c_\kappa(t)}^{\frac{q+1}{q-1}}_{L^{4}}+C(q)\abs{\bu_\kappa(t) }_{\nsH}^{\frac{q+1}{q-1}}+C(\theta,q)\\
&\leq C(q)r^{\frac{q+1}{q}}_n\abs{n}^{q+1}_V+\abs{\xi(t)}^{q+1}_{L^{q+1}}+C\abs{\nabla c_\kappa(t)}^{4}_{L^{4}}+C\abs{\bu_\kappa(t) }_{\nsH}^{4}+C(\theta,q)\\
&\leq C_0\abs{n}^{\alpha^*}_V+f(t)\\
&\leq (C_0\abs{n}^{\alpha^*}_V+f(t))(1+\abs{n}^2_\CH).
\end{align*}

Observe, as before,
that $f\in  L^1([0,T]\times \Omega;dt\otimes\PP)$.

In this way, we have proved that the Hypothesis (H1), (H2$^{\prime}$), (H3), and  (H4$^{\prime}$) are satisfied.
The existence and uniqueness of $n_\kappa$ such that  for all $p\geq 2$,

$$\EE\sup_{0\le s\le  {T} }\abs{\nk(s)}_{H^{-1}_2}^p<\infty,$$

follows by an application of Theorem 5.1.3 in <cit.>.

To calculate the bound,
we apply the Itô formula to the function $\phi(\nk)=  |\nk|_{H^{-1}_2}^2$ and obtain

\begin{align*}
&|\nk(t)|_{H^{-1}_2}^2+2r_n\int_0^{t} \abs{\nk(s)}^{q+1}_{L^{q+1}} ds\notag\\
&\quad-2\chi\int_0^{t}(\nabla^{-1} \nk (s),\nabla^{-1} (\Div (\xi(s)\nabla \ckk(s))))ds
-2\int_0^{t}(\nabla^{-1} \nk(s),
\nabla^{-1} \lk( {\bf u}_\kappa(s) \cdot\nabla \xi(s)
\rk))ds
\\
&\quad{}+2\int_0^{t}\left(\nk (s),g_{\gamma_1}(\nk(s))\,dW_1(s)\right)_{H^{-1}_2}+ \int_0^t|g_{\gamma_1}(\nk(s))|_{L_{\text{HS}}(H_1,H_2^{-1})}^2\,ds,
\end{align*}

where $\nabla^{-1}:=-(-\Delta)^{-1/2}$.

Using the  the fact that $L^{q+1}(\CO)\hookrightarrow H^{-1}_2(\CO)$ we obtain

\begin{align}
(\nabla^{-1}\xi,\nabla^{-1}\nk)\leq\abs{\xi}_{H^{-1}_2} \abs{\nk}_{H^{-1}_2}\leq C\abs{\xi}_{L^{q+1}} \abs{\nk}_{H^{-1}_2}\leq \frac{1}{2}\abs{\xi}^2_{L^{q+1}} +C\abs{\nk}^2_{H^{-1}_2}\notag.
\end{align}

By the Young inequality, we have

\begin{align*}
2\chi(\nabla^{-1} \nk ,\nabla^{-1} (\Div (\xi\nabla \ckk)))&\leq C\abs{\nk}_{H^{-1}_2} \abs{\Div (\xi\nabla \ckk)}_{H^{-1}_2}\\
&\leq C\abs{\xi}_{L^4}\abs{\nabla\ck}_{L^4}\abs{\nk}_{H^{-1}_2} \\
&\leq C\abs{\nk}_{H^{-1}_2}\abs{\xi}_{L^4}^2+C\abs{\nk}_{H^{-1}_2}\abs{\nabla\ck}_{L^4}^2 \\
&\leq C\abs{\nk}_{H^{-1}_2}\abs{\xi}_{L^{q+1}}^2+C\abs{\nk}_{H^{-1}_2}\abs{\ck}_{H_2^2}^2
\end{align*}

\begin{align}
2(\nabla^{-1} \nk,
\nabla^{-1} \lk( {\bf u}_\kappa \cdot\nabla \xi
\rk))&\leq2\abs{\nabla \xi\cdot\bu_\kappa}_{H^{-1}_2}\abs{\nk}_{H^{-1}_2}\notag\\
&\leq C\abs{\bu_\kappa}_{\nsH}\abs{\xi }_{L^4}\abs{\nk}_{H^{-1}_2}
\leq \abs{\bu_\kappa}_{\nsH}^2\abs{\xi }^2_{L^{q+1}}+C\abs{\nk}_{H^{-1}_2}^2\notag\\
&\leq \frac{1}{2}\abs{\xi }^2_{L^{q+1}}+\frac{1}{2}\abs{\bu_\kappa}_{\nsH}^4+C\abs{\nk}_{H^{-1}_2}^2.\notag
\end{align}

Using the inequality <ref>,  we obtain

\begin{equation*}
|g_{\gamma_1}(\nk)|_{L_{\text{HS}}(H_1,H_2^{-1})}^2\leq C\abs{\nk}_{H^{-1}_2}^2.
\end{equation*}

Combining these inequalities, we derive  that

\begin{align}
&|\nk(t)|_{H^{-1}_2}^2+2r_n\int_0^{t} \abs{\nk(s)}^{q+1}_{L^{q+1}} ds\notag\\
&\leq|n_0|_{H^{-1}_2}^2+\int_0^t\abs{\xi(s)}^2_{L^{q+1}}ds+ C\int_0^t\abs{\nk(s)}_{H^{-1}_2}^2ds+\frac{1}{2}\int_0^t\abs{\bu_\kappa(s)}_{\nsH}^4ds\notag\\
&\quad +2\int_0^{t}\left(\nk (s),g_{\gamma_1}(\nk(s))\,dW_1(s)\right)_{H^{-1}_2}.\notag
\end{align}

This implies that

\begin{align}
&\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{4m^*}{r^*}+(2r_n)^\frac{2m^*}{r^*}\EE\left(\int_0^{T} \abs{\nk(s)}^{q+1}_{L^{q+1}} ds\right)^\frac{2m^*}{r^*}\notag\\
&\quad+ C\EE\left(\int_0^T\abs{\nk(s)}_{H^{-1}_2}^2ds\right)^\frac{2m^*}{r^*}+C\EE\left(\int_0^T\abs{\nk(s)}_{H^{-1}_2}\abs{\xi(s)}_{L^{q+1}}^2ds\right)^\frac{2m^*}{r^*}\label{3.27}\\
&\quad+C\EE\left(\int_0^T\abs{\nk(s)}_{H^{-1}_2}\abs{\ck(s)}_{H_2^2}^2ds\right)^\frac{2m^*}{r^*}+C\EE\sup_{0\le t\le T}\abs{\int_0^{t}\left(\nk (s),g_{\gamma_1}(\nk(s))\,dW_1(s)\right)_{H^{-1}_2}}^\frac{2m^*}{r^*}.\notag
\end{align}

We remark that we get  by estimate (<ref>)

\begin{align}
\EE\left(\int_0^T\abs{\bu_\kappa(s)}_{\nsH}^4ds\right)^\frac{2m^*}{r^*}&\leq C(T)\EE\sup_{0\le s\le  {T} }\abs{\bu_\kappa(s)}_{\nsH}^\frac{8m^*}{r^*}\notag\\
&\leq C\EE\abs{\bu_0}^{8\frac{m^*}{r^*}}_\nsH+ CR^\frac{4}{q+1}+C.\notag
\end{align}

We note that a two times application of  the Hölder inequality  gives

\begin{align}
\EE\left(\int_0^T\abs{ \xi(s)}^2_{L^2}ds\right)^\frac{2m^*}{r^*}\leq &C\EE\left(\int_0^T\abs{ \xi(s)}^{q+1}_{L^{q+1}}ds\right)^\frac{4m^*}{r^*(q+1)}\notag\\
&\leq C(T)\left(\EE\left(\int_0^T\abs{ \xi(s)}^{q+1}_{L^{q+1}}ds\right)^\frac{2m^*}{r^*}\right)^\frac{2}{q+1}\notag\\
&\leq CR^\frac{2}{q+1},\notag
\end{align}

\begin{equation}
\EE\left(\int_0^T\abs{\nk(s)}_{H^{-1}_2}^2ds\right)^\frac{2m^*}{r^*}\leq C(T,m^*,r^*)\EE\int_0^T\abs{\nk(s)}_{H^{-1}_2}^\frac{4m^*}{r^*}ds\notag,
\end{equation}

as well as

\begin{align}
C\EE\left(\int_0^T\abs{\nk(s)}_{H^{-1}_2}\abs{\xi(s)}_{L^{q+1}}^2ds\right)^\frac{2m^*}{r^*}	&\leq C\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{2m^*}{r^*}\left(\int_0^T\abs{\xi(s)}_{L^{q+1}}^2ds\right)^\frac{2m^*}{r^*}\notag\\
&\leq \frac{1}{6}\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{4m^*}{r^*}+C\EE\left(\int_0^T\abs{\xi(s)}_{L^{q+1}}^2ds\right)^\frac{4m^*}{r^*}\notag\\
&\leq \frac{1}{6}\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{4m^*}{r^*}+C\EE\left(\int_0^T\abs{\xi(s)}_{L^{q+1}}^{q+1}ds\right)^\frac{8m^*}{r^*(q+1)}\notag\\
&\leq \frac{1}{6}\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{2m^*}{r^*}+C\left(\EE\left(\int_0^T\abs{\xi(s)}_{L^{q+1}}^{q+1}ds\right)^\frac{2m^*}{r^*}\right)^\frac{4}{q+1}\notag\\
&\leq \frac{1}{6}\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{2m^*}{r^*}+CR^\frac{4}{q+1}\notag.
\end{align}

In a similar way we get

\begin{align}
&\leq C\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{2m^*}{r^*}\left(\int_0^T\abs{\ck(s)}_{H^{2}_2}^2ds\right)^\frac{2m^*}{r^*}\notag\\
&\leq \frac{1}{6}\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{2m^*}{r^*}+C\EE\left(\int_0^T\abs{\ck(s)}_{H_2^{2}}^2ds\right)^\frac{4m^*}{r^*}\notag.
\end{align}

Using the estimate (<ref>), we derive that

\begin{equation}
\EE\left(\int_0^T\abs{ \ck(s)}^2_{H^2_2}ds\right)^\frac{4m^*}{r^*}\leq C\EE\abs{c_0}^\frac{4m^*}{r^*}_{H_2^1}+CR^\frac{4}{q+1}.\notag
\end{equation}

By the Burkholder-Gundy-Davis inequality and the inequality <ref>, we have

\begin{align}
&C\EE\sup_{0\le t\le T}\abs{\int_0^t(\nk(s),g_{\gamma_1}(\nk(s)) dW_1(s))_{H_2^{-1}}}^\frac{2m^*}{r^*}\notag\\
&\leq C\EE\left(\int_0^T\abs{\nk}^2_{H^{-1}_2}\abs{g_{\gamma_1}(\nk(s))}^2_{L_{\text{HS}}(H_1,H_2^{-1})}ds\right)^\frac{m^*}{r^*}\notag\\
&\leq C\EE\sup_{0\le t\le T}\abs{\nk(t)}^\frac{2m^*}{r^*}_{H^{-1}_2}\left(\int_0^T\abs{g_{\gamma_1}(\nk(s))}^2_{L_{\text{HS}}(H_1,H_2^{-1})}ds\right)^\frac{m^*}{2r^*}\notag\\
&\leq\frac{1}{6}\EE\sup_{0\le t\le T}\abs{\nk(T)}^\frac{4m^*}{r^*}_{H^{-1}_2}+C\EE\left(\int_0^T\abs{g_{\gamma_1}(\nk(s))}^2_{L_{\text{HS}}(H_1,H_2^{-1})}ds\right)^\frac{2m^*}{r^*}\notag\\
&\leq\frac{1}{6}\EE\sup_{0\le t\le T}\abs{\nk(t)}^\frac{4m^*}{r^*}_{H^{-1}_2}+C(T)\EE\int_0^T\abs{\nk(s)}_{H^{-1}_2}^\frac{4m^*}{r^*}ds.\notag
\end{align}

From these a priori estimates, we infer from (<ref>) that

\begin{align}
&\frac{1}{2}\EE\sup_{0\le t\le T}\	|\nk(t)|_{H^{-1}_2}^\frac{4m^*}{r^*}+(2r_n)^\frac{2m^*}{r^*}\EE\left(\int_0^{T} \abs{\nk(s)}^{q+1}_{L^{q+1}} ds\right)^\frac{2m^*}{r^*}\notag\\
\end{align}

The Gronwall lemma implies (<ref>) and ends the proof of Claim  <ref>.

Verification of condition (<ref>) of Theorem <ref>:
Now, we shall tackle the continuity of the operator $\CV_\MA^\kappa$ in the norm induced by $\CM^{m^\ast}_\MA(\X)$. In fact, in the next proposition we show that  assumption (<ref>) of Theorem <ref> is satisfied.

Let us assume that the assumptions of Theorem <ref> are valid.
Let  $\xi_1,\xi_2\in \CX_\MA(R)\subset\CM^{m^\ast}_\MA(\BX)$ be arbitrary.

Then the following statements are valid.

* Let $\uk^{(1)}$ and $\uk^{(2)}$ be the solutions to system
dn^(j)_κ(t)     =    (r_nΔ|n^(j)_κ(t)|^q-1n^(j)_κ(t)+θξ(t)-χ( ξ_j(t) ∇^(j)(t)))  dt

+  ∇ξ_j(t)^(j)_κ(t)dt +g_γ_1(^(j)(s))dW_1,|
with  $\uk ^{(j)}(0)=n_0, j=1,2$. Then,

for any $R>0$, for any $\kappa\in\NN$,
there exist  constants $C=C(R,\kappa)>0$ such that  for any $\xi\in \CM^{m^\ast}_{\mathfrak{A}}(\mathbb{X})\cap\CX(R)$,
we have

\begin{align}
&\EE\sup_{0\le t\le T}|\nk^{(1)}(t)-\nk ^{(2)}(t) |_{H^{-1}_2}^\frac{2m^*}{r^*}+\EE\left(\int_0^{T}\abs{\nk^{(1)}(s)-\nk ^{(2)}(s)}^{q+1}_{L^{q+1}}ds\right)^\frac{m^*}{r^*}\notag\\
C(R)	\	\left(\EE \sup_{0\leq s\leq T}\abs{\ckk^{(1)}(s)-\ckk^{(2)}(s)}_{L^{2}}^{2\frac{m^*}{r^*}}\right)^{\frac{r'}{4(q+1)}}+
C(R)	\left(\EE \sup_{0\leq s\leq T}\abs{\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)}_{\nsH}^{4\frac{m^*}{r^*}}\right)^{\frac{r'}{4q+4}}\notag.
\end{align}

* Let $\ck^{(1)}$ and $\ck^{(2)}$ be the solutions to system

d{\ckk}^{(j)}(t) +\Theta_\kappa(\ukk^{(j)},t)\ukk^{(j)}(t)\nabla c^{(j)}(t)\, dt=\lk( r_c \Delta \ckk^{(j)}(t)  -\alpha \ckk^{(j)}(t)+ \beta \xi_j(t) \rk)\, dt +g_{\gamma_2}(\ck^{(j)}(s)) dW_2,

where $j=1,2$,
and   $\ck ^{(j)}(0)=c_0, j=1,2$. Then,

for any $R>0$, for any $\kappa\in\NN$,
there exist  constants $C=C(R,\kappa)$ such that for any $\xi\in \CM^{m^\ast}_{\mathfrak{A}}(\mathbb{X})\cap\CX(R)$
we have

\begin{align}
&\EE\sup_{0\le s\le  {T}  }\abs{{\ckk^{(1)}-\ckk^{(2)}}(s) }^\frac{2m^*}{r^*}_{L^2}+\alpha_0^\frac{m^*}{r^*}\EE\left(\int_0^T\abs{{\ckk^{(1)}(s)-\ckk^{(2)}}(s) }^2_{H^1_2}ds\right)^\frac{m^*}{r^*}\notag\\
&\leq C\left(\EE\abss{\xi_1-\xi_2}^{m^*}_{\mathbb{X}}\right)^\frac{2}{r^*}+	C(\kappa,R)\left(\EE\sup_{0\le s\le  {T}  }\abs{\ukk^{(1)}(s)-\ukk^{(2)}(s)}_{\nsH}^{4\frac{m^*}{r^*}}\right)^\frac{1}{2} \notag.
\end{align}

* Let $\ukk^{(1)}$ and $\ukk^{(2)}$ be the solutions to system
d ^(j) _κ(t)-Δ^(j)_κ(t)dt=ξ_j(t)⋆Φ  dt+ σ_3 (t) dW_3(t),|
with  $\ukk ^{(j)}(0)=\bu_0, j=1,2$.  Then,

for any $R>0$, for any $\kappa\in\NN$,
there exist  constants $C=C(R,\kappa)>0$ such that for any $\xi\in \CM^{m^\ast}_{\mathfrak{A}}(\mathbb{X})\cap\CX(R)$,
we have

\begin{equation*}
\frac{1}{2}\EE\sup_{0\le s\le  {T}  }|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{\nsH}^{2\frac{m^*}{r^*}}+(2r)^\frac{m^*}{r^*}_u \EE\left(\int_0^{T}|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{\nsV}^2\, ds\right)^\frac{m^*}{r^*}\leq C(\delta_1,\delta_2)\left(\EE\abss{\xi_1-\xi_2}^{m^*}_{\mathbb{X}}\right)^\frac{2}{r^*}.
\end{equation*}

The continuity of the operator $\CV_\MA^\kappa: \CX_\MA(R)\to \CX_\MA(R)$  follows by an interpolation argument.

Let us assume that the assumptions of Theorem <ref> are valid.
Let  $\xi_1,\xi_2\in\mathcal{X}_\MA(R)\subset\CM^{m^\ast}_\MA(\BX)$ be arbitrary.
Then there exists some constant $C>0$ such that

\EE \abss{ \CV_\MA^\kappa(\xi_1)-\CV_\MA^\kappa(\xi_2)}^{m^\ast}_\BX
\le C\EE \abss{\xi_1-\xi_2}^{m^\ast}_\BX.

The proof is a combination of points (i), (ii) and (iii) of Proposition <ref> with an application of Proposition <ref>. We have to verify if the parameter $r=r^\ast$, $s=s^{\ast\ast}$, and $m=m^\ast$,
satisfy the constrains
$r\in(2,q+1)$, $m\in(q+1,\infty)$, $s\in(0,1)$, with  $\frac 1r\ge\frac 1m+\frac s2$. However, since $q>4$, $s^{**}=\frac{2}{q+1}$,
and  the  numbers $r^\ast$ and $m^\ast$ satisfy
$q\leq r^*<q+1$ and $ m^*\geq 2q+2$, the constrains are satisfied and we can apply Proposition <ref>.

Firstly, we consider the difference $\bu_\kappa^{(1)}-\bu_\kappa^{(2)}$ which is the solution of the following random equation

\begin{equation}
d(\bu_\kappa^{(1)}-\bu_\kappa^{(2)})(t)-\Delta (\bu_\kappa^{(1)}-\bu_\kappa^{(2)})(t)dt=(\xi_1(t)-\xi_2(t))\star\Phi dt, \ \text{ with } \  (\bu_\kappa^{(1)}-\bu_\kappa^{(2)})(0)=0.\notag
\end{equation}

Testing this equation by $-\Delta(\bu_\kappa^{(1)}-\bu_\kappa^{(2)})$, we arrive at

\begin{align}
&\sup_{0\le s\le  {T}  }|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{\nsH}^2+2r_u \int_0^{t}|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{\nsV}^2 ds\notag\\
&= -2\int_0^{T} \left(\Delta(\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)),(\xi_1(s)-\xi_2(s))\star \Phi\right) ds\notag\\
&\leq \frac{1 }{2}\sup_{0\le s\le  {T}  }|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{\nsH}^2+C\sum_{i=1}^2 \int_0^{T}|\nabla((\xi_1(s)-\xi_2(s))\star \phi_i)|_{\mathbf{L}^2}^2 ds.\notag
\end{align}

This implies that

\begin{align}
&\f{1}{2}\EE\sup_{0\le s\le  {T}  }|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{L^2}^{2\frac{m^*}{r^*}}+(2r)^\frac{m^*}{r^*}_u \EE\left(\int_0^{T}|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{H^1_2}^2\, ds\right)^\frac{m^*}{r^*}\notag\\
&\leq C\sum_{i=1}^2\EE\left( \int_0^{T}|\nabla((\xi_1(s)-\xi_2(s))\star \phi_i)|_{\mathbf{L}^2}^2 ds\right)^\frac{m^*}{r^*}.\label{3.29}
\end{align}

From semi-group theory (see, e.g. <cit.>), we obtain that for $i=1,2$,

\begin{align}
|\nabla((\xi_1-\xi_2)\star \phi_i)|_{\mathbf{L}^2}&\leq \sum_{i=1}^2	|(\xi_1-\xi_2)\star \phi_i|^2_{H^1_2}= \sum_{i=1}^2	\abs{e^{-\delta_i\Delta}(\xi_1-\xi_2)}_{H^1_2}^2\notag\\
&\leq C \sum_{i=1}^2\abs{e^{-\delta_i\Delta}}_{L(H^{-1}_2;H^1_2)}^2\abs{(\xi_1-\xi_2)}^2_{H^{-1}_2}\notag\\
&\leq  C\abs{(\xi_1-\xi_2)}^2_{H^{-1}_2}\sum_{i=1}^2\abs{-\Delta e^{-\delta_i\Delta}}^2_{L(H^{-1}_2;H^{-1}_2)}\notag\\
&\leq C\abs{(\xi_1-\xi_2)}^2_{H^{-1}_2}\sum_{i=1}^2\frac{C}{\delta_i^{1/2}}\notag\\
&\leq C(\delta_i)\abs{\xi_{1}-\xi_{2}}_{H^{-1}_2}\notag.
\end{align}

Using the embedding of $H^{-s^{**}}_2(\CO)$ into $H^{-1}_2(\CO)$, we infer that  for $i=1,2$,

\begin{align*}
\EE\left( \int_0^{T}|\nabla((\xi_1(s)-\xi_2(s))\star \phi_i)|_{\mathbf{L}^2}^2 ds\right)^\frac{m^*}{r^*}&\leq C(\delta_i)\EE\left( \int_0^{T}|(\xi_1(s)-\xi_2(s))|_{H^{-s^{**}}_2}^2 ds\right)^\frac{m^*}{r^*}\\
\end{align*}

Since $m^*>2$ and $2<r^*$ we have

\begin{equation}
\EE\abs{\xi_1-\xi_2}^{2\frac{m^*}{r^*}}_{L^2(0,T;H_2^{-s^{**}})}\leq C\EE\abs{\xi_1-\xi_2}^{2\frac{m^*}{r^*}}_{L^{m^*}(0,T;H_{r^*}^{-s^{**}})}\leq C\EE\abs{\xi_1-\xi_2}^{2\frac{m^*}{r^*}}_{\mathbb{X}}\leq C\left(\EE\abs{\xi_1-\xi_2}^{m^*}_{\mathbb{X}}\right)^\frac{2}{r^*}.\notag
\end{equation}

We then infer from the inequality (<ref>) that

\begin{equation*}
\frac{1}{2}\EE\sup_{0\le s\le  {T}  }|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{\nsH}^{2\frac{m^*}{r^*}}+(2r)^\frac{m^*}{r^*}_u \EE\left(\int_0^{T}|\bu_\kappa^{(1)}(s)-\bu_\kappa^{(2)}(s)|_{\nsV}^2\, ds\right)^\frac{m^*}{r^*}\leq C(\delta_1,\delta_2)\left(\EE\abs{\xi_1-\xi_2}^{m^*}_{\mathbb{X}}\right)^\frac{2}{r^*}.
\end{equation*}

and the estimate for the difference  $\bu_\kappa^{(1)}-\bu_\kappa^{(2)}$ follows.

Secondly, we consider the difference $\ckk^{(1)}-\ckk^{(2)}$ which satisfies the following equation

\begin{align*}
&d({\ckk^{(1)}-\ckk^{(2)}})(t) +\left(\Theta_\kappa(\ukk^{(1)},t)\ukk^{(1)}(t)\nabla c^{(1)}(t)-\Theta_\kappa(\ukk^{(2)},t)\ukk^{(2)}(t)\nabla c^{(2)}(t)\right) dt\\
&=\lk( r_c \Delta (\ckk^{(1)}-\ckk^{(2)})(t)  -\alpha(\ckk^{(1)}-\ckk^{(2)})(t)+ \beta \xi_1(t)-\xi_2(t) \rk)\, dt +g_{\gamma_2}(\ckk^{(1)}(t)-\ckk^{(2)}(t)) dW^t_2.
\end{align*}

Setting $\alpha_0=\min (r_c,\alpha)$ and applying the Itô formula we arrive at

\begin{align}
&\sup_{0\le s\le  {T}  }\abs{{\ckk^{(1)}(s)-\ckk^{(2)}}(s) }^2_{L^2}+2\alpha_0\int_0^T\abs{{\ckk^{(1)}(s)-\ckk^{(2)}}(s) }^2_{H^1_2}ds\notag\\
&\leq2\int_0^T(\xi_1(s)-\xi_2(s),  \ckk^{(1)}(s)-\ckk^{(2)}(s))ds\notag\\
&\qquad+2\int_0^T\left(\Theta_\kappa(\ukk^{(1)},s)\ukk^{(1)}(s)\nabla \ckk^{(1)}(s)-\Theta_\kappa(\ukk^{(2)},s)\ukk^{(2)}(s)\nabla c^{(2)}(s), \ckk^{(1)}-\ckk^{(2)}(s)\right) ds\notag\\
&\qquad+\int_0^T\abs{g_{\gamma_2}(\ckk^{(1)}-\ckk^{(2)}(s))}^2_{L_{\text{HS}}(H_1,L^{2})}ds+2\sup_{0\le t\le  {T}  }\int_0^t(\ckk^{(1)}(s)-\ckk^{(2)}(s),g_{\gamma_2}(\ckk^{(1)}(s)-\ckk^{(2)}(s)) dW_2(s)).\notag
\end{align}

Since $s^{**}<1$, we have

\begin{align}
(\xi_1-\xi_2,  \ckk^{(1)}-\ckk^{(2)})&\leq \abs{\xi_1-\xi_2}_{H_2^{-s^{**}}} \abs{\ckk^{(1)}-\ckk^{(2)}}_{H_2^{s^{**}}}\notag\\
&\leq \abs{\xi_1-\xi_2}_{H_2^{-s^{**}}} \abs{\ckk^{(1)}-\ckk^{(2)}}_{H_2^{1}}\notag\\
&\leq \frac{\alpha_0}{3}\abs{\ckk^{(1)}-\ckk^{(2)}}^2_{H_2^{1}}+C\abs{\xi_1-\xi_2}_{H_2^{-s^{**}}} ^2.\notag
\end{align}

For the second integrand, we have

\begin{align}