\section{Introduction}

Let us consider the following nonlinear damped stochastic Schr\"{o}dinger equation
\begin{equation}
\label{ProblemStratonovich}
\left\{
\begin{aligned}
\df u(t)&= -\left[\im A u(t)+\im  F(u(t)) +\beta u(t) \right] dt  -\im B u(t) \circ \df W(t)-\im G(u(t)) \,\df \newW(t) ,\qquad t> 0,\\
u(0)&=u_0.
\end{aligned}\right.
\end{equation}
Here $A$ is a linear self-adjoint  non-negative operator,
$\beta$  is a damping constant (usually considered not negative),
$B$ is a linear bounded operator; $F$  and $G$  are nonlinear terms. Moreover
$W$ and $\newW$  are two independent  Wiener processes;
the first  stochastic differential is in the Stratonovich form and the other one is  in the It\^o form.

A basic example of the operator  $A$ is the negative Laplace-Beltrami operator $-\Delta_g$ on a compact Riemannian
manifold $(M,g)$ without boundary; this appears in some previous papers on the nonlinear
Schr\"{o}dinger
equation, see for instance \cite{Burq+G+T_2004}, \cite{BrzezniakStrichartz}, \cite{BHW-2019}.
However, we can deal as well with  the negative Laplacian $-\Delta$ on a
relatively compact  smooth domain in  $\Rd$ with Neumann or Dirichlet boundary conditions.
We will consider $F$ to be a  power-type defocusing nonlinearity.

The nonlinear Schr\"odinger equation  occurs as a basic model in many areas of physics:
hydrodinamics, plasma physics, nonlinear optics, molecular biology. It describes the propagation of waves media with both
nonlinear and dispersive responses, see, e.g.,  \cite{SulSul} where many physical models are discussed.
A lot of attention has been paid
recently to the influence of a noise on the dynamics described by the equation:
the noise acts as a random potential to incorporate spatial and temporal fluctuations of certain parameters in a physical
model. Typically the noise depends on the solution itself and, according to the physical situation one aims at describing,
the It\^o or the Stratonovich forms of the noise can be taken into account.
These different types of stochastic differential lead in fact to different properties of solutions.
For instance in the multiplicative noise model the mass is preserved only
when the noise is taken in the Stratonovich sense of a particular type.
This provides the same property as for  the deterministic equation.
However, with  both  a Stratonovich and  an It\^o noise, the energy is not conserved anymore in general.
For this reason, when addressing the problem of the existence of an invariant measure, some dissipative terms (e.g.: our model with $\beta>0$) are usually involved in the stochastic case.

The question of the existence and/or the uniqueness of solutions for nonlinear Schr\"{o}dinger equations  with additive or linear
multiplicative noise was previously addressed
in the $\Rd$-case by De Bouard and Debussche \cite{BouardLzwei, BouardHeins},
Barbu, R\"{o}ckner and Zhang \cite{BarbuL2, BarbuH1, Barbu17},
Cui, Hong and Sun in \cite{Cui}
and F. Hornung  \cite{Hornung-F_PhD} \adda{and \cite{Hornung_2018}}. In the case of compact two dimensional Riemannian manifolds there are results
 by Brze{\'{z}}niak and Millet \cite{BrzezniakStrichartz} and by  Brze{\'{z}}niak, Hornung and Weis
\cite{BHW-2019}.

So far  the existence  of an invariant measure has been obtained for this equation with  a damping term
and an additive It\^o  noise, i.e.  $\beta>0$, $B=0$ and $G$ constant in equation \eqref{ProblemStratonovich}.
In this framework, in the papers by Kim \cite{Kim_2006} and by Ekren,  Kukavica and  Ziane \cite{Ekren_2017} the result is proved in the full space $\mathbb{R}^d$, $d\ge 1$. Debussche and Odasso \cite{Odasso} obtain instead the result on a bounded one-dimensional domain (dealing with the cubic 
focusing Schr\"odinger equation) as  a byproduct of the proof of the mixing property,  see Remark 1.2 therein. In the same paper the authors also solve the corresponding uniqueness problem. This is the only result in the literature concerning the uniqueness of the invariant measure for the stochastically forced and damped nonlinear Schr\"odinger equation. 
Some numerical approximations of invariant measures can be found in the book by Hong and Wang \cite{Hong+Wang_2019_invariant}.
The aim of our paper is to generalize the previously cited papers by considering a more general stochastic forcing term: we consider 
 a linear multiplicative Stratonovich noise $B$, which conserves the $H$-norm,  and a nonlinear It\^o noise $G$. 
When we reduce to the case of a pure additive noise we get the existence of an invariant measure when $\beta>0$. This finding is in line with \cite{Kim_2006} and \cite{Ekren_2017} that obtain the same conclusions working on the full space. We emphasize here that, as far as we know, our result is the first one providing the existence of an invariant measure in the case of a two-dimensional compact Riemannian manifold without boundary and of a relatively compact  smooth domain
with either the Dirichlet or the Neumann boundary conditions.




The paper is structured as follows. In the first part of the paper we shall construct a martingale solution of problem $\eqref{ProblemStratonovich}$
in the energy space ${V}=\mathcal{D}(A^{\frac{1}{2}})$ by a modified
Faedo-Galerkin's approximation, following the lines
of a previous paper by the first author and collaborators \cite{BHW-2019}. One should mention here that a very recent  paper
\cite{Hornung_2020} provides another generalization \cite{BHW-2019} in the direction of stochastic NSL on unbounded domains and non-compact manifolds. It would be interested to see if it is possible to combine
our present paper and a recent paper \cite{Hornung_2020}  by proving the existence and uniqueness of solutions for the more general noise as well as by investigation the existence of invariant measures.
A proof of that result with a more general noise would interesting and challenging. Indeed,  for the approach  from \cite{Hornung_2020},  the conservative noise was crucial to overcome the lack of compactness in the limit procedure.

In the second part of the paper we shall prove the pathwise uniqueness of the solutions.
Hence the existence and the uniqueness of a strong solution will follow.
 As far as the existence of an invariant measure is concerned, in the last  part of the paper  we proceed differently from  \cite{Kim_2006}  and
\cite{Ekren_2017}. Following the proof of the existence of a martingale solution  we prove that the corresponding Markov semigroup is
sequential weak Feller in ${V}$.
Moreover we show a tightness result in the space ${V}$ equipped with the weak topology, when the damping coefficient is
sufficiently large.
With these two latter properties we prove the existence of at least one invariant measure, by means of the method introduced
 by Maslowski and Seidler \cite{Mas-Seid}, as a version of the classical Krylov-Bogoliubov technique reset with weak
 topologies. This method has been successful to prove existence of invariant measures for other SPDE's, as the stochastic
 nonlinear beam and wave equations \cite{BOndSeid}, the Navier-Stokes equations in unbounded domains
 \cite{BMO,BFer}, the
stochastic Landau-Lifshitz-Bloch equation \cite{BGL}, the stochastic damped Euler equation \cite{BesFer}.

The paper is organized as follows.
In Section \ref{s-2} we present notation and main assumptions. In Section \ref{Section3MR} we state our main result. In Section
\ref{CompactnessSection} we collect some compactness results. Section \ref{sec-exis} deals with the existence of
martingale solutions. Pathwise uniqueness is proved in Section \ref{UniquenessSection}.
The two last sections \ref{sec-weakCont} and \ref{s-inv} are concerned with the sequential weak  Feller property  and
the existence of invariant measures. In Section \ref{sect-unique}  we consider the particular case of multiplicative noise,
where there is also uniqueness of the invariant measure.
In Appendix \ref{app_Lapl} we recall some facts about Laplacian-type operators on manifolds and on bounded domains
with Dirichlet/Neumann boundary conditions and derive the needed Strichartz estimates; this is very different from the
setting in $\mathbb R^d$ considered in many papers. In Appendices \ref{App_B} and \ref{App_C} we collect the proofs of
some results. In Appendix \ref{varie} we collect some remarks on the assumptions.







\section{Mathematical setting and Assumptions}  \label{s-2}

In this Section, we fix the notation, explain the assumptions and formulate the framework for our problem.
Let $\left(X,\Sigma,\mu\right)$ be a $\sigma$-finite measure space, such that the Borel $\sigma$-field is contained in $\Sigma$, with metric $\rho$. Let $D \subseteq X$.
Our framework covers the following cases
\begin{footnote}{From now on we will denote by $\mathcal{O}$ a subset of $\mathbb{R}^2$ and by $M$ a two-dimensional manifold. We will use the letter $D$ when we need to deal with the two cases above at the same time.
}\end{footnote}
:
\begin{itemize}
\item $X=\mathbb{R}^2$ with the Euclidean distance $\rho$ and the Lebesgue measure $\mu$ and $D=\mathcal{O}$ is a  relatively compact  smooth, i.e. with $C^{\infty}$ boundary, domain of $\mathbb{R}^2$.
\item $D=X=M$ is a two-dimensional compact Riemaniann manifold without boundary with the geodesic distance $\rho$ and the canonical volume measure $\mu$ on $X$.
\end{itemize}

By $L^q(D)$, for $q\in [1,\infty]$, we denote the space of equivalence classes of $\mathbb{C}$-valued $q-$Lebesgue integrable functions.
We abbreviate $L^q:=L^q(D)$. For $q\in [1,\infty],$ let  $q':=\frac{q}{q-1}\in [1,\infty]$ be the conjugate exponent. We further abbreviate ${H}:=L^2$.
This is a complex Hilbert space with inner product $(u,v)_H=\int_D u \overline{v} \,{\rm d}\mu$.
However,  we often interpret $H$ as  a real Hilbert space with the inner product $\operatorname{Re}(u,v)_H$.
They are different but  in one-to-one correspondence: $(u,v)_H=\operatorname{Re} (u,v)_H+i\operatorname{Re} (u,iv)_H$.
These inner products introduce the same norms and hence both spaces are topologically equivalent.

By $H^{s,q}(D)$ we denote the fractional Sobolev space of regularity $s\in \R$ and integrability $q\in (1,\infty)$. We abbreviate $H^{s,q}:=H^{s,q}(D)$ and we shortly write $H^s:=H^{s,2}$.
For a definition of these spaces see Appendix \ref{app_Lapl}.

In the sequel, given two Banach spaces $E$ and $F$, we denote by $\mathscr{L}(E,F)$ the space of all linear bounded
operators $B: E\to F$ and abbreviate $\mathscr{L}(E):=\mathscr{L}(E,E).$  Furthermore, we write $E\hookrightarrow F$, if
$E$ is continuously embedded in $F$, i.e. $E\subset F$ with natural embedding  $j\in \mathscr{L}(E,F)$.
For a Hilbert space $H$ and a Banach space $E$, $\gamma(H,E)$ denotes the spaces of $\gamma$-radonifying operators from $H$ to $E$. If $E$ is a Hilbert space, this is indeed the space of  Hilbert-Schmidt operators from $H$ to $E$.
The space $C^{1,2}([0,T]\times E,F)$ consists of all functions $\varPhi\colon [0,T]\times E\to F$ such that
$\varPhi(\cdot,x)\in C^1([0,T],F)$ for every $x\in E$ and $\varPhi(t,\cdot)\in C^2(E,F)$ for every $t\in[0,T]$.
Given the Hilbert space $H$, $C_w([0,T];H)$ stands for  the space
of all continuous functions from the interval $[0,T]$ to  the space $H$ endowed with the weak topology.

If functions  $a,b\ge 0$ satisfy the inequality $a \le C(A) b$ with a constant $C(A)>0$ depending on the expression $A$, we write $a \lesssim_A b$; for a generic constant we put no subscript.
 If we have $a \lesssim_A b$ and $b \lesssim_A a$, we write $a \simeq_A  b$.

\subsection{Assumptions on the operator $A$}

\begin{Assumption}
\label{spaceAssumptions}
The operator $A$ that appears in equation \eqref{ProblemStratonovich} is a Laplacian-type operator. We consider
 $A$ to be as one of the following:
\begin{itemize}
\item [i)] the negative Laplace-Beltrami operator $-\Delta_g$ on a compact two-dimensional Riemannian manifold
$(M,g)$ without boundary, equipped with a Lipschitz metric $g$. In this case $\mu$ is the canonical volume measure
\item [ii)] the negative Laplacian with Dirichlet boundary conditions $-\Delta_D$ on a smooth,
i.e. $C^{\infty}$, relatively compact  domain $\mathcal{O}$ of $\mathbb{R}^2$.
\item [iii)] the negative Laplacian with Neumann boundary conditions $-\Delta_N$ on a smooth,
i.e. $C^{\infty}$, relatively compact  domain $\mathcal{O}$ of $\mathbb{R}^2$.
\end{itemize}
\end{Assumption}

Some classical results, see e.g., \cite{Ouh_2009},  ensure that the operator $A$, in any of the forms given in Assumption \ref{spaceAssumptions}, is a non-negative self-adjoint operator on $H$. We denote by $\mathcal{D}(A)$ its domain. We set ${V}:=\mathcal{D}(A^{\frac{1}{2}})$ and note that $V$  is a Hilbert space when equipped with the inner product
		\begin{align*}
		\skp{u}{v}_{{V}}:=\skpH{u}{v}+\skpH{A^{\frac{1}{2}} u}{A^{\frac{1}{2}} v},\qquad u,v\in {V}.
		\end{align*}
We call it the \emph{energy space} and we call the induced norm $\norm{\cdot}_{{V}}$ the  \emph{energy norm} associated to $A$.
For a characterization of the energy spaces associated to the operators that appear in Assumption \ref{spaceAssumptions},
see Remark \ref{B4} and Proposition \ref{propB6}(i). We denote the dual space of ${V}$ by ${V}^*$ and abbreviate the
duality with $\duality{\cdot}{\cdot}$,
where the complex conjugation is taken over the second variable of the duality. Note that $\left({V}, H, {V}^*\right)$ is a
Gelfand triple, i.e. ${V}\hookrightarrow H \cong H^* \hookrightarrow {V}^*$.

Notice that, thanks of the geometry of the domain $D$, the condition $\alpha \in (1, \infty)$ ensures that the embedding $V \subset L^{\alpha+1}$ is continuous and compact. Hence we can extend the ${V}-{V}^*$  duality $\duality{\cdot}{\cdot}$
to the couple ${L^{\alpha+1}}-{L^{\frac{\alpha+1}{\alpha}}}$, since $({L^{\alpha+1}})^*={L^{\frac{\alpha+1}{\alpha}}}$.

It can be proved, see e.g. \cite[Lemma 2.3(a)]{BHW-2019}, that there exists a non-negative self-adjoint operator $\hat{A}$
on ${V}^*$ with $\mathcal{D}(\hat{A})={V}$ and $\hat{A}=A$ on $H$.
In most cases where this does not cause ambiguity or confusion, we also use the notation $A$ for $\hat{A}$.

In the following Lemma we introduce the operator $S$ and state some properties of it. $S$ will play the role of an auxiliary operator to cover the different cases we consider in an unified framework.
\begin{Lemma}
\label {S_properties}
Given $A$ as in Assumption \ref{spaceAssumptions}, there exists an operator $S$ on $H$ such that
\begin{itemize}
\item [i)]$S$ is strictly positive and self-adjoint. $S$ commutes with $A$ and   satisfies
		$\mathcal{D}(S^k)\hookrightarrow {V}$
		for sufficiently large $k.$ Moreover, $S$
			satisfies the upper Gaussian estimate i.e. for all $t >0$ there is a measurable function $p(t, \cdot, \cdot):D \times D \rightarrow \mathbb{R}$ with
	\begin{equation*}
\left(e^{-tS}f\right)(x)= \int_Dp(t,x,y)f(y)\, \mu({\rm d}y), \quad t >0, \quad \text{a.e.} \ x \in D,
	\end{equation*}
for all $f \in H$ and with constants $c,C>0$ and $m \ge 2$
\begin{equation}
\label{Gauss_est}
|p(t,x,y)| \le \frac{C}{\mu(B(x, t^{\frac 1m})}
\exp\left(-c \left(\frac{\rho(x,y)^m}{t} \right)^{\frac{1}{m-1}} \right),
\end{equation}
for all $t>0$ and a.e. $(x,y) \in D \times D$.
\item [ii)] $S$ has compact resolvent. In particular, there is an orthonormal basis $\left(h_n\right)_{n\in \N}$ of $H$ and a nondecreasing sequence $\left(\lambda_n\right)_{n\in\N}$ with $\lambda_n>0$ and  $\lambda_n\to \infty$ as $n\to \infty$ such that
		\begin{align}
		\label{spectral_S}
		S x=\sum_{n=1}^\infty \lambda_n \skpH{x}{h_n} h_n, \quad x\in \mathcal{D}(S)=\left\{x\in H: \sum_{n=1}^\infty \lambda_n^2 \vert \skpH{x}{h_n}\vert^2<\infty\right\}.
		\end{align}
\end{itemize}
\end{Lemma}
\begin{proof}
For $A=-\Delta_g$, we choose $S:=I-\Delta_g$, for $A=-\Delta_D$, we choose $S=A=-\Delta_D$, for $A=-\Delta_N$, we fix $\varepsilon>0$ and choose $S=\varepsilon I-\Delta_N$. For these choices of $S$ all the  statements of the Lemma are verified: for a proof see \cite[Sections 3.2 - 3.3, Remark 2.2(b) and Lemma 2.3(c)]{BHW-2019} and the therein references.
\end{proof}


\begin{Remark}
The operator $S$ plays a crucial role in the construction of our Galerkin approximations.
The Gaussian estimate \eqref{Gauss_est} is used in the proof of Proposition \ref{PaleyLittlewoodLemma} where spectral multiplier theorems for $S$ are employed (for further details one can consult \cite[Chapter 7]{Ouh_2009}).
\end{Remark}

\subsection{Assumptions on the nonlinear term $F$}
We continue with the assumptions on the nonlinear term $F$ of our problem. We deal with power-type defocusing
nonlinearities.

\begin{Assumption}
\label{F_def}
Given $\alpha \in (1, \infty)$, we set
\begin{equation*}
F(u):=|u|^{\alpha-1}u.
\end{equation*}
\end{Assumption}

The function $F$ satisfies a set of properties that we summarize in the following Lemma (for a proof see e.g. \cite[Proposition 3.25 and Remark 3.16]{Cazenave} and \cite[Proposition 3.1]{BHW-2019}).
It is important to remark that the embedding $V \subset L^{\alpha+1}$ is continuous and compact. Hence
 ${V}\hookrightarrow{L^{\alpha+1}}$ implies $({L^{\alpha+1}})^*={L^{\frac{\alpha+1}{\alpha}}}\hookrightarrow {V^*}$
		is a bounded map.

\begin{Lemma}
\label{nonlinearAssumptions}
	Let $\alpha \in (1, \infty)$.
	\begin{itemize}
	\item[i)] The map $F:L^{\alpha +1} \to {L^{\frac{\alpha+1}{\alpha}}}$ satisfies,   for any $ u,v\in {L^{\alpha+1}}$
		\begin{align}\label{nonlinearityEstimate}
		\norm{F(u)}_{L^{\frac{\alpha+1}{\alpha}}} = \norm{u}_{L^{\alpha+1}}^\alpha,
		\end{align}
	%
		\begin{align}\label{nonlinearityLocallyLipschitz}
                 \norm{  F(u)-F(v)}_{{L^{\frac{\alpha+1}{\alpha}}}}
                &\lesssim
                \left(\norm{u}_{L^{\alpha+1}}+\norm{v}_{L^{\alpha+1}}\right)^{\alpha-1} \norm{u-v}_{L^{\alpha+1}}.
                \end{align}	
                %
      	Moreover, $F: {V} \to {V^*}$, $F(0)=0$ and
                \begin{align}\label{nonlinearityComplex}
		\operatorname{Re} \duality{\im u}{F(u)}=0, \quad u\in {L^{\alpha+1}} 	,	\end{align}
		%
		\begin{align}\label{ineq-dissipativity}
		 \duality{F(u)}{u} = \norm{u}_{L^{\alpha+1}}^{\alpha+1} ,\quad u\in {L^{\alpha+1}}.
		\end{align}
			%
	\item[ii)] The map $F: {L^{\alpha+1}}\to {L^{\frac{\alpha+1}{\alpha}}}$ is continuously
	 Fr\'{e}chet differentiable with
		\begin{align}\label{deriveNonlinearBound}
		\Vert F^{\prime}[u]\Vert_{L^{\alpha+1}\to L^\frac{\alpha+1}{\alpha}} \le \alpha \norm{u}
		 _{L^{\alpha+1}}^{\alpha-1}, \quad u\in {L^{\alpha+1}}.
		\end{align}
	\item[iii)]The map $F$ is defocusing, that is it admits the real non-negative antiderivative
	\begin{footnote}
	{
	We recall that, if there exists a Fr\'echet differentiable map $\hat F:L^{\alpha+1} \rightarrow \mathbb{R}$ with $\hat F'[u]h=\text{Re}\langle F(u),h\rangle$, for every $u,h \in L^{\alpha+1}$, $\hat F$ is called the antiderivative of $F$.
	}
	\end{footnote}$\hat{F}: {L^{\alpha+1}}\to \R$ given by
\begin{align}
\label{boundantiderivative}\hat F(u)=\frac 1{\alpha+1}\|u\|_{L^{\alpha+1}}^{\alpha+1}.
\end{align}			
\end{itemize}
\end{Lemma}


\begin{Remark}
\label{remnew}
It follows from \eqref{nonlinearityEstimate} and  \eqref{boundantiderivative}  that
		\begin{align}\label{ineq-F}
		\norm{F(u)}_{L^{\frac{\alpha+1}{\alpha}}}
		=\norm{u}_{L^{\alpha+1}}^\alpha
		\lesssim[\hat{F}(u)]^{\frac{\alpha}{\alpha+1}}, \; \quad u\in {L^{\alpha+1}}.
		\end{align}
It follows from \eqref{deriveNonlinearBound} and  \eqref{boundantiderivative} that
		\begin{align}\label{ineq-Fp}
		\|F'[u]\|_{L^{\alpha+1}\rightarrow L^{\frac{\alpha+1}{\alpha}}} \le\alpha\|u\|^{\alpha-1}_{L^{\alpha+1}}
		= \alpha (\alpha+1)^{\frac{\alpha-1}{\alpha+1}} [\hat F(u)]^{\frac{\alpha-1}{\alpha+1}} .
		\end{align}
\end{Remark}

\subsection{Assumptions on the stochastic terms}

For a probability space $\left(\Omega, \mathscr{F}}%{\mathcal{F}, \mathbb{P}\right),$ the law of a random variable $X: \Omega \to E$ is denoted by Law$_\mathbb{P}(X)$.

\begin{Assumption}\label{stochasticAssumptions}
	We assume the following.
	\begin{itemize}
		\item[i)] Let $(\Omega,\mathscr{F}}%{\mathcal{F},\mathbb{P})$ be a probability space, $\Yb$ and $\Yg$ two separable real Hilbert spaces,
		with an orthonormal bases  $(f_m)_{m\in\N}$ and $(e_m)_{m \in \mathbb{N}}$ respectively, and $W$ and
		$\newW$ two independent, $\Yb$  respectively
		$\Yg$, canonical cylindrical Wiener processes adapted to a filtration
		$\mathbb{F}$ satisfying the usual conditions.
		\item[ii)] 		Let $B: {H} \to \gamma(\Yb,{H})$ be a linear
		operator and set $B_m u:=B(u)f_m$ for $u\in {H}$ and $m\in \N.$ Additionally, we assume that
		$B_m\in\mathscr{L}(H)$ is self-adjoint for every $m\in\N$ and the following stronger assumption, needed to make sense of the Stratonovich correction terms,
is satisfied
		\begin{align}
  \label{noiseBoundsH}
		\sumM \norm{B_m}_{\mathscr{L}({H})}^2<\infty.
		\end{align}
Moreover we assume that $B_m\in \mathscr{L}({V})$ and  $B_m\in \mathscr{L}({L^{\alpha+1}})$ for all  $m\in\N$  and
		\begin{align}\label{noiseBoundsEnergy}
		&\sumM\norm{B_m}_{\mathscr{L}({V})}^2<\infty,\\
\label{noiseBounds-L^alpha}
		&\sumM \norm{B_m}_{\mathscr{L}(L^{\alpha+1})}^2<\infty.
		\end{align}	
\item[(iii)]
Let $G:{H} \to \gamma(\Yg,{H})$ be Lipschitz continuous,  i.e.
 \begin{equation}
 \label{Lipschitz_G}
\exists \ L_G >0 : \quad  \|G(v_1)-G(v_2)\|_{\gamma(\Yg,H)} \le L_G\|v_1-v_2\|_H \qquad \forall v_1, v_2 \in H.
\end{equation}
Moreover the following  ``restrictions'' of $G$, i.e.
\begin{align*}
G: {V} \to \gamma(\Yg,{V}) \mbox{ and } G: {L^{\alpha+1}} \to \gamma(\Yg,{L^{\alpha+1}})\end{align*}
are measurable, see \cite[Section 2]{Brz+Rana_2021} for a reasonably thorough discussion of this issue,  and of at most  linear growth, i.e.
  for some non negative constants $C_2, \tilde C_2, C_3, \tilde C_3$, the following inequalities hold
 \begin{equation}\label{crescitaGEA}
   \|G(u)\|_{\gamma(\Yg,{V})} \le C_2+\tilde C_2\|u\|_{{V}}\qquad \forall u \in {V},
\end{equation}
and
 \begin{equation}\label{crescitaGL}
\|G(u)\|_{\gamma(\Yg, {L^{\alpha+1}} )} \le C_3+\tilde C_3\|u\|_{ {L^{\alpha+1}} }\qquad \forall u \in  {L^{\alpha+1}}.
\end{equation}
 \newline
Finally,  also assume the following weak continuity assumption of the diffusion coefficient $G$.
For every $m \in \mathbb{N}$ the map
\[ H \ni \varphi \mapsto G(\varphi)e_m \in  H \]
extends uniquely to a continuous  map from $V^\ast$ to $V^\ast$, i.e.

\begin{equation}
 \label{cont_V^*}
 \text{the map} \ V^\ast \ni \varphi \mapsto G(\varphi)e_m \in V^\ast \ \text{is continuous, }  m \in \mathbb{N}.
 \end{equation}
\end{itemize}

\end{Assumption}
Examples of operator $B$ are in \cite[Section 3.5]{BHW-2019}.
The self-adjointness of $B$ is crucial there, see \cite[Remark 3.7]{BHW-2019}.
Concerning the second operator $G$,
in the case it is linear (and examples are the same as for $B$) we do not require it to be self-adjoint.


\begin{Remark}\label{Rem-linear growth} It is well known that the Lipschitz assumption \eqref{Lipschitz_G} implies the following linear growth condition. There exist positive constants $C_1, \tilde C_1$ such that
\begin{equation}\label{crescitaG}
\|G(u)\|_{\gamma(\Yg,H)} \le C_1+\tilde C_1\|u\|_H\qquad \forall u \in H.
\end{equation}
 We mention this obvious fact since  we explicitly use this estimate \eqref{crescitaG} in  many  computations in the following sections.
\end{Remark}

\begin{Remark}
By assumption  \ref{stochasticAssumptions} i), we can represent the Wiener processes as
\[
W(t)=\sumM f_m \beta_m(t) \qquad\qquad \newW(t)=\sumM e_m {\bm\beta}_m(t)
\]
for two sequences of independent standard real Wiener processes $\{\beta_m\}_m$ and $\{\bm\beta_m\}_m$.
\end{Remark}
\begin{Remark}
\label{HS}
	The estimates $\eqref{noiseBoundsH}$, $\eqref{noiseBoundsEnergy}$ and \eqref{noiseBounds-L^alpha} imply
	\begin{align*}
	B\in \mathscr{L}({H},\gamma(\Yb,{H})),\quad B\in \mathscr{L}({V},\gamma(\Yb,{V})),\quad B\in \mathscr{L}({L^{\alpha+1}},\gamma(\Yb,{L^{\alpha+1}})).
	\end{align*}
\end{Remark}



\section{Statement of the main result}
\label{Section3MR}



We rewrite equation \eqref{ProblemStratonovich} in the It\^o form. We have, see e.g. \cite{BE-loops,TwN},
\begin{align*}
-\im B u(t) \circ\df W(t)&=-\im B u(t) \df W(t)+\frac12  \sumM -\im B'[u]\left(-\im B(u(t))f_m\right)f_m \df t\\
&=-\im Bu(t)\,{\rm d}W(t)- \frac12\sumM B(Bu(t)f_m)f_m\, {\rm d}t
\\
&= -\im B u(t) \df W(t)-\frac 12 \sumM B_m^2 u(t) \df t.
\end{align*}
Hence,  equation $\eqref{ProblemStratonovich}$ will be understood in the following It\^o form 	
\begin{equation}\label{Problem}
\left\{
\begin{aligned}
\df u(t)&= -\left[\im A u(t)+\im  F(u(t)+\beta u(t)-b(u(t)) \right] \df t
\\
&\,\qquad\qquad\qquad-\im B u(t) \, \df W(t) -\im G(u(t)) \,\df \newW(t)
,\hspace{0,3 cm} t>0,\\
u(0)&=u_0,
\end{aligned}\right.
\end{equation}
where
\begin{align}
\label{Strat_cor}
b(u) := -\frac{1}{2} \sumM B_m^2 u,\qquad u\in{H},
\end{align}
is the Stratonovich correction term.
Notice that from \eqref{noiseBoundsH}, \eqref{noiseBoundsEnergy}
we know that $b$ is a linear bounded operator in $H$ as well as in ${V}$ and ${L^{\alpha+1}}$.

In the special  case $\beta=0$ and $G=0$, our problem reduces to the one studied in the recent paper  \cite{BHW-2019}.


\begin{Definition}\label{MartingaleSolutionDef}
	Let $T>0$ and $u_0\in {V}.$ A \emph{martingale solution} of the equation $\eqref{Problem}$ is a system $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{W},\tilde{\newW},\tilde{\mathbb{F}},u\right)$ consisting of
	\begin{itemize}
		\item a probability space $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}}\right);$
		\item  independent  $\Yb$-cylindrical  Wiener  process $\tilde{W}$ and $\Yg$-cylindrical  Wiener  $\tilde{\newW}$ on  $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}}\right);$
		\item  a filtration $\tilde{\mathbb{F}}=\left(\tilde{\mathscr{F}}%{\mathcal{F}}_t\right)_{t\in [0,T]}$ satisfying the usual conditions;
		\item a continuous, $\tilde{\mathbb{F}}$-adapted, ${V^*}$-valued process $u$ with $u \in L^2(\Omega \times[0,T];H)$		and $\tilde{\mathbb{P}}$-almost all paths in $C_w([0,T],\EA)$,
	such that the equality
	\begin{align}\label{ItoFormSolution}
	u(t)=  u_0- \int_0^t \left[\im A u(s)+\im F(u(s))+\beta  u(s) -b(u(s))\right] \df s     \nonumber
	\\
	- \im \int_0^t B u(s) \df \tilde{W}(s) -\im \int_0^t G(u(s)) \,\df \tilde{\newW}(s)
	\end{align}
	holds in ${V^*}$ for all $t\in [0,T]$, $\tilde{\mathbb{P}}$-almost surely.
\end{itemize}
\end{Definition}

If the probability space and the Wiener processes are given apriori, we say that a process with the specified regularity and fulfilling equation \eqref{ItoFormSolution}
is a strong solution. We will prove existence of a martingale solution and pathwise uniqueness. This implies that the martingale solution is actually a strong solution.

Denoting by  $\{u(t;u_0)\}_{t\ge 0}$
the unique solution with initial data $u_0\in V$, we shall define the family of operators  $\{P_t\}_{t\ge 0}$ by
\[
P_t\phi(u_0)=\mathbb E[\phi(u(t);u_0)]
\]
and prove that this is a Markov semigroup, see Section \ref{sec-weakCont}, which is sequential weak Feller in $V$.
Then we say that a Borel probability measure $\pi$ on  $V$ is an invariant measure for equation \eqref{Problem} iff
\[
\int_V P_t \phi\ d\pi=\int_V \phi\  d\pi
\]
for  all $t\ge 0$ and all  bounded functions $\phi:V\to\mathbb R$ which are
 sequentially continuous with respect to the
weak topology on $V$.

Here is our main result.
\begin{Theorem}\label{mainTh}
Under the Assumptions \ref{spaceAssumptions}, \ref{F_def} and \ref{stochasticAssumptions},
 \begin{itemize}
  \item [i)] for any $T>0$ and  $u_0\in {V}$,
   there exists a martingale solution  $u$ to equation \eqref{Problem}, which satisfies
 	\begin{align}
		\tilde \mathbb{E}[ \|u\|^q_{{L^\infty(0,T;\EA)}}]<\infty
	\end{align}
      for all $q\in [1,\infty)$;
 \item [ii)]
 the solution is pathwise unique;
 \item [iii)]   there exists at least one invariant measure for equation \eqref{Problem} provided
      \begin{equation}\label{condizione-beta}
\beta>\max\left(\tilde{C}_1^2+\tilde{C}_2^2+\|B\|^2_{\mathscr{L}(V,\gamma(Y_1,V))},
                     \frac{\alpha +1}{2}\|B\|^2_{\mathscr{L}(L^{\alpha+1},\gamma(Y_1,L^{\alpha+1}))} +\alpha\tilde{C}^2_3  \right).
      \end{equation}
      \end{itemize}
\end{Theorem}


\begin{Remark} $\quad$
\begin{itemize}
\item [i)] Statement (i)  of Theorem \ref{mainTh} holds in a more general setting, see Remark \ref{FR1}. Since our focus is on the existence of an invariant measure, we restrict our analysis to the less general setting.
This is done also in \cite{BHW-2019} to get the uniqueness of solutions.
\item[ii)] For a discussion concerning the regularity assumptions we impose on the domain $\mathcal{O}$ in the case of Assumption \ref{spaceAssumptions}(ii)-(iii), see Remark \ref{FR2}.
\item [iii)]
The condition \eqref{condizione-beta} on $\beta$ does not depend on the coefficients
$C_1$, $C_2$ and $C_3$ characterizing the second noise term.
When $B=0$ and $\tilde{C}_1=\tilde{C}_2=\tilde{C}_3=0$, that is when we consider a multiplicative noise
$G(u) \,\df \tilde{\newW}$ with bounded covariance, the condition \eqref{condizione-beta} reduces to $\beta>0$.
Therefore, in the particular case of additive noise  (i.e., $B=0$ and $G$ independent of $u$), we recover the same condition
$\beta>0$ as  in \cite{Kim_2006} and \cite{Ekren_2017}.
\item [iv)] In condition \eqref{condizione-beta} there is the constant $\tilde{C}_1$ (which somehow measures the
intensity of the noise $G$ in the $H$-norm)  but not the analogue for the noise $B$,  that is the term
$\|B\|^2_{\mathscr{L}(H,\gamma(Y_1,H))}$. This asymmetry is due to the fact that the Stratonovich noise $B$
 (in the absence of damping) preserves the $H$-norm, whereas the noise $G$ does not.
 In other words, the correction term $b$ of the Stratonovich noise cancels with the term
 $\|B\|^2_{\mathscr{L}(H,\gamma(Y_1,H))}$ acting as a sort of damping term which perfectly balance the intensity
 of the noise $B$ in the $H$-norm.

A similar reasoning can be done to explain also why we have the different constants $\alpha$ and $ \frac{\alpha +1}{2}$
multiplying the terms $\tilde{C}^2_3$ and $\|B\|^2_{\mathscr{L}(L^{\alpha+1},\gamma(Y_1,L^{\alpha+1}))}$ respectively.
We have $\frac{\alpha+1}{2}<\alpha$: the correction term $b$ provides part of the dissipation in the $L^{\alpha +1}$ norm.
\end{itemize}
\end{Remark}

For the purely multiplicative noise,  the  uniqueness  of an invariant measure will be given in Corollary \ref{cordelta0}.






\section{Compactness and Tightness Criteria}	\label{CompactnessSection}

This Section is devoted to recalling the compactness results, which will be used  in Section \ref{sec-exis} to obtain a
martingale solution as  limit of
the Faedo-Galerkin approximation and in Section \ref{sec-weakCont} to prove the continuous  dependence of the solutions on the initial data.
We only recall the results from \cite{BHW-2019}, for the proofs the reader can consult that paper.

Let $\alpha>1$ and $A$ be chosen according to Assumption $\ref{spaceAssumptions}$.
We consider  the  Banach spaces ${C([0,T],\EAdual)}$ and ${L^{\alpha+1}(0,T;\LalphaPlusEins)}$ ,
and the  locally convex space $C_w([0,T],\EA)$.
So we define the space
\[
Z_T= {C([0,T],\EAdual)}\cap{L^{\alpha+1}(0,T;\LalphaPlusEins)}\cap C_w([0,T],\EA)
\]
with the topology $\mathcal{T}$ given by the supremum of the corresponding topologies. Hence we associate  the Borel
$\sigma$-algebra, i.e. the $\sigma$-algebra generated by the open sets in the locally convex topology of $Z_T$.

In the next Proposition, we give a criterion for compactness in $Z_T.$
\begin{Prop}\label{CompactnessDeterministic}
	Let $K$ be a subset of $Z_T$ such that
	\begin{enumerate}
		\item[a)] $
		\sup_{u\in K} \Vert u\Vert_{{L^\infty(0,T;\EA)}}< \infty;
		$
		\item[b)] $K$ is equicontinuous in ${C([0,T],\EAdual)},$ i.e.
		\begin{align*}
		\lim_{\delta \to 0} \sup_{u\in K} \sup_{\vert t-s\vert\le \delta} \Vert u(t)-u(s)\Vert_{{V^*}}=0.
		\end{align*}
	\end{enumerate}
	Then, $K$ is relatively compact in $Z_T.$
\end{Prop}
Now we want to obtain a criterion for tightness in $Z_T.$ Therefore, we introduce the Aldous condition.
\begin{Definition}\label{DefinitionAldous}
	Let $(X_n)_{n\in\N}$ be a sequence of  stochastic processes taking values in a Banach space $E$.
	 Assume that for every $\varepsilon>0$ and $\eta>0$ there is $\delta>0$ such that for every sequence
	 $(\tau_n)_{n\in\N}$ of $[0,T]$-valued stopping times one has
	\begin{align}\tag{[A]}
	\sup_{n\in\N} \sup_{0<\theta \le \delta} \mathbb{P} \left\{ \Vert X_n((\tau_n+\theta)\land T)-X_n(\tau_n)\Vert_E\ge \eta \right\}\le \varepsilon.
	\end{align}
	In this case, we say that $(X_n)_{n\in\N}$
	satisfies the Aldous condition $[A].$
\end{Definition}


The following Lemma, see \cite[Lemma A.7]{Motyl},  gives us a useful consequence of the Aldous condition $[A].$

\begin{Lemma} \label{AldousLemma}			
	Let $(X_n)_{n\in\N}$ be a sequence of continuous stochastic processes in a separable Banach space $E$,
	which satisfies the Aldous condition $[A].$ Then, for every $\varepsilon>0$ there exists a measurable subset
	$A_\varepsilon \subset C([0,T],E)$ such that
	\begin{align*}
	\mathbb{P}^{X_n}(A_\varepsilon)\ge 1-\varepsilon,\qquad
	\lim_{\delta\to 0} \sup_{u\in A_\varepsilon} \sup_{\vert t-s\vert\le \delta} \Vert u(t)-u(s)\Vert_E=0.
	\end{align*}
\end{Lemma}


The deterministic compactness result in Proposition \ref{CompactnessDeterministic} and Lemma \ref{AldousLemma} can be used to get the following criterion for tightness in $Z_T.$
\begin{Prop}\label{TightnessCriterion}
	Let $(X_n)_{n\in\N}$ be a sequence of continuous adapted ${V^*}$-valued processes satisfying the Aldous condition $[A]$ in ${V^*}$ and
	\begin{align*}
	\sup_{n\in\N} \mathbb{E} \left[\Vert X_n\Vert_{L^\infty(0,T;\EA)}^2\right] <\infty.
	\end{align*}
	Then the sequence $\left({\mathbb{P}}^{X_n}\right)_{n\in\N}$ is tight in $Z_T,$ i.e. for every $\varepsilon>0$ there is a compact set $K_\varepsilon\subset Z_T$ with
	\begin{align*}
	\inf_{n \in \mathbb{N}}\mathbb{P}^{X_n}(K_\varepsilon)\ge 1- \varepsilon.
	\end{align*}
\end{Prop}


In metric spaces, one can apply the Prokhorov  and the  Skorohod Theorems to deduce convergence from
tightness. Since the space $Z_T$ is a locally convex space, in this case one can use the Jakubowski
generalization of the Skorohod Theorem to non-metric spaces, see \cite{Jak86}, \cite{Jak98}.
This generalization yields the final result of this Section.
\begin{corollary}\label{corollaryEstimatesToASconvergence}
	Let $(X_n)_{n\in\N}$ be a sequence of adapted ${V^*}$-valued processes satisfying the Aldous condition $[A]$ in ${V^*}$ and
	\begin{align*}
	\sup_{n\in\N} \mathbb{E} \left[\Vert X_n\Vert_{L^\infty(0,T;\EA)}^2\right] <\infty.
	\end{align*}
Then, there are a subsequence $(X_{n_k})_{k\in\N}$ and $Z_T$-valued  random variables
$\{\tilde{X}_k\}_{k \in \mathbb{N}}$, $\tilde{X}$ defined on a second probability space
$(\tilde{\Omega},\tilde{\mathbb{F}},\tilde{\Prob})$ with $\tilde{\Prob}^{\tilde{X}_k}=\mathbb{P}^{X_{n_k}}$ for any $k\in\N$, such that
$\tilde{X}_k \to \tilde{X}$ $\tilde{\Prob}$-almost surely in $Z_T$ for $k\to \infty.$
\end{corollary}











\section{Existence of a martingale solution}\label{sec-exis}

In this Section we prove the existence of at least one martingale solution, see Definition \ref{MartingaleSolutionDef}. This is our main result.
\begin{Theorem}\label{existTheorem}
Let $T>0$ and $u_0\in {V}$.
Under Assumptions \ref{spaceAssumptions},  \ref{F_def}, \ref{stochasticAssumptions},
there exists a martingale solution  $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{W},\tilde{\newW}, \tilde{\mathbb{F}},u\right)$
 of equation \eqref{Problem}, which satisfies
 	\begin{align}\label{propertySolution}
	u\in L^q\left(\tilde{\Omega},{L^\infty(0,T;\EA)}\cap C([0,T];H)\right)
	\end{align}
for all $q\in [1,\infty)$.
\newline
Moreover, if $\beta$ satisfies condition \eqref{condizione-beta}, then
\begin{equation}\label{Ebound}
\sup_{t\ge 0} \mathbb E \|u(t)\|^2_{{V}}<\infty.
\end{equation}
\end{Theorem}

The proof of the existence part  is based on a technique already used in  \cite{BM} and \cite{BHW-2019}.
We present the basic steps: the estimates on the Galerkin approximation and its  convergence
to a process which is a martingale solution. To be more precise, in Proposition \ref{MassEstimateGalerkinSolution} we prove
that there exists a unique global solution of the approximated problem and we obtain the apriori estimates in the space $H$.
In Proposition \ref{EstimatesGalerkinSolution} we obtain the apriori estimates for the energy functional and prove the Aldous condition.
These results lead to Corollary \ref{aprioriEA} which together with the to Aldous condition imply the tightness.
In Section \ref{Section_Convergence} we prove the convergence of the Galerkin approximations to the martingale solution of our problem.  The result concerning the
continuity in time with values in $H$ will be a byproduct of Proposition \ref{mild_prop}.
The estimate \eqref{Ebound} will be proved first for the Galerkin approximation and then for the limit too.

More general assumptions can be considered only to prove the existence of a martingale solutions;
some remarks are in Appendix \ref{varie}.






\subsection{The Galerkin Approximation and apriori estimates}  \label{sectionGalerkin}	

In this Section we introduce the Galerkin approximation. We prove the well-posedness of the approximated equation
and the uniform estimates for the solutions, that are sufficient to apply Corollary \ref{corollaryEstimatesToASconvergence}.

Let us recall that the operator $S$ was introduced in Lemma \ref{S_properties}. By the functional calculus we define the operators
$P_n:H \rightarrow H$ by $P_n:=\pmb{1}_{(0,2^{n+1})}(S)$ for $n \in \mathbb{N}_0$. Since $S$ has the representation
\eqref{spectral_S}, we observe that  $P_n$ is the orthogonal projection from $H$ to
$H_n:=\operatorname{span}\left\{h_m: m\in\N, \lambda_m< 2^{n+1}\right\}$ and
\begin{align*}
P_n x = \sum_{\lambda_m< 2^{n+1}} \skpH{x}{h_m}h_m, \qquad x\in {H}.
\end{align*}
Note that we have $h_m\in \bigcap_{k\in\N}\mathcal{D}(S^k)$ for $m\in\N$. Since by Lemma \ref{spectral_S}(i)
$\mathcal{D}(S^k)\hookrightarrow {V}$
for some $k\in\N$, we infer that $H_n$ is a closed subspace of ${V}$ for $n\in\N$. In particular, $H_n$ is
a closed subspace of ${V^*}$.
The fact that the operators $S$ and $A$ commute implies that $P_n$ and $A^\frac{1}{2}$ commute. Thus we get
\begin{align}\label{PnHeinsKontraktiv}
\norm{P_n x}_{{V}}^2=\norm{P_n x}_{H}^2+\norm{A^{\frac{1}{2}} P_n x}_{H}^2
=\norm{P_n x}_{H}^2+\norm{ P_n A^{\frac{1}{2}} x}_{H}^2
\le \norm{x}_{{V}}^2,\qquad x\in {V}.
\end{align}
Moreover
\[
\|P_nx\|^2_H \le \|x\|^2_H, \quad x \in H,
\qquad \quad \text{and} \qquad
\norm{P_n x}_{{V^*}}\le \norm{x}_{V^*}.
\]
By density, we can extend $P_n$ to an operator $P_n: {V^*}\to H_n$ with $\norm{P_n}_{{V^*}\to {V^*}}\le 1$ and
\begin{align}\label{PnInEAdual}
\duality{v}{P_n v}\in \R, \qquad \duality{v}{P_n w}=\skpH{P_n v}{w}, \qquad v\in {V^*}, \quad w\in {V}.
\end{align}

Unfortunately, the operators $P_n$, $n\in\N$,
are not uniformly bounded from ${L^{\alpha+1}}$ to ${L^{\alpha+1}}$. This property is crucial in the proof of the apriori estimates of the stochastic terms. To overcome this deficit, in the following Proposition we construct the sequence $\left(S_n\right)_{n\in\N}$ which enjoys the needed properties.

\begin{Prop}\label{PaleyLittlewoodLemma}
	There exists a sequence $\left(S_n\right)_{n\in\N}$ of self-adjoint operators $S_n: H \to H_n$ for $n\in\N$ such that
	$S_n \psi \to \psi$ in ${V}$ for $n\to \infty$ and $\psi \in {V}$  and the uniform norm estimates
	\begin{align}\label{SnUniformlyBounded}
	\sup_{n\in\N}\norm{S_n}_{{\mathscr{L}(H)}}\le 1, \quad \sup_{n\in\N} \norm{S_n}_{\mathscr{L}({V})}\le 1,
	\quad \sup_{n\in\N} \norm{S_n}_{\mathscr{L}(L^{\alpha+1})}<\infty
	\end{align}
	hold.
\end{Prop}

\begin{Remark}
Somehow, $S_n$ represents a smoothed version of the indicator function $p_n:=\pmb{1}_{(0,2^{n+1})}$ used to
define the operator $P_n$. This allows to use the spectral multiplier theorems to prove the uniform
$L^{\alpha +1}$- boundedness of the sequence $(S_n)_{n \in \mathbb{N}_0}$.
In \cite{BHW-2019} the same result is proved by means of the abstract Littlewood-Paley theory rather than
spectral multipliers theorems. Our  proof follows the lines of  \cite[Proposition 10]{BHM-2020} and [Lemma 3.14 and Remark 4.15]\cite{Hornung-F_PhD}
with the difference that here we use the classical estimate from Ouhabaz \cite{Ouh_2009} instead of the results from Kunstmann and Uhl \cite{KU_2015}.
\end{Remark}


\begin{proof}[Proof of Proposition \ref{PaleyLittlewoodLemma}]
We take a function $\rho \in C^{\infty}_c(0, \infty)$ with supp$(\rho)\in \left[ \frac 12 , 2\right]$ and $\sum_{m \in \mathbb{Z}}\rho(2^{-m}t)=1$ for all $t >0$. For a fixed $n \in \mathbb{N}_0$ we introduce the function
\begin{equation*}
s_n: (0, \infty) \rightarrow \mathbb{C}, \qquad \quad s_n(\lambda):=\sum_{m=-\infty}^{n}\rho\left(2^{-m}\lambda\right)
\end{equation*}
and we see that
\begin{equation*}
s_n(\lambda)=
\begin{cases}
\ 1 &\lambda \in (0, 2^n)
\\
\rho(2^{-n}\lambda) & \lambda \in [2^n, 2^{n+1})
\\
0 & \lambda \ge 2^{n+1}.
\end{cases}
\end{equation*}



We define $S_n:=s_n(S)$ via the functional calculus for self-adjoint operators.
In particular, by Lemma \ref{S_properties}(ii), we have the representation
\begin{equation}
\label{Sn_rep}
S_n x= \sum_{\lambda_m<2^n}(x,h_m)_Hh_m+ \sum_{\lambda_m\in[2^n,2^{n+1})}
\rho(2^{-n}\lambda_m)(x,h_m)_Hh_m, \qquad x \in H,
\end{equation}
from which immediately follows that the range of $S_n$ is contained in $H_n$.
 Since $s_n$ is real-valued and bounded by $1$, the operator $S_n$ is self-adjoint with $\|S_n\|_{\mathscr{L}(H)}\le1$. Moreover, since by Lemma \ref{S_properties}(i), $S_n$ and $A$ commute, we obtain $\|S_n\|_{\mathscr{L}(V)}\le1$ and $S_n \psi \rightarrow \psi$ for all $\psi \in V$, by the convergence property of the functional calculus.

 Finally, the uniform estimate in $L^{\alpha +1}$ is a consequence of the spectral multiplier Theorem \cite[Theorem 7.23]{Ouh_2009} and the Marcinkiewicz interpolation Theorem. It is sufficient to show, see \cite[equation (7.69)]{Ouh_2009}, that $s_n$ satisfies
 \begin{equation*}
 \sup_{\lambda>0}|\lambda ^ks_n^{(k)}(\lambda)|< \infty, \qquad \quad \ k=0,1,2. \end{equation*}
 We have
 \begin{equation*}
  \sup_{\lambda>0}|\lambda^ks_n^{(k)}(\lambda)|
  = \sup_{\lambda \in [2^n, 2^{n+1})}|\lambda^ks_n^{(k)}(\lambda)|
  =\sup_{\lambda \in [2^n, 2^{n+1})}|\lambda^k\frac{{\rm d}^k}{{\rm d}\lambda^k}\rho(2^{-n}\lambda)|
  \le
  2^k\sup_{\lambda >0}|\rho^{(k)}(\lambda)|< \infty,
     \end{equation*}
 for all $k \in \mathbb{N}$. This completes the proof of Proposition \ref{PaleyLittlewoodLemma}.
\end{proof}


Using the operators $P_n$ and $S_n,$ $n\in\N,$ we approximate our original problem $\eqref{Problem}$ by the stochastic differential equation in $H_n$ given by	
\begin{equation*}
\left\{
\begin{aligned}
\df u_n(t)&=- \left[\im A u_n(t)+\im P_n F(u_n(t)) +\beta u_n(t) \right] \df t-\im  S_n B(S_n u_n(t)) \circ \df W(t)
\\
&\qquad\qquad -\im  S_n G(S_n u_n(t)) \,\df \newW(t)
\\
u_n(0)&=P_n u_0.
\end{aligned}\right.
\end{equation*}
With the Stratonovich correction term
\begin{align*}
b_n := -\frac{1}{2} \sumM \left(S_n B_m S_n\right)^2,
\end{align*}
the approximated problem can be written in the following  It\^o form 	
\begin{equation}\label{galerkinEquation}
\left\{
\begin{aligned}
\df u_n(t)&= -\left[\im A u_n(t)+\im P_n F(u_n(t))+\beta u_n(t)- b_n (u_n(t)) \right] \df t-\im  S_n B (S_n u_n(t)) \df W(t)
\\
&\qquad\qquad  -\im  S_n G(S_n u_n(t)) \,\df \newW(t) \\
u_n(0)&=P_n u_0.
\end{aligned}\right.
\end{equation}
By the well known theory of finite dimensional stochastic differential equations with locally Lipschitz coefficients,
we get a local well-posedness result for $\eqref{galerkinEquation}.$


\begin{Prop}\label{localSolutionGalerkin}
Suppose  Assumptions $\ref{spaceAssumptions}$, $\ref{F_def}$, $\ref{stochasticAssumptions}$ hold.
Then for each $n\in\N$ there is a unique local solution $u_n$ of $\eqref{galerkinEquation}$ with continuous paths in $H_n$
and maximal existence time $\tau_n,$ which is a blow-up time in the sense that
\[\displaystyle\limsup_{ t \nearrow \tau_n(\omega)}\norm{u_n(t,\omega)}_{H_n}=\infty\]
for almost all $\omega\in\Omega$ with  $\tau_n(\omega)<\infty.$
\end{Prop}

Now we introduce a technical lemma used in many instances for apriori estimates, see \cite[Lemma 5.6]{BHW-2019}.
\begin{Lemma}\label{LemmaYOmegaNachLzweiZeit}
	Let $r\in [1,\infty),$ $\varepsilon>0,$ $T>0$ and $X\in L^r(\Omega,L^\infty(0,T)).$ Then,
	\begin{align*}
	\norm{X}_{L^r(\Omega,L^2(0,t))}\le \varepsilon \norm{X}_{L^r(\Omega,L^\infty(0,t))} +\frac{1}{4\varepsilon}\int_0^t \norm{X}_{L^r(\Omega,L^\infty(0,s))} \df s,\qquad t\in[0,T].
	\end{align*}
\end{Lemma}

We prove apriori estimates in the space $H$ so to get global existence. We work on any  bounded time interval.


\begin{Prop}\label{MassEstimateGalerkinSolution}
Suppose Assumptions $\ref{spaceAssumptions}$, $\ref{F_def}$  and  $\ref{stochasticAssumptions}$
hold. If $u_0\in H$ then for each $n\in\N$  there exists a unique global solution $u_n$ of $\eqref{galerkinEquation}$
with continuous paths in $H_n$.
	\newline
	Moreover, for every finite $T>0$ and $q \in [1, \infty)$
		\begin{equation}
	\label{esHnew}
	\sup_{n \in \mathbb{N}}\mathbb{E} \left[\sup_{s \in [0,T]}\|u_n(s)\|_H^{2q} \right] <\infty.
\end{equation}
\end{Prop}
\begin{proof}
\emph{Step 1:} We fix $n\in\N$ and take the unique maximal  solution $(u_n,\tau_n)$ from Proposition $\ref{localSolutionGalerkin}$.
We prove that the solution is global appealing to the Khasmiskii's test for non explosion, see \cite{Kah}.
Let us introduce a sequence $\{\tau_{n,k}\}_{k\in\N}$ of stopping times defined by
	\begin{align*}
	\tau_{n,k}:=\inf \left\{t\ge0: \norm{u_n(t)}_{H_n}\ge k\right\},\qquad k\in\N.
	\end{align*}
In order to prove that $\tau_n=+ \infty$ $\mathbb{P}$-a.s. it is sufficient to find a Liapunov function $\mathcal{V}:H \rightarrow \mathbb{R}$ satisfying
\begin{align}
 &\mathcal{V} \ge 0  \qquad \text{on} \quad H \label{K1}\\
 &q_k:= \inf_{\|v\|_{H}\ge k}\mathcal{V}(v) \rightarrow \infty, \qquad \text{as} \quad k \rightarrow \infty \label{K2}\\
 &\mathbb{E}[\mathcal{V}(u_n(0)]< \infty  \label{K3}
\end{align}
such that
\begin{equation}
\label{K4}
\mathbb{E}[\mathcal{V}(u(t\wedge \tau_{n,k}))] \le \mathbb{E}[\mathcal{V}(u_n(0))]+ C\int_0^t (1+\mathbb{E}[\mathcal{V}(u(s\wedge \tau_{n,k}))])\, {\rm d}s
\end{equation}
for a constant $C< \infty$ and all $t\ge 0$ and $k \in \mathbb{N}$.
The idea is the following: once such a function $\mathcal{V}$  is found, by the Gronwall's lemma we infer
\begin{equation*}
\mathbb{E}[\mathcal{V}(u_n(t \wedge \tau_{n,k})] \le e^{Ct} (1+ \mathbb{E}[\mathcal{V}(u_n(0))]), \quad t \ge 0,
\end{equation*}
which implies
\begin{equation*}
\mathbb{P}(\tau_{n,k}<t) \le \frac{1}{q_k}\mathbb{E}\left[\mathbbm{1}_{\{\tau_{n,k}<t\}}\mathcal{V}(u_n(t \wedge \tau_{n,k})) \right]
\le \frac{1}{q_k}e^{Ct}\left(1+\mathbb{E}[\mathcal{V}(u_n(0))]\right).
\end{equation*}
Passing to the limit, we get
\begin{equation*}
\lim_{k \rightarrow \infty}\mathbb{P}(\tau_{n,k}<t)\le e^{Ct}(1+\mathbb{E}[\mathcal{V}(u(0))])\lim_{k \rightarrow \infty}\frac{1}{q_k}=0,
\end{equation*}
for every fixed $t\ge 0$. Therefore $\mathbb{P}(\tau_n<t)=\displaystyle\lim_{k \rightarrow \infty}\mathbb{P}(\tau_{n,k}<t)=0$ for every fixed $t \ge 0$, which means $\mathbb{P}(\tau_n=+\infty)=1$.
\newline
Set
\begin{equation}
\label{V_Kas}
\mathcal{V}(v):=\|v\|^2_H, \qquad v \in H_n.
\end{equation}
The function $\mathcal{V} \in C^2(H)$, $\mathcal{V}$ is uniformly continuous on bounded sets and satisfies \eqref{K1} and \eqref{K2}.
Moreover, $\mathbb{E}[\mathcal{V}(P_nu_0)]< \infty$ is equivalent to $\|P_nu_0\|^2_H< \infty$.
\newline
In order to derive inequality \eqref{K4} we use the It\^o formula.
The function $\mathcal{V}: H_n \to \R$  defined  in \eqref{V_Kas} is  twice continuously Fr\'{e}chet-differentiable with
	\begin{align*}
	\mathcal{V}'[v]h_1&= 2 \operatorname{Re} \skpH{v}{ h_1}, \qquad
	\mathcal{V}^{\prime\prime}[v] \left[h_1,h_2\right]= 2 \operatorname{Re} \skpH{ h_1}{h_2},
	\end{align*}
	for $v, h_1, h_2\in H_n.$
We look for estimates for $\norm{u_n}_{H}^2$. The It\^o formula for this real process involves only real quantities,
expressed by means of the real part of the complex scalar product in $H$.

For a fixed $v\in H_n$ and $m\in\N$, we have some basic relationships:
	\begin{align*}
	\operatorname{Re} \skpH{ v}{ -\im A v}&=\operatorname{Re} \left[\im \Vert A^{\frac{1}{2}} v\Vert_{H}^2\right]=0,		\\
	\operatorname{Re} \skpH{ v}{ -\im P_n F\left( v\right)}&=\operatorname{Re} \duality{\im v}{   F\left( v\right)}=0, \\
	2\operatorname{Re} \skpH{ v}{b_n(v)}&= -\sumM\operatorname{Re}  \skpH{ v}{\left(S_n B_m S_n\right)^2 v}=-\sumM  \Vert{ S_n B_m S_n v}\Vert_{H}^2,
	\\
\operatorname{Re} \skpH{v}{ -\im S_n B(S_n v)f_m}&
= \operatorname{Re} \left[\im\skpH{ v}{S_n   B_m S_n v}\right]=0,	
	\end{align*}
where we used $\eqref{PnInEAdual}$ and \eqref{nonlinearityComplex} for the second term and the fact that the operator $S_n B_mS_n $ is self-adjoint for the third and four terms.
		
Therefore, by the It\^o formula we get, for $t\ge 0$,
\begin{align*}
	\norm{u_n(t\wedge \tau_{n,k})}_{H}^2=&\norm{P_n u_0}_{H}^2-2 \beta\int_0^{t\wedge \tau_{n,k}}  \|u_n(s)\|_H^2
 \df s
 +\int_0^{t\wedge \tau_{n,k}}
	\Vert  S_n G\bigl(S_n u_n(s)\bigr) \Vert_{\gamma(Y_2,H)}^2\df  s
\\
&
+2 \int_0^{t\wedge \tau_{n,k}} \operatorname{Re} \skpH{u_n(s)}{ -\im S_n G( S_n u_n(s))  \df \textbf{W}(s)}.
	\end{align*}
To estimate the term in the RHS of the above equation we introduce the  stochastic process
\begin{equation*}
\alpha_k(t):=
\pmb{1}_{(t<\tau_{n,k})}.
\end{equation*}
and we notice that
\begin{equation}
\label{trick1}
\int_0^{t \wedge \tau_{n,k}}\|u_n(s)\|^2_H\, {\rm d}s
= \int_0^t \|\alpha_k(s)u_n(s)\|^2_H\,{\rm d}s
\le \int_0^t \|u_n(s \wedge \tau_{n,k})\|^2_H\, {\rm d}s.
\end{equation}	
Therefore, by Proposition \ref{PaleyLittlewoodLemma} and  \eqref{crescitaG} we get
\begin{equation*}
\Vert  S_n G\bigl(S_n u_n(s)\bigr) \Vert_{\gamma(Y_2,H)}^2 \le 2C_1^2+ 2\tilde{C}_1^2\|u_n(s)\|^2_H.
\end{equation*}
So we obtain
\begin{align*}
	\norm{u_n(t\wedge \tau_{n,k})}_{H}^2\le
	&\norm{P_n u_0}_{H}^2+2\int_0^t (C_1^2+(-\beta+\tilde{C}_1^2)
	 \|u_n(s\wedge \tau_{n,k})\|_H^2)\df  s
\\
&
+2 \int_0^{t\wedge \tau_{n,k}} \operatorname{Re} \skpH{u_n(s)}{ -\im S_n G( S_n u_n(s))  \df \textbf{W}(s)}, \qquad t\ge 0.
\end{align*}
Taking now the expected value on both sides, we obtain
\[
	\mathbb{E}\left[\norm{u_n(t\wedge \tau_{n,k})}_{H}^2\right]
	\le
	\norm{P_n u_0}_{H}^2+2\int_0^t \Big(C_1^2+(-\beta+\tilde{C}_1^2)
	 \mathbb{E}\left[\|u_n(s\wedge \tau_{n,k})\|_H^2\right]\Big)\df  s .
\]
This proves \eqref{K4} and so we conclude the proof of the global existence of the solution.
\newline
\emph{Step 2:}
        We now prove estimate \eqref{esHnew}.
Let us fix $T>0$. We start from equality
\begin{equation}\label{stima-2q-Galerkin}
\begin{split}
	\norm{u_n(t)}_{H}^2=&\norm{P_n u_0}_{H}^2-2 \beta\int_0^{t}  \|u_n(s)\|_H^2
 \df s
 +\int_0^t
	\Vert  S_n G\bigl(S_n u_n(s)\bigr) \Vert_{\gamma(Y_2,H)}^2\df  s
\\
&
+2 \int_0^t \operatorname{Re} \skpH{u_n(s)}{ -\im S_n G( S_n u_n(s))  \df \textbf{W}(s)}
	\end{split}\end{equation}
for $t\in [0,T]$, and we apply the $L^q(\Omega, L^{\infty}(0,T))$-norm to this identity.
From Proposition \ref{PaleyLittlewoodLemma} and  \eqref{crescitaG}  we immediately get
\begin{equation*}
\left \Vert \int_0^\cdot
	\Vert  S_n G\bigl(S_n u_n(s)\bigr) \Vert_{\gamma(Y_2,H)}^2\df  s \right \Vert_{L^q(\Omega, L^{\infty}(0,T))}	\le 2C_1^2T + 2\tilde C_1^2 \left\Vert\int_0^T
	\|u_n(s)\|_H^2\df  s \right\Vert_{L^q(\Omega)}.\
	\end{equation*}
The Minkowski inequality yields
\begin{align*}
	\left \Vert \int_0^T\|u_n(s)\|^2_H\, {\rm d}s\right \Vert_{L^q(\Omega)}
&\le\left \Vert \int_0^{T}\sup_{r \in[0,s]}\|u_n(r)\|^2_H\, {\rm d}s\right \Vert_{L^q(\Omega)}
\\
&\le \int_0^T\left \Vert \sup_{r \in[0,s]}\|u_n(r)\|^2_H\right \Vert_{L^q(\Omega)} \, {\rm d}s
= \int_0^T\left\Vert \|u_n\|^2_H \right\Vert_{L^q(\Omega,L^{\infty}(0,s))}\, {\rm d}s.
\end{align*}
By means of the Burkholder-Davis-Gundy and the Young inequalities, Lemma \ref{LemmaYOmegaNachLzweiZeit} and \eqref{crescitaG} we obtain,  for some constant $C$ (depending on $q$)
\[\begin{split}
	  \Big\Vert\int_0^{\cdot} \operatorname{Re} & \skpH{u_n(s)}{ -\im S_n G(S_n u_n(s)) \df \textbf{W}(s)} \Big\Vert_{L^q(\Omega, L^{\infty}(0,T))}	
	  \\
           &\le C	 \left\Vert \left(\int_0^T\sum_{m=1}^{\infty} \skpH{u_n(s)}{ -\im S_n G(S_n u_n(s))e_m}^2 ds\right)^{1/2}\right\Vert_{L^q(\Omega)}	
	 \\
           &  \le C \left\Vert\|G(S_nu_n)\|_{\gamma(Y_2,H)}\|u_n\|_H \right\Vert_{L^q(\Omega, L^2(0,T))}	
           \\
            &   \le C \left\Vert   (C_1+\tilde C_1 \|u_n\|_H) \|u_n\|_H \right\Vert_{L^q(\Omega, L^2(0,T))}	
	 \\
           &\lesssim  \left\Vert   1+ \|u_n\|_H^2    \right\Vert_{L^q(\Omega, L^2(0,T))}	
	 \\
	  &\le1+T + \left\Vert\|u_n\|^2_H\right\Vert_{L^q(\Omega, L^2(0,T))}
	  \\
	  &
	 \le 1+T + \varepsilon \left\Vert\|u_n\|^2_H\right\Vert_{L^q(\Omega, L^{\infty}(0,T))}
	        + \frac{1}{4 \varepsilon} \int_0^T \left\Vert\|u_n\|^2_H \right\Vert_{L^q(\Omega, L^{\infty}(0,s))}\,{\rm d}s,
	 \end{split}\]	
for any $\varepsilon>0$.
	
Collecting the above estimates, we obtain
\begin{align*}
\left\Vert \|u_n\|_H^{2}\right\Vert_{L^q(\Omega, L^{\infty}(0,T))}	
\le \|u_0\|^2_H+
 C+CT+
+ \varepsilon C \left\Vert\|u_n\|^2_H\right\Vert_{L^q(\Omega, L^{\infty}(0,T))}
\\
+\left(2|\beta|+ \frac{C}{4\varepsilon}\right)
\int_0^T\left\Vert\|u_n\|^2_H \right\Vert_{L^q(\Omega, L^{\infty}(0,s))}\,{\rm d}s.
\end{align*}
If we choose $\varepsilon>0$ small enough, we can apply the Gronwall's lemma and we get that there exists a positive constant $C$, independent of $n$ (but depending on $T$ and other parameters)  such that
\begin{equation*}
\left\Vert \|u_n\|_H^2\right\Vert_{L^q(\Omega, L^{\infty}(0,T))}\le C,\qedhere
\end{equation*}
for any $n \in \mathbb{N}$.
	\end{proof}
	

From the proof we see that the result holds true also for random initial data and working on any finite time interval.
\begin{corollary}
Let $[t_0,T]\subset [0,+\infty)$ and $q \in [1, \infty)$.
Under the assumptions of Proposition \ref{MassEstimateGalerkinSolution}, if $u_0\in L^{2q}(\Omega,H)$ is $\mathscr{F}}%{\mathcal{F}_{t_0}$-measurable, then there exists
 a unique global solution $u_n$ of $\eqref{galerkinEquation}$ with continuous paths in $H_n$ and
 \[
\sup_{n \in \mathbb{N}}\mathbb{E} \left[\sup_{s \in [t_0,T]}\|u_n(s)\|_H^{2q} \right] <\infty.
\]
\end{corollary}


The  next goal  is to find uniform energy estimates for the global solutions of the equation $\eqref{galerkinEquation}.$  Recall that the nonlinearity $F$ has a real antiderivative denoted by $\hat{F}.$

\begin{Definition}
	We define the energy
	\begin{align}
	\label{energy_def}
	\mathcal{E}(u):= \frac{1}{2} \Vert A^{\frac{1}{2}} u \Vert_{H}^2+\hat{F}(u),\qquad u\in {V}.
	\end{align}	
\end{Definition}
Note that $\mathcal{E}(u)$ is well defined by the  embedding ${V} \hookrightarrow L^{\alpha+1}.$



The next Proposition is the key step to show that we can apply Corollary \ref{corollaryEstimatesToASconvergence} to the sequence of solutions $(u_n)_{n\in\N}$ of the equation \eqref{galerkinEquation}. We  work on any  bounded time interval.
	
\begin{Prop}\label{EstimatesGalerkinSolution}
	 Under Assumptions $\ref{spaceAssumptions}$, $\ref{F_def}$
	 and  $\ref{stochasticAssumptions}$ the following assertions hold.
	\begin{enumerate}
		\item[a)]   If $u_0 \in {V}$, then for every finite $T>0$ and $q \in [1, \infty)$
				\begin{equation}\label{stimaEnergy}
		                 \sup_{n\in\N}\mathbb{E} \Big[ \sup_{t \in [0,T]} \mathcal{E}(u_n(t)) ^q\Big] <\infty.
		                  \end{equation}
		\item[b)] The sequence $(u_n)_{n\in\N}$ satisfies the Aldous condition $[A]$ in ${V^*}.$
		\item[c)]
		                If $\beta$ satisfies condition \eqref{condizione-beta}, then
                                 \begin{equation}\label{Galerkin-Ebound}
                                   \sup_{n \in \mathbb{N}} \sup_{t\ge 0} \mathbb E \|u_n(t)\|^2_{{V}}<\infty.
                                  \end{equation}
	\end{enumerate}
\end{Prop}
\begin{proof}
\emph{ad a):}
By Lemma $\ref{nonlinearAssumptions}$, the restriction of the energy $\mathcal{E}: H_n \to \R$ is twice
continuously Fr\'{e}chet-differentiable with
	\begin{align*}
	\mathcal{E}'[v]h_1=&\operatorname{Re} \duality{Av+F(v)}{ h_1}; \\
	\mathcal{E}^{\prime\prime}[v] \left[h_1,h_2\right]=& \operatorname{Re} \skpH{A^{\frac{1}{2}} h_1}{A^{\frac{1}{2}} h_2}+\operatorname{Re} \duality{F^{\prime}[v]h_2}{h_1}
	\end{align*}
	for $v, h_1, h_2\in H_n.$
We look for estimates on $\mathcal{E}(u_n)$.
		
	Notice that
\begin{align*}
	\operatorname{Re} \left[\duality{A v}{ -\im P_n F(v)}+\duality{F(v)}{ -\im A v}\right]
	&=\operatorname{Re} \left[-\duality{A v}{ \im F(v)}+\overline{\duality{  A v}{\im F(v)}}\right]=0,
	\end{align*}
	\begin{align*}
	\operatorname{Re} \skpH{A v}{ -\im A v}=\operatorname{Re} \left[\im \norm{A v}_{H}^2\right]=0,
	\end{align*}
	for all $v\in H_n$,	
and we can use $\eqref{PnInEAdual}$ for
	\begin{align*}
	\operatorname{Re} \duality{F(v)}{ -\im P_n F(v)}=\operatorname{Re} \left[\im \duality{F(v)}{  P_n F(v)}\right]=0.
	\end{align*}
Therefore, the It\^o formula leads to the identity
\begin{equation}\label{ItoEnergyWithoutExponent}
   \begin{split}
	\mathcal{E}\left(u_n(t)\right) &=	\mathcal{E}\left(P_n u_0\right)	
	+\int_0^t \operatorname{Re} \duality{A u_n(s)+F(u_n(s))}{ b_n(u_n(s))-\beta u_n(s)} \df s	
	\\
	&+\int_0^t \operatorname{Re} \duality{A u_n(s)+F(u_n(s))}{ -\im \left(S_n B S_n u_n(s)\right)\df W(s)}
	\\
	&+\int_0^t \operatorname{Re} \duality{A u_n(s)+F(u_n(s))}{ -\im S_n G \left(S_n u_n(s)\right)\df \newW(s)}
	 \\
	&+\tfrac{1}{2} \int_0^t  \Vert A^{\frac{1}{2}} S_n B S_n u_n(s)\Vert_{\gamma(Y_1,H)}^2\df s
	+\tfrac{1}{2} \int_0^t  \Vert A^{\frac{1}{2}} S_n G S_n u_n(s)\Vert^2_{\gamma{(Y_2,H)}}\df s
	\\
	&+\tfrac{1}{2}\int_0^t \sumM \operatorname{Re} \duality{F^{\prime}[u_n(s)] \left(S_n B S_n u_n(s)\right)f_m}{ (S_n B S_n u_n(s))f_m} \df s
	\\
	&+\tfrac{1}{2}\int_0^t \sumM \operatorname{Re} \duality{F^{\prime}[u_n(s)] \left(S_n G( S_n u_n(s))e_m\right)}{ S_n G(S_n u_n(s))e_m} \df s,
\end{split}\end{equation}	
almost surely for all $t\in [0,T].$

Let us introduce the short notation
\begin{equation}\label{def-z-somma}
Z(u):= \|u\|^2_H + 2\mathcal{E}(u)\equiv \|u\|^2_V+2\hat{F}(u), \quad u \in V.
\end{equation}
We will estimate the various terms that appear in the RHS of
\eqref{ItoEnergyWithoutExponent} in the $L^q(\Omega, L^{\infty}(0,T))$-norm.
To slightly simplify the proof we will neglect here the dissipation term by assuming $\beta \ge 0$.
The case $\beta<0$ can be treated as in the proof of Proposition \ref{EstimatesGalerkinSolution} (c), see Appendix \ref{App_B}.
We set $I_{1,n}(u)(t)=2\int_0^t  \operatorname{Re} \duality{A u(s)+F(u(s))}{ b_n(u(s))} \df s$. We have
\[
\left \Vert I_{1,n}(u_n) \right \Vert_{L^q(\Omega,L^{\infty}(0,T))}
\le 	\left\Vert \int_0^T\left| \sumM  \operatorname{Re} \duality{A u_n(s)+F(u_n(s))}{(S_nB_mS_n)^2u_n(s)} \right|\df s\right \Vert_{L^q(\Omega)}.
\]
Using the bounds $\eqref{SnUniformlyBounded}$ and Assumptions \ref{stochasticAssumptions}(ii), see also Remark \ref{HS}, by means of the Young inequality we get
\[\begin{split}
\left| \sumM  \operatorname{Re} \duality{A u_n}{(S_nB_mS_n)^2u_n} \right|
&\le  \|A^{\frac 12}u_n\|_H  \sumM \|A^{\frac 12}(S_nB_mS_n)^2u_n)\|_H
\\
&\le \|A^{\frac 12}u_n\|_H  \|B\|^2_{\mathscr{L}({V}, \gamma(Y_1,{V}))}   \|u_n\|_{{V}}
\\
&\le \|B\|^2_{\mathscr{L}({V}, \gamma(Y_1,{V}))}  \left(
\|A^{\frac 12}u_n\|^2_H+\|u_n\|^2_H\right)
\\&
\lesssim\|B\|^2_{\mathscr{L}({V}, \gamma(Y_1,{V}))}Z(u_n).
\end{split}\]
On the other hand, $\eqref{SnUniformlyBounded},$  Assumptions \ref{stochasticAssumptions}(ii),
$\eqref{nonlinearityEstimate}$ and $\eqref{boundantiderivative}$ lead to
\[\begin{split}
 \left| \sumM \operatorname{Re} \duality{F(u_n)}{(S_nB_mS_n)^2u_n}\right|
 &\le \|F(u_n)\|_{L^{\frac{\alpha+1}\alpha}} \sumM \|(S_nB_mS_n)^2u_n\|_{L^{\alpha+1}}
 \\
 &\lesssim
 \hat F(u_n)\|B\|^2_{\mathscr{L}(L^{\alpha+1},\gamma(Y_1,L^{\alpha+1}))}
 \\
 &\lesssim
 \|B\|^2_{\mathscr{L}(L^{\alpha+1},\gamma(Y_1,L^{\alpha+1}))}Z(u_n).
\end{split}\]
Therefore we obtain
\begin{multline}
\label{e1}
\left \Vert I_{1,n}(u_n)  \right \Vert_{L^q(\Omega,L^{\infty}(0,T))}
\\
\lesssim
\left(\|B\|^2_{\mathscr{L}({V}, \gamma(Y_1,{V}))} + \|B\|^2_{\mathscr{L}(L^{\alpha+1},\gamma(Y_1,L^{\alpha+1}))}\right)\int_0^T\|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))}\, {\rm d}s.
\end{multline}

We set $I_{2,n}(u)(t)=2\int_0^t \operatorname{Re} \duality{A u(s)+F(u(s))}{ -\im S_n B \left(S_n u(s)\right)\df W(s)}$.
We employ the Burkholder-Davis-Gundy inequality to get
\[
\left \Vert   I_{2,n}(u_n)     \right \Vert_{L^q(\Omega,L^{\infty}(0,T))}
\le 2
\left \Vert \left(\sumM|\operatorname{Re} \duality {A u_n+F(u_n)}{ -\im (S_n B S_n u_n)f_m}|^2 \right)^{\frac 12}\right \Vert_{L^q(\Omega,L^2(0,T))} .
\]
Using the bounds $\eqref{SnUniformlyBounded},$  Assumptions \ref{stochasticAssumptions}(ii) and the Young inequality, we estimate
\[\begin{split}
\left(  \sumM  |\operatorname{Re} \duality{A u_n}{ -\im \left( S_n B S_n u_n\right) f_m}|^2 \right)^{\frac 12}
&\le \|A^{\frac 12}u_n\|_H \|B(S_nu_n)\|_{\gamma(Y_1, {V})}
\\&
\le \|A^{\frac 12}u_n\|_H  \|B\|_{\mathcal L ({V},\gamma(Y_1, {V}))}   \|u_n\|_{{V}}
\\&
\le \|B\|_{\mathscr{L}({V},\gamma(Y_1, {V}))}  \left( \|A^{\frac 12}u_n\|^2_H+\|u_n\|^2_H\right)
\\&
\lesssim \|B\|_{\mathscr{L}({V},\gamma(Y_1, {V}))}  Z(u_n).
\end{split}\]
On the other hand $\eqref{SnUniformlyBounded},$  Assumptions \ref{stochasticAssumptions}(ii), $\eqref{nonlinearityEstimate}$ and $\eqref{boundantiderivative}$ lead to
\[\begin{split}
 \left(\sumM|\operatorname{Re} \duality {F(u_n)}{ -\im \left(S_n B S_n u_n\right))f_m}|^2 \right)^{\frac 12}
 &\lesssim \|B\|_{\mathscr{L}(L^{\alpha+1}, \gamma(Y_1, L^{\alpha+1}))} \hat F(u_n)
 \\
&\lesssim \|B\|_{\mathscr{L}(L^{\alpha+1}, \gamma(Y_1, L^{\alpha+1}))} Z(u_n).
 \end{split}\]
Therefore, the latter three estimates with Lemma $\ref{LemmaYOmegaNachLzweiZeit}$
 yield
\begin{align}
\label{e2}
 \Vert I_{2,n}(u) \Vert_{L^q(\Omega,L^{\infty}(0,T))}
&
\lesssim
\left(\|B\|_{\mathscr{L}({V}, \gamma(Y_1,{V}))} + \|B\|_{\mathscr{L}(L^{\alpha+1},\gamma(Y_1,L^{\alpha+1}))}\right)\|Z(u_n)\|_{L^q(\Omega,L^2(0,T))}
\notag \\&
\lesssim \varepsilon \|Z(u_n)\|_{L^q(\Omega, L^{\infty}(0,T))}+\frac{1}{4\varepsilon}\int_0^T \|Z(u_n)\|_{L^q(\Omega, L^{\infty}(0,s))}\, {\rm d}s,
\end{align}
for any $\varepsilon>0$.

We set $I_{3,n}(u)(t)=2\int_0^t   \operatorname{Re} \duality{A u(s)+F(u(s))}{ -\im S_n G \left(S_n u(s)\right)\df \newW(s)} $.
Also for this stochastic integral we employ  the Burkholder-Davis-Gundy's inequality to get
\[
\left \Vert  I_{3,n}(u_n)\right \Vert_{L^q(\Omega,L^{\infty}(0,T))}
\le 2 \left \Vert \left(\sumM|\operatorname{Re} \duality {A u_n+F(u_n)}{ -\im S_n G \left(S_n u_n\right))e_m}|^2 \right)^{\frac 12}\right \Vert_{L^q(\Omega,L^2(0,T))}.
\]
Using the bounds $\eqref{SnUniformlyBounded}$ and Assumptions \ref{stochasticAssumptions}(iii), exploiting the Young inequality we obtain
\[\begin{split}
 \left(\sumM|\operatorname{Re} \duality{A u_n}{ -\im \left( S_n G (S_n u_n)\right)e_m} |^2 \right)^{\frac 12}
 &\lesssim \|A^{\frac12}u_n\|_H \|G(S_nu_n)\|_{\gamma(Y_2, {V})}
 \\&
 \lesssim \|A^{\frac12 }u_n\|_H \left(C_2+ \tilde C_2\|u_n\|_{{V}}\right)
\\&
 \lesssim 1+ Z(u_n).
 \end{split}\]
Moreover, $\eqref{SnUniformlyBounded},$  Assumptions \ref{stochasticAssumptions}(iii), $\eqref{nonlinearityEstimate}$,
$\eqref{boundantiderivative}$, \eqref{ineq-F} and the Young inequality lead to
 \[\begin{split}
 \left(\sumM|\operatorname{Re} \duality {F(u_n)}{ -\im S_n G \left(S_n u_n\right)e_m}|^2 \right)^{\frac 12}
 &\le \|F(u_n)\|_{L^{\frac{\alpha+1}{\alpha}}}\|G(S_nu_n)\|_{\gamma(Y_2,L^{\alpha+1})}
 \\
 &\lesssim \|u_n\|^{\alpha}_{L^{\alpha+1}}\left(C_3+ \tilde C_3\|u_n\|_{L^{\alpha+1}} \right)
 \\
 &\lesssim [\hat F(u_n)]^{\frac{\alpha}{\alpha+1}}+ \hat F(u_n)
  \\
 &\lesssim 1+ \hat F(u_n)
 \\
 &\lesssim  1 +Z(u_n).
  \end{split}\]
 Therefore, collecting the previous estimates, by means of Lemma $\ref{LemmaYOmegaNachLzweiZeit}$ we obtain
\begin{equation} \begin{split}
 \label{e3}
 \left \Vert   I_{3,n}(u_n) \right \Vert_{L^q(\Omega,L^{\infty}(0,T))}
 &
\lesssim
\|1+Z(u_n)\|_{L^q(\Omega, L^2(0,T))}
\\&
\lesssim 1+ \varepsilon \|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,T))}+ \frac{1}{4 \varepsilon}\int_0^T \|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))} \, {\rm d}s,
\end{split}\end{equation}
for any $\varepsilon>0$.


We set $I_{4,n}(u)(t)= \int_0^t  \Vert A^{\frac{1}{2}} S_n B S_n u(s)\Vert_{\gamma(Y_1,H)}^2\df s $.
 Exploiting Remark \ref{HS} and bounds $\eqref{SnUniformlyBounded}$, we easily obtain
\begin{equation}
\label{e4}
  \left\Vert I_{4,n}(u_n) \right\Vert_{L^q(\Omega, L^{\infty}(0,T))}
 \lesssim \|B\|^2_{\mathscr{L}({V},\gamma(Y_1, {V}))}\int_0^T \|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))}\,{\rm d}s.
\end{equation}


We set $I_{5,n}(u)(t)=\int_0^t \Vert A^{\frac{1}{2}} S_n G (S_n u(s))\Vert_{\gamma(Y_2,H)}^2\df s $.
Similarly, estimate \eqref{crescitaGEA} and bounds \eqref{SnUniformlyBounded} yield
\[
\Vert A^{\frac{1}{2}} S_n G (S_n u_n)\Vert_{\gamma(Y_2,H)}^2\le (C_2 +\tilde C_2 \|u_n\|_V)^2
 \lesssim 1+ Z(u_n) .
\]
Hence
\begin{equation}
\label{e5}
 \left\Vert  I_{5,n}(u_n)   \right\Vert_{L^q(\Omega, L^{\infty}(0,T))}
 \lesssim 1+ \int_0^T \|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))}\,{\rm d}s.
\end{equation}

We set $I_{6,n}(u)(t)=\int_0^t \sumM \operatorname{Re} \duality{F^{\prime}[u(s)] \left(S_n B (S_n u(s))f_m\right)}{ S_n B (S_n u(s))f_m} \df s$.
From \eqref{deriveNonlinearBound}, Remark \ref{HS}, \eqref{SnUniformlyBounded}  and \eqref{boundantiderivative} we get
\[\begin{split}
\left| \sumM \operatorname{Re} \duality{F^{\prime}[u_n] \left(S_n B S_n u_n\right)f_m}{ (S_n B S_n u_n) f_m} \right|
& \lesssim
  \|B\|^2_{\mathscr{L}(L^{\alpha+1}(M),\gamma(Y_1,L^{\alpha+1})} \|u_n\|^{\alpha+1}_{L^{\alpha+1}}
\\&
 \lesssim  \|B\|^2_{\mathscr{L}(L^{\alpha+1},\gamma(Y_1,L^{\alpha+1})} \hat F(u_n).
\end{split}\]
Hence,
\begin{equation}\label{e6}
\left\Vert   I_{6,n}(u_n)    \right\Vert_{L^q(\Omega, L^{\infty}(0,T))}
 \lesssim \|B\|^2_{\mathscr{L}(L^{\alpha+1},\gamma(Y_1,L^{\alpha+1}))}
 \int_0^T \|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))}\,{\rm d}s.
\end{equation}

We set $I_{7,n}(u)(t)=\int_0^t \sumM \operatorname{Re} \duality{F^{\prime}[u(s)] \left(S_n G( S_n u(s))e_m\right)}{ S_n G(S_n u(s))e_m} \df s $.
By means of   \eqref{ineq-Fp}, \eqref{crescitaGL}, \eqref{SnUniformlyBounded} and the Young inequality
 we get
\[\begin{split}
\left|  \sumM \operatorname{Re} \duality{F^{\prime}[u_n] \left(S_n G( S_n u_n)e_m\right)}{ S_n G(S_n u_n)e_m}    \right|
&\le
\|F'[u_n]\|_{L^{\alpha+1} \rightarrow L^{\frac{\alpha+1}{\alpha}}}
\sumM \|S_n G( S_n u_n)e_m\|^2 _{L^{\alpha+1}}
\\
&\lesssim
\hat F(u_n)^{\frac{\alpha-1}{\alpha+1}} \|G( S_n u_n)\|^2 _{\gamma(Y_2,L^{\alpha+1})}
\\
&\le
\hat F(u_n)^{\frac{\alpha-1}{\alpha+1}}  (C_3+\tilde C_3 \|u_n\|_{L^{\alpha+1}})^2
\\
&\lesssim \hat F(u_n)^{\frac{\alpha-1}{\alpha+1}}+\hat F(u_n)
\\
&\lesssim 1+\hat F(u_n) \le 1+ Z(u_n).
\end{split}\]
Therefore,
\begin{equation}
\label{e7}
\left\Vert I_{7,n}(u_n)  \right\Vert_{L^q(\Omega, L^{\infty}(0,T))}
\lesssim
1+\int_0^T \|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))}\,{\rm d}s.
\end{equation}

We now go back to \eqref{ItoEnergyWithoutExponent}; using \eqref{e1}-\eqref{e7} we finally obtain
\[\begin{split}
\|\mathcal{E}(u_n)\|_{L^q(\Omega,L^{\infty}(0,T))}
&\le
\mathcal{E}(P_nu_0)+C
+C  \int_0^T \|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))}\, {\rm d}s
+2 \varepsilon \|Z(u_n)\|_{L^q(\Omega,L^{\infty}(0,T))}
\\
&\le
\mathcal{E}(P_nu_0)+C
+
C  \int_0^T 2\|\mathcal{E}(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))}\, {\rm d}s
+4 \varepsilon \|\mathcal{E}(u_n)\|_{L^q(\Omega,L^{\infty}(0,T))}
\\
&\quad+(CT+2 \varepsilon)   \left\Vert\|u_n\|^2_H\right\Vert_{L^q(\Omega,L^{\infty}(0,T))}
\end{split}\]
for some positive constant $C$, independent of $n$.
If we choose $\varepsilon$ sufficiently small and bear in mind  the apriori estimate \eqref{esHnew} of Proposition \ref{MassEstimateGalerkinSolution}, we get
\[
\|\mathcal{E}(u_n)\|_{L^q(\Omega,L^{\infty}(0,T))} \lesssim 1+  \int_0^T \|\mathcal{E}(u_n)\|_{L^q(\Omega,L^{\infty}(0,s))}\, {\rm d}s.
\]
By the Gronwall Lemma we deduce the assertion of Proposition \ref{EstimatesGalerkinSolution}, part a).

\emph{ad b):}
We choose and fix $T>0$. Let us now prove the Aldous condition.
The proof of this part is an extension of the proof of part (b) in \cite[Proposition 5.7]{BHW-2019}.
We will provide just the main steps.
We have
	\begin{align*}
	u_n(t)- P_n u_0=&  -\im \int_0^t A u_n(s) \df s-\im \int_0^t P_n F(u_n(s)) \df s+\int_0^t b_n(u_n(s)) \df s
	-\beta \int_0^t u_n(s)\, {\rm d}s	\\
	\hspace{1cm}&- \im \int_0^t S_n B (S_n u_n(s)) \df W(s)
	-\im \int_0^t S_nG(S_n u_n(s))\, {\rm d}\textbf{W}(s)
	\\
	=&:J_1(t)+J_2(t)+J_3(t)+J_4(t)+J_5(t)+J_6(t),
	\end{align*}
	in $H_n$ almost surely for all $t\in [0,T]$ and therefore
	\begin{align*}
	\Vert u_n((\tau_n+\theta)\land T)-u_n(\tau_n)\Vert_{V^*}\le\sum_{k=1}^{6} \Vert J_k((\tau_n+\theta)\land T)-J_k(\tau_n)\Vert_{{V^*}}
	\end{align*}
	for each sequence $\left(\tau_n\right)_{n\in\N}$ of stopping times and $\theta>0.$
	Hence, we get
	\begin{align}\label{AldousStartingEstimate}
	\mathbb{P} \left\{\Vert u_n((\tau_n+\theta)\land T)-u_n(\tau_n)\Vert_{V^*}\ge \eta \right\}\le \sum_{k=1}^6\mathbb{P} \left\{\Vert J_k((\tau_n+\theta)\land T)-J_k(\tau_n)\Vert_{{V^*}}\ge \frac{\eta}{6}\right\},
	\end{align}
	for a fixed $\eta>0$. We aim to apply the Chebyshev inequality and estimate the expected value of each term in the sum. 	
	Proceeding as in the proof of part (b) in \cite[Proposition 5.7]{BHW-2019} one obtains the following estimates:
	\begin{align*}
	\mathbb{E}\Vert J_1((\tau_n+\theta)\land T)-J_1(\tau_n)\Vert_{{V^*}}&\le  \theta K_1,
	\end{align*}
	\begin{align*}
	\mathbb{E}\Vert J_2((\tau_n+\theta)\land T)&-J_2(\tau_n)\Vert_{{V^*}}
	\le \theta K_2,
	\end{align*}
	\begin{align*}
	\mathbb{E}\Vert J_3((\tau_n+\theta)\land T)-J_3(\tau_n)\Vert_{{V^*}}
	\le \theta K_3,
	\end{align*}
\begin{align*}
	\mathbb{E}\Vert J_5((\tau_n+\theta)\land T)-J_5(\tau_n)\Vert_{{V^*}}^2&
	\le \theta K_5.
	\end{align*}
We use part a) to estimate
\begin{align*}
	\mathbb{E}\Vert J_4((\tau_n+\theta)\land T)&-J_4(\tau_n)\Vert_{{V^*}}
	\le \beta\mathbb{E} \left\Vert\int_{\tau_n}^{(\tau_n+\theta)\wedge T} u_n(s)\, {\rm d}s\right\Vert_{H}
	\\
	&\le \beta\mathbb{E}\int_{\tau_n}^{(\tau_n+\theta)\wedge T} \|u_n(s)\|_{H}\, {\rm d}s
	\lesssim \theta \beta\mathbb{E} \left[\sup_{s \in [0,T]}\|u_n(s)\|_H\right] \le \theta K_4.
		\end{align*}
The It\^o isometry and \eqref{crescitaG} yield
\begin{align*}
\mathbb{E}\Vert J_6((\tau_n+\theta)\land T)&-J_6(\tau_n)\Vert^2_{{V^*}}
	\le \mathbb{E} \left\Vert \int_{\tau_n}^{(\tau_n+\theta)\wedge T}S_nG(S_nu_n(s))\, {\rm d}\newW(s)\right\Vert^2_H
	\\
	& = \mathbb{E}\int_{\tau_n}^{(\tau_n+\theta)\wedge T}\|S_nG(S_nu_n(s))\|^2_{\gamma(Y,H)}\, {\rm d}s
	\\
	&\le \mathbb{E}\int_{\tau_n}^{(\tau_n+\theta)\wedge T}\left(2C_1^2+2\tilde C_1^2\|u_n(s)\|^2_H \right)\, {\rm d}s
	\\
	&\lesssim \mathbb{E}\int_{\tau_n}^{(\tau_n+\theta)\wedge T}\left(1+\|u_n(s)\|^2_H\right)\, {\rm d}s
	\lesssim \theta + \theta \mathbb{E} \left[ \sup_{s \in [0,T]}\|u_n(s)\|_H^2\right] \lesssim \theta K_6.
\end{align*}
	By the Chebyschev inequality, we obtain for any given $\eta>0$
	\begin{align}\label{AldousEins}
	\mathbb{P} \left\{\Vert J_k((\tau_n+\theta)\land T)-J_k(\tau_n)\Vert_{{V^*}}\ge \frac{\eta}{6}\right\}\le \frac{6}{\eta} \mathbb{E}\Vert J_k((\tau_n+\theta)\land T)-J_k(\tau_n)\Vert_{{V^*}}\le \frac{ 6K_k \theta}{\eta}
	\end{align}
	for $k\in \{1,2,3,4\}$ and
	\begin{align}\label{AldousZwei}
	\mathbb{P} \left\{\Vert J_k((\tau_n+\theta)\land T)-J_k(\tau_n)\Vert_{{V^*}}\ge \frac{\eta}{6}\right\}\le \frac{36}{\eta^2} \mathbb{E}\Vert J_k((\tau_n+\theta)\land T)-J_k(\tau_n)\Vert_{{V^*}}^2\le \frac{36 K_k \theta}{\eta^2}
	\end{align}
	for $k \in \{5,6\}$.
	Let us fix $\varepsilon>0$ and $\eta>0$. Due to  estimates $\eqref{AldousEins}$ and $\eqref{AldousZwei}$ we can choose $\delta_1,\dots,\delta_{6}>0$ such that
	\begin{align*}
	\mathbb{P} \left\{\Vert J_k((\tau_n+\theta)\land T)-J_k(\tau_n)\Vert_{{V^*}}\ge \frac{\eta}{6}\right\}\le \frac{\varepsilon}{6}
	\end{align*}
	for $0<\theta\le \delta_k$ and $k=1,\dots,6.$ With $\delta:= \min \left\{\delta_1,\dots,\delta_{6}\right\},$  using $\eqref{AldousStartingEstimate}$ we get
	\begin{align*}
	\mathbb{P} \left\{\Vert J_k((\tau_n+\theta)\land T)-J_k(\tau_n)\Vert_{{V^*}}\ge \eta\right\}\le \varepsilon
	\end{align*}
	for all $n\in\N$, $k=1,...6$ and $0<\theta\le \delta$ and therefore, the Aldous condition $[A]$ holds in $V^*.$	

\emph{ad c):}
This point has some similarities with point (a) proved above.
We prove it in Appendix \ref{App_B}.
\end{proof}

As an immediate consequence of Propositions \ref{MassEstimateGalerkinSolution} and  \ref{EstimatesGalerkinSolution} we obtain
\begin{corollary}
\label{aprioriEA}
Let $[t_0,T]\subset [0,+\infty)$ and $q \in [1, \infty)$.
Under Assumptions $\ref{spaceAssumptions}$, $\ref{F_def}$, $\ref{stochasticAssumptions}$,
 if $u_0\in L^{q}(\Omega,{V})$ is $\mathscr{F}}%{\mathcal{F}_{t_0}$-measurable,
 then
		\begin{align*}
		\sup_{n\in\N}\mathbb{E} \Big[\sup_{t\in[0,T]} \norm{u_n(t)}_{V}^{q}\Big]<\infty.
		\end{align*}
\end{corollary}


\subsection{Convergence. Proof of the first part of Theorem \ref{existTheorem}}
\label{Section_Convergence}



In this Section we prove part (i) of Theorem \ref{existTheorem}, that is the existence of a martingale solution of \eqref{Problem} which satisfies $u \in L^q(\tilde{\Omega},L^{\infty}(0,T;V))$ in \eqref{propertySolution}. The additional regularity $u \in L^q(\tilde{\Omega}, C([0,T];H))$ will follow as a byproduct by Proposition \ref{mild_prop}.


We construct a solution to equation \eqref{Problem} by a suitable limiting process in the Galerkin equation \eqref{galerkinEquation}, exploiting the results of the previous sections.
Proposition \ref{EstimatesGalerkinSolution}(b) and Corollary \ref{aprioriEA} provide the tightness to pass to the limit.
One proceed as in \cite{BHW-2019} and \cite{BM}. We will just provide the main steps of the proof and refer to these papers for more details.

Let us recall from Section \ref{CompactnessSection} the definition of the space
\begin{align*}
Z_T={C([0,T],\EAdual)}\cap {L^{\alpha+1}(0,T;\LalphaPlusEins)} \cap C_w([0,T],\EA).
\end{align*}
Proposition \ref{EstimatesGalerkinSolution} (b)  and Corollary \ref{aprioriEA}
provide the apriori estimates on the Galerkin approximation sequence; hence, this is tight  in $Z_T$ thanks to Proposition \ref{TightnessCriterion}.
Then by means of Corollary \ref{corollaryEstimatesToASconvergence}  we get the convergence.
More precisely,  there exist a subsequence $\left(u_{n_k}\right)_{k\in\N}$, a probability space $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}}\right)$ and random variables $v_k, v:\tilde{\Omega} \rightarrow Z_T$ with $\tilde{\mathbb{P}}^{v_k}=\mathbb{P}^{u_{n_k}}$ such that
\begin{equation}
\label{convergencev_n}
v_k\to v  \ \quad \tilde{\mathbb{P}}-\text{a.s.} \quad  \text{in $Z_T$ for $k\to \infty$}.
\end{equation}
Moreover, $v_k \in  C\left([0,T],H_k\right)$ $\tilde{\mathbb{P}}$-a.s. and for all $r\in [1,\infty)$ we have
\begin{align}
\label{STAR}
\sup_{k\in\N} \tilde{\E} \left[ \norm{v_k}_{L^\infty(0,T;\EA)}^r\right]<\infty.
\end{align}
Let us remark that by considering again a subsequence, if necessary, we also get
\begin{equation}\label{confronto-norme}
\tilde{\E} \left[ \norm{v}_{L^\infty(0,T;\EA)}^r\right] \le \liminf_k \tilde{\E} \left[ \norm{v_k}_{L^\infty(0,T;\EA)}^r\right]< \infty.
\end{equation}
Since each $v_n$ has the same law as $u_{n_k}$, it is a martingale solution to equation \eqref{galerkinEquation}: expanding the arguments in \cite[Lemma 6.3]{BHW-2019} one can easily prove that each process $N_n: \tilde{\Omega} \times [0,T] \rightarrow H_n$ defined by
\begin{align}
\label{Nn}
N_n(t)=-v_n(t)+ P_n u_0&+ \int_0^t \left[-\im A v_n(s)-\im P_n F(v_n(s))+b_n(v_n(s))-\beta v_n(s)\right] \df s
\end{align}
for $n\in \N$ and $t\in[0,T]$ is  an ${H}$-valued continuous square integrable martingale w.r.t the filtration $\tilde{\mathscr{F}}%{\mathcal{F}}_{n,t}:=\sigma \left(v_n(s): s\le t\right)$.
As far as its quadratic variation process is concerned, in order to exploit classical results presented in a real  Hilbert space setting, see e.g., \cite{DPZ1},
we work with the real inner product in $H$ (and $H_n$). This means that the  quadratic variation process $\quadVar{N_n}$ is defined through
the property that for any $\psi,\phi \in H_n$ the process
\[
\skpHReal{N_n(t)}{\psi}\skpHReal{N_n(t)}{\phi}-\skpHReal{\quadVar{N_n}_t\psi }{\phi}, \qquad t \in [0,T],
\]
is a martingale.
Therefore we find that
	\begin{multline}
	\label{Nnqv}
	\quadVar{N_n}_t\psi
	= \sumM \int_0^t \im S_n B_m S_n v_n(s) \skpHReal{\im S_n B_m S_n v_n(s)}{\psi} \df s
	\\
	+\sumM \int_0^t \im S_n G(S_n v_n(s))e_m \skpHReal{\im S_n G(S_n v_n(s))e_m}{\psi} \df s
	\end{multline}
	for all $\psi \in {H}$ and $t\in[0,T]$.
The martingale property can be rephrased as
\begin{align}
\label{1lim}
	\tilde{\E} \left[ \skpHReal{N_n(t)-N_n(s)}{\psi} h(v_n|_{[0,s]})\right]=0
	\end{align}
	and
	\begin{align}
	\label{2lim}
	\tilde{\E} \Bigg[ \Bigg(&\skpHReal{N_n(t)}{\psi}\skpHReal{N_n(t)}{\varphi}-\skpHReal{N_n(s)}{\psi}\skpHReal{N_n(s)}{\varphi}
	\notag\\
	&\hspace{1 cm}-\sumM \int_s^t  \skpHReal{\im S_n B_m S_n v_n(r)}{\psi} \skpHReal{\im S_n B_m S_n v_n(r)}{\varphi} \df r
	\notag\\
	&\hspace{1 cm}-\sumM \int_s^t  \skpHReal{\im S_n G (S_n v_n(r))e_m}{\psi} \skpHReal{\im S_n G(S_n v_n(r))e_m}{\varphi} \df r
	\Bigg) h(v_n|_{[0,s]})\Bigg]=0,
	\end{align}		
	for all $0<s<t\le T$,  $\psi, \varphi \in {H}$ and bounded continuous functions $h$ on $C([0,T],H_n).$

It is useful at this point to introduce the following notation. Let $\iota: {V} \hookrightarrow {H}$ be the usual embedding,
$\iota^*: {H} \rightarrow {V}$ its Hilbert-space-adjoint,
i.e. $\skpH{\iota u}{v}=\skp{u}{\iota^* v}_{V}$ for $u\in{V}$ and $v\in{H}.$
Further, we set $L:=\left(\iota^*\right)': {V^*} \rightarrow {H}$  as the dual operator of $\iota^*$ with respect to the
Gelfand triple ${V}\hookrightarrow H\eqsim H^*\hookrightarrow{V^*}.$
	\newline
Let us introduce the process
\begin{align*}
N(t):=-v(t)+ u_0+ \int_0^t \left[-\im A v(s)-\im  F(v(s))+b(v(s))-\beta v(s)\right] \df s, \quad t\in[0,T],
\end{align*}
which has ${V^*}$-valued continuous paths. Moreover, $N(t) \in L^2(\tilde \Omega,{V}^*)$.

We now use the martingale property of $N_n$ for $n\in\N$ and a passage to the limit in \eqref{1lim} and \eqref{2lim}
to show that $L N$ is an ${H}$-valued continuous square integrable martingale with respect to the filtration
$\tilde{\mathbb{F}}=\left(\tilde{\mathscr{F}}%{\mathcal{F}}_t\right)_{t\in[0,T]},$ where $\tilde{\mathscr{F}}%{\mathcal{F}}_{t}:=\sigma \left(v(s): s\le t\right)$,
with quadratic variation given by
	\begin{align}
	\label{qvN}
	\quadVar{L N}_t\zeta
	=\sumM \int_0^t\left[\im  L B_m v(s) \skpHReal{\im L B_m v(s)}{\zeta} + \im  L G(v(s))e_m
	\skpHReal{\im L G( v(s))e_m}{\zeta}\right]\df s,
		\end{align}
	for all $\zeta \in {H}.$	
We just provide the main steps of the limiting process (for a detailed proof see \cite{BHW-2019})
and the computations of what is new.

Taking the limit as $n \rightarrow \infty$ in \eqref{1lim}, for $\psi \in {V}$, we obtain
\begin{footnote}{
For the proof see \cite[Lemma 6.2, Lemma 6.4 Steps 1-2]{BHW-2019}. Here, in addition, we have to consider the
convergence of the damping term but the needed estimates can be obtained rather easily.}
\end{footnote}
\begin{equation}
\label{martH}
\tilde{\E} \left[\langle N(t)-N(s),\psi\rangle h(v|_{[0,s]}) \right]=0.
\end{equation}
Then we take the limit as $n\to\infty$  in \eqref{2lim}, for $\phi, \psi \in {V}$, and obtain
	\begin{align}
	\label{martes2}
	\tilde{\E} \Bigg[ \Bigg(
	\dualityReal{N(t)}{\psi}&\dualityReal{N(t)}{\varphi}-\dualityReal{N(s)}{\psi}\dualityReal{N(s)}{\varphi}	\\ \notag
	&-\sumM \int_s^t  \dualityReal{ B_m v(r)}{\psi} \dualityReal{B_m v(r)}{\varphi} \df r
	\\\notag
	&-\sumM \int_s^t  \dualityReal{G v(r)e_m}{\psi} \dualityReal{G v(r)e_m}{\varphi} \df r
	\Bigg) h(v|_{[0,s]})\Bigg]=0.
	\end{align}
To prove the convergence of the first three terms in \eqref{2lim} one proceeds as in \cite[Lemma 6.4 Steps 3-4]{BHW-2019}.
The convergence of the fourth term is proved in the following lemma whose proof is postponed
to Appendix \ref{App_C}.
\begin{Lemma}
 \label{convquadvari}
 Under Assumption \ref{stochasticAssumptions}(iii), for all $0\le s\le t \le T$, $\psi, \varphi\in {V}$, $h$ a bounded continuous function on ${C([0,T],\EAdual)}$, we have
 \begin{multline*}
\lim_{n \rightarrow \infty}\tilde{\E}
 \Bigg[
\Bigg(
\sumM \int_s^t  \skpHReal{S_n G (S_n v_n(r))e_m}{\psi} \skpHReal{S_n G (S_n v_n(r))e_m}{\varphi} \df r
\Bigg) h(v|_{[0,s]})
\Bigg]
\\
=
\tilde{\E} \Bigg[
\Bigg(
\sumM \dualityReal{ G  v(r)e_m}{\psi} \dualityReal{G  v(r)e_m}{\varphi}  \df r
\Bigg) h(v|_{[0,s]})
\Bigg].
\end{multline*}
\end{Lemma}

Now let $\eta, \zeta \in H$. Then $i^*\eta$, $i^*\zeta \in {V}$ and for every $z$ in ${V}^*$ we have $\skpHReal{Lz}{\eta}=\dualityReal{z}{i^*\eta}$.
Thus from \eqref{martH} and \eqref{martes2} we deduce
\begin{equation}
\label{martH2}
\tilde{\E} \left[\left( LN(t)-LN(s),\psi\right)_H h(v|_{[0,s]}) \right]=0.
\end{equation}
and
\begin{align}
	\label{martes2a}
	\tilde{\E} \Bigg[ \Bigg(\skpHReal{L N(t)}{\eta}&\skpHReal{L N(t)}{\zeta}-\skpHReal{L N(s)}{\eta}\skpHReal{L N(s)}{\zeta}\\ \notag
	&-\sumM \int_s^t  \skpHReal{L  B_m v(r)}{\eta} \skpHReal{L B_m v(r)}{\zeta} \df r
	\\\notag
	&-\sumM \int_s^t  \skpHReal{L  G( v(r)e_m)}{\eta} \skpHReal{L G( v(r)e_m)}{\zeta} \df r
	\Bigg) h(v|_{[0,s]})\Bigg]=0.
	\end{align}
	Hence, from \eqref{martH2} and \eqref{martes2a}, we infer that $L N$ is a continuous, square integrable martingale in ${H}$ with respect to  $\tilde{\mathscr{F}}%{\mathcal{F}}_{t}:=\sigma \left(v(s): s\le t\right)$ and quadratic variation given by \eqref{qvN}.		
			
Therefore, with the usual Martingale Representation Theorem, see  \cite[Theorem 8.2]{DPZ1},
we can conclude that there exist two  cylindrical Wiener processes $\tilde{W}$ and $\tilde{\newW}$ on $Y$
defined on a probability space $
	\left(\Omega',\mathscr{F}}%{\mathcal{F}^{\prime},\mathbb{P}^{\prime}\right)=\left(\tilde{\Omega} \times \tilde{\tilde{\Omega}}, \tilde{\mathscr{F}}%{\mathcal{F}}\otimes \tilde{\tilde{\mathscr{F}}%{\mathcal{F}}},  \tilde{\mathbb{P}}\otimes\tilde{\tilde{\mathbb{P}}}\right)$
	with
	\begin{align}
	\label{LN}
	L N(t)
	=\int_0^t \im L B\left( v(s)\right) \df \tilde{W}(s)+\int_0^t \im L G\left( v(s)\right) \df \tilde{\newW}(s)
	\end{align}
	for $t\in [0,T].$
	Thanks to 	\eqref{confronto-norme}, the estimates
	\begin{multline*}
	\norm{B v}_{L^2([0,T]\times \Omega,\gamma(Y_1,{V^*}))}^2
	=\mathbb{E} \int_0^T \sumM \norm{B_m v(s)}_{V^*}^2 \df s
	\lesssim\mathbb{E} \int_0^T \sumM \norm{B_m v(s)}_{V}^2 \df s\\
	\le \mathbb{E} \int_0^T \left(\sumM \norm{B_m}_{{\mathscr{L}({V})}}^2\right) \norm{v(s)}_{V}^2 \df s
	\lesssim \mathbb{E} \int_0^T  \norm{v(s)}_{V}^2 \df s
	\lesssim \norm{v}_{L^2(\Omega,{L^\infty(0,T;\EA)})}^2 <+\infty	
	\end{multline*}
	and
	\begin{multline*}
	\norm{G (v)}_{L^2([0,T]\times \Omega,\gamma(Y_2,{V^*}))}^2
	=\mathbb{E} \int_0^T \sumM \norm{G (v(s))e_m}_{V^*}^2 \df s
	\lesssim\mathbb{E} \int_0^T \sumM \norm{G (v(s))e_m}_{V}^2 \df s
	\\
	= \mathbb{E} \int_0^T \|G(v(s))\|^2_{\gamma(Y_2, {V})}\, {\rm d}s
	\lesssim 1+ \norm{v}_{L^2(\Omega,{L^\infty(0,T;\EA)})}^2<\infty ,
	\end{multline*}
yield that $LN$ in \eqref{LN} is a continuous martingale in $H$ and using the continuity of the operator $L$, we get
	\begin{align*}
	\int_0^t \im L B\left( v(s)\right) \df \tilde{W}(s)  +\int_0^t \im L B\left( v(s)\right) \df \tilde{\newW}(s)
	=
	L \left(\int_0^t \im  B\left( v(s)\right) \df \tilde{W}(s) +\int_0^t \im G\left(v(s)\right) \df \tilde{\newW}(s) \right)
	\end{align*}
	for all $t\in [0,T].$
	The definition of $N$ and the injectivity of $L$ yield the equality
	\begin{align}\label{vIsSolution}		
	\int_0^t \im B  v(s) \df \tilde{W}(s) +\int_0^t \im G (v(s)) \df \tilde{\newW}(s)=-v(t)+ u_0+ \int_0^t \left[-\im A v(s)-\im  F(v(s))+b(v(s))\right] \df s
	\end{align}
		in ${V^*}$ for $t\in [0,T].$

The weak continuity of the paths of $v$ in ${V}$ and the estimates for property $\eqref{propertySolution}$ have
already been shown at the beginning of the proof. Hence, the system
$\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{W},\tilde{\mathbb{F}},v\right)$ is a martingale solution of equation
$\eqref{ProblemStratonovich}$.

It remains to prove Lemma \ref{convquadvari}. This is done in Appendix \ref{App_C}.



\subsection{Proof of the second part of Theorem \ref{existTheorem}}

Inequality \eqref{Ebound} is a consequence of the same inequality for the Galerkin approximation
\eqref{Galerkin-Ebound} and the property \eqref{confronto-norme} for the limit.






\section{Pathwise Uniqueness}\label{UniquenessSection}

In this section we study the pathwise uniqueness for solutions to \eqref{Problem}.
We work under  Assumptions \ref{spaceAssumptions}, \ref{F_def} and \ref{stochasticAssumptions}.

The proof of  the pathwise uniqueness is based on an additional regularity of the solution which we obtain by applying the deterministic and stochastic Strichartz estimates derived in Section \ref{Str_est_sec}. These estimates are based on the homogeneous Strichartz estimates due to Blair, Smith and Sogge, see \cite{Blair+Sogge_2008}. These estimates allow us to work at the same time with the Laplace-Beltrami operator on a two-dimensional compact Riemannian manifold $M$ and the realization of the negative Laplace operator with Dirichlet or Neumann boundary
conditions on a smooth relatively compact  domain $\mathcal{O} \subset \mathbb{R}^2$.
This is a first difference with respect to \cite{BHW-2019} where the authors start from the Strichartz estimates due to Bernicot and Samoyeau, see \cite{Bernicot} and  \cite[Lemmas B.3 and B.4]{BHW-2019}.
Moreover, differently to \cite{BHW-2019}, in order to prove the  pathwise uniqueness we cannot work pathwise since our noise is not conservative. We address this issue by appealing to a classical argument contained in \cite{Sch1997}.

As usual, when we write $L^q$, $H^{s,q}$ without specifying the domain, we mean either $L^q(M)$, $H^{s,q}(M)$ or $L^q(\mathcal{O})$, $H^{s,q}(\mathcal{O})$.

We start with the following result about the nonlinear term.
\begin{Lemma}\label{lem-F-continity}
Assume that $\alpha>1$
and $\vartheta \in (0,1)$. Then $F$ maps the space $V$ to $\mathcal{D}(A^{\frac{\vartheta }{2}})$ and
\begin{align}\label{eqn-F-growth}
  \|F(u) \|_{\mathcal{D}(A^{\frac{\vartheta }{2}})} &\lesssim  \|u \|_{V}^\alpha, \qquad u \in V.
  \end{align}
\end{Lemma}
\begin{proof}
For the case of $(M,g)$ a compact manifold without boundary equipped with a Lipschitz metric g and $-A$ equal to the Laplace-Beltrami operator we refer to \cite[Lemma 7.1]{BHW-2019}.
Hence we have to  prove \eqref{eqn-F-growth} when $-A$ is the Laplace operator with either Dirichlet or Neumann
boundary conditions on a smooth relatively compact  subset $\mathcal{O}$ of $\mathbb{R}^2$.

Let us fix $\vartheta \in (0,1)$ and choose $r \in (1,2)$ such that
\begin{equation}
\label{beta_r}
\vartheta  < \frac{2(r-1)}{r}.
\end{equation}
 We start by proving that
\begin{equation}
\label{ineq_1r}
\|F(u)\|_{W^{1,r}(\mathcal{O})} \lesssim \|u\|_{H^1(\mathcal{O})}^{\alpha},\;\; u\in H^1(\mathcal{O}).
\end{equation}
In order to prove \eqref{ineq_1r} we compute the weak derivative of $F(u)$:
\begin{equation*}
\nabla F(u)= \left( \frac{\alpha -1}{2}\right)|u|^{\alpha -3}\left(\bar u\nabla u+u \nabla \bar u\right)u +|u|^{\alpha-1}\nabla u, \qquad  \text{for an arbitrary} \ u \in H^1(\mathcal{O}).
\end{equation*}
The H\"older inequality and the Sobolev embedding $H^1(\mathcal{O}) \subset L^q(\mathcal{O})$ with $q=\frac{2r(\alpha-1)}{2-r}$, see Proposition \ref{B1}(i), yield
\begin{equation*}
\|\nabla F(u)\|_{L^r(\mathcal{O})}
\lesssim \| |u|^{\alpha-1}\|_{L^{\frac{2r}{2-r}}(\mathcal{O})}\|\nabla u\|_{L^2(\mathcal{O})}
\lesssim \|u\|^{\alpha-1}_{L^q(\mathcal{O})}\|\nabla u\|_{L^2(\mathcal{O})} \lesssim \|u\|^{\alpha}_{H^1(\mathcal{O})}.
\end{equation*}
Similarly, the Sobolev embedding $H^1(\mathcal{O})\subset L^{r\alpha}(\mathcal{O})$ yields
\begin{equation*}
\|F(u)\|_{L^r(\mathcal{O})} \simeq \|u\|_{L^{r\alpha}(\mathcal{O})}^{\alpha} \lesssim \|u\|^{\alpha}_{H^1(\mathcal{O})},\;\; u\in H^1(\mathcal{O})
\end{equation*}
and thus \eqref{ineq_1r} immediately follows. By the choice of $r$ in \eqref{beta_r}, Proposition \ref{B1}(iii) ensures that Sobolev embedding $H^{1,r}(\mathcal{O})\subset H^{\vartheta }(\mathcal{O})$ holds and thus
\begin{equation}
\label{general_est}
\|F(u)\|_{H^{\vartheta }(\mathcal{O})} \lesssim \|u\|^{\alpha}_{H^1(\mathcal{O})}.
\end{equation}

Observe now that, when on $\mathcal{O}$ we consider the Neumann boundary conditions, thanks to \eqref{realization_sN}, $H^{\vartheta }=\mathcal{D}(A_N^{\frac{\vartheta }{2}})$ and thus from \eqref{general_est}, \eqref{eqn-F-growth} immediately follows.

In the case of the Dirichlet boundary conditions, from \eqref{realization_sD} and \eqref{general_est}, \eqref{eqn-F-growth}
immediately follows when $\vartheta  \in \left(0, \frac 12\right)$. When $\vartheta  \in \left(\frac 12,1\right)$, \eqref{eqn-F-growth}
is obtained by a density argument: for $\vartheta  \in \left(\frac 12,1\right)$, by \eqref{realization_sD},
$\mathcal{D}(A_D^{\frac{\vartheta }{2}})=H_0^{\vartheta }$, where $H_0^{\vartheta }$ is the closure of $C^{\infty}_0$ w.r.t. the norm
$\|\cdot\|_{H^{\vartheta ,2}}$. One can easily verify that $F$ maps $C^{\infty}_0$ in itself and thus, by \eqref{general_est} we
deduce \eqref{eqn-F-growth} when $\vartheta  \in \left(\frac 12,1\right)$.
\end{proof}

We now reformulate problem \eqref{Problem} in the mild form to show additional regularity properties of solutions to \eqref{Problem}.
\begin{Prop}\label{mild_prop}
Fix $T>0$. Let $u_0 \in V$. Assume that $\alpha >1$,  $r>1$ and  $\gamma \ge  \max\{\alpha,2\}$. Let
$\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{W},\tilde{\newW},\tilde{\mathbb{F}},u\right)$ be  a solution to
\eqref{Problem}  such that
\begin{equation}
\label{originalSmoothnessSolutions}
u \in L^{\alpha r}(\tilde \Omega; L^{\gamma}(0,T;V)).
\end{equation}
If $2<p,q<\infty$ with  $\frac 2p+\frac 2q=1$ and  $\vartheta  \in \left(\frac{4}{3p},1\right)$, then
there exists a modification of the process $u$, denoted also by $u$, such that $\tilde{\mathbb{P}}$-a.s., the equality
 \begin{align}
\label{mild_form}
\im u(t)&= \im e^{-\im tA}u_0+ \int_0^t e^{-\im(t-\tau)A}F(u(\tau))\, {\rm d}\tau
\notag\\
&+\im \int_0^t e^{-\im(t-\tau)A}b(u(\tau))\, {\rm d}\tau - \im \beta   \int_0^t e^{-\im(t-\tau)A}u(\tau)\, {\rm d}\tau
\notag\\
&+\int_0^t e^{-\im(t-\tau)A}B(u(\tau))\, {\rm d}\tilde{W}(\tau)+\int_0^t e^{-\im(t-\tau)A}G(u(\tau))\, {\rm d}\tilde{\newW}(\tau)
\end{align}
is satisfied
in $\mathcal{D}(A^{\frac{\vartheta }{2}})$ for all $t\in [0,T]$, and
\begin{equation}
\label{more_regularity}
u \in L^r\left(\tilde{\Omega};C([0,T];\mathcal{D}(A^{\frac \vartheta  2}))\cap L^p(0,T;\mathcal{D}(A_q^{\frac \vartheta  2-\frac{2}{3p}})) \right)
\end{equation}
$\tilde{\mathbb{P}}$-a.s..
\end{Prop}
\begin{Remark}
Let us note the following difference between our result and Proposition 7.2 in \cite{BHW-2019}. Here in \eqref{originalSmoothnessSolutions}  we assume the $V$-regularity of the solution while in the paper \cite{BHW-2019} only $H^s$-regularity  was assumed, see assumption (7.6).   We have to make a stronger assumption here because the Strichatz estimates for  a  boundaryless manifold  are stronger
than the Strichatz estimates for a bounded domain with smooth boundary, see Remark \ref{rem-Strichartz comparison}. However, this stronger assumption  is fully sufficient for our purposes.  \\
Note that in the proof below we use the  inclusion  $\gamma(Y;V) \subset \gamma(Y;D(A^{\frac{\vartheta }{2}}))$ which is a consequence of
$V \subset D(A^{\frac{\vartheta }{2}})$.
\end{Remark}
\begin{proof}
Let us fix and choose  $T>0$, $\alpha >1$,  $r>1$,  $\gamma \geq \max\{\alpha,2\}$ and  a solution
   $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{W},\tilde{\newW},\tilde{\mathbb{F}},u\right)$  to \eqref{Problem}
   satisfying \eqref{originalSmoothnessSolutions}.
\dela{Let us  observe  that  because of \eqref{originalSmoothnessSolutions}, the equality \eqref{mild_form} makes sense in
$\mathcal{D}(A^{\frac{\vartheta }{2}})$.
}
Let us at first show  the equality \eqref{mild_form} makes sense in
$\mathcal{D}(A^{-\frac 32})$.  Let us notice that for  $\sigma=-\frac32$,
 the semigroup $(e^{-\im tA})_{t\ge 0}$ on $L^2$ extends to a $C_0$-semigroup $(T_{\sigma}(t))_{t\ge 0}$ on $\mathcal{D}(A^{-\sigma})$ with the generator
 $-iA_\sigma$,  where  $\mathcal{D}(A_\sigma)=\mathcal{D}(A^{-\sigma+1})$, i.e.  $-iA_\sigma$ is a suitable extension of  $-iA$.
 To keep the notation simple we will denote this semigroup  by $(e^{-itA})_{t \ge 0}$.\\
Let us choose and fix $t \in (0,T]$.
We apply the It\^o formula, see \cite[Theorem 2.4]{BvNVW},  to the process $\im f(\tau, u(t-\tau))$, $\tau \in [0,t]$,  where $f$ is the function defined as
\begin{equation*}
f : [0,t]\times \mathcal{D}(A^{-\frac 12}) \ni  (\tau,x) \mapsto e^{-\im (t-\tau)A}x \in \mathcal{D}(A^{-\frac 32}).
\end{equation*}
Obviously $f$ is of  $C^{1,2}$-class and since it  follows from assumptions that the assumptions of \cite[Theorem 2.4]{BvNVW} are satisfied,    we deduce that $\tilde{\mathbb{P}}$-a.s.,
\begin{align*}
\im u(t)
&= \im e^{-\im tA}u_0+ \int_0^t e^{-\im(t-\tau)A}F(u(\tau))\, {\rm d}\tau
\\
&+\im \int_0^t e^{-\im(t-\tau)A}b(u(\tau))\, {\rm d}\tau - \im \beta  \int_0^t e^{-\im(t-\tau)A}u(\tau)\, {\rm d}\tau
\\
&+\int_0^t e^{-\im(t-\tau)A}B(u(\tau))\, {\rm d}\tilde{W}(\tau)+\int_0^t e^{-\im(t-\tau)A}G(u(\tau))\, {\rm d}\tilde{\newW}(\tau) \;\mbox{ in $\mathcal{D}(A^{-\frac 32})$}.
\end{align*}
We now use the Strichartz estimates from Lemma \ref{Strichartz_lemma} to improve the regularity of the solution.

Let us additionally choose and fix $p,q \in (2,\infty)$ such that $\frac 2p+\frac 2q=1$ and $\vartheta  \in \left(\frac{4}{3p},1\right)$.
Estimating the terms in the RHS of the above expression we will prove \eqref{more_regularity} and that identity
\eqref{mild_form} holds a.s. in $\mathcal{D}(A^{\frac{\vartheta }{2}})$.
\newline
Let us fix and introduce a Banach  space $Y^\vartheta _T$ defined by
\begin{equation*}
Y^\vartheta _T:=L^p(0,T;\mathcal{D}(A_q^{\frac{\vartheta }2-\frac{2}{3p}})) \cap L^\infty(0,T;\mathcal{D}(A^{\frac{\vartheta }{2}})),
\end{equation*}
endowed with a  norm $\|\cdot\|_{Y_T^{\vartheta }}=\|\cdot\|_{L^p(0,T;\mathcal{D}(A_q^{\frac{\vartheta }2-\frac{2}{3p}})))}+ \|\cdot\|_{L^\infty(0,T;\mathcal{D}(A^{\frac{\vartheta }{2}}))}$.
Notice that  by assumptions the exponent $\frac{\vartheta }2-\frac{2}{3p}$ is positive.
Thanks to the homogeneous Strichartz estimate \eqref{eqn-Strichartz-211} we get
\begin{equation*}
\|e^{-\im tA}u_0\|_{Y^{\vartheta }_T}\lesssim_T \|u_0\|_{\mathcal{D}(A^{\frac{\vartheta }{2}})} \lesssim \|u_0\|_V < \infty.
\end{equation*}
The inhomogeneous Strichartz estimate \eqref{eqn-Strichartz-212}, Lemma \ref{lem-F-continity} and assumption \eqref{originalSmoothnessSolutions} yield
\begin{align*}
\left \Vert\int_0^{\cdot} e^{-\im(\cdot-\tau)A} F(u(\tau))\ {\rm d}\tau\right\Vert_{L^r(\tilde{\Omega};Y^{\vartheta }_T)}
&\lesssim_T
\|F(u)\|_{L^r(\tilde{\Omega},L^1(0,T;\mathcal{D}(A^{\frac{\vartheta }{2}})))}
\\
&\lesssim \|u\|^{\alpha}_{L^{\alpha r}(\tilde{\Omega},L^{\alpha}(0,T;V))}< \infty.
\end{align*}
Since we assume $\alpha>1$, $\gamma= \max\{\alpha, 2\}$ and $\vartheta <1$, from the inhomogeneous Strichartz estimate \eqref{eqn-Strichartz-212} and assumption \eqref{originalSmoothnessSolutions} we infer that
\begin{align*}
\left \Vert   \int_0^{\cdot} e^{-\im(\cdot-\tau)A} u(\tau)\ {\rm d}\tau\right\Vert_{L^r(\tilde{\Omega};Y^{\vartheta }_T)}
&\lesssim
\dela{\beta} \|u\|_{L^r(\tilde{\Omega},L^1(0,T;\mathcal{D}(A^{\frac{\vartheta }{2}})))}
\lesssim \dela{\beta} \|u\|_{L^{\alpha r}(\tilde{\Omega},L^{\gamma}(0,T;V))}< \infty.
\end{align*}
The It\^o correction term can be estimated as follows. Recalling \eqref{Strat_cor}, by the inhomogeneous Strichartz estimate \eqref{eqn-Strichartz-212}, and \eqref{noiseBoundsEnergy} and \eqref{originalSmoothnessSolutions}
	\begin{align*}
	\left \Vert\int_0^{\cdot} e^{-\im (\cdot-\tau)A}b(u(\tau))\df \tau\right\Vert_{L^r(\tilde{\Omega},Y^{\vartheta }_T)}
	&\lesssim \norm{b(u)}_{L^r(\tilde{\Omega}, L^1(0,T; \mathcal{D}(A^{\frac{\vartheta }{2}})))}
	\\
&\lesssim \norm{b(u)}_{L^{\alpha r}(\tilde{\Omega}, L^\gamma(0,T; V))}
	\\&\lesssim \norm{u}_{L^{\alpha r}(\tilde{\Omega}, L^\gamma(0,T; V)}<\infty.
	\end{align*}
For what concerns the stochastic convolution term involving the operator $B$, by means of the stochastic Strichartz estimate \eqref{eqn-Strichartz-213}, and \eqref{noiseBoundsEnergy}  and
\eqref{originalSmoothnessSolutions}, we obtain
	\begin{align*}
	\left\Vert \int_0^{\cdot} e^{-\im (\cdot-\tau)A}B(u(\tau))\df \tilde{W}(\tau)\right\Vert_{L^r(\tilde{\Omega},Y_T^{\vartheta })}
	&	
	\lesssim \norm{B(u)}_{L^r(\tilde{\Omega}, L^2(0,T; \gamma(Y_1,\mathcal{D}(A^{\frac{\vartheta }{2}}))))}
	\dela{\\	&\lesssim \mathbb{E} \left[ \left( \int_0^T\sum_{m=1}^{\infty}\|B_m\|^2_{\mathscr{L}(\mathcal{D}(A^{\frac{\vartheta }{2}}))}\|u(\tau)\|^2_{\mathcal{D}(A^{\frac{\vartheta }{2}})}\, {\rm d}\tau\right)^{\frac r2}\right]
}	\\
\lesssim \norm{B(u)}_{L^r(\tilde{\Omega}, L^2(0,T; \gamma(Y_1,V)))}
	&\lesssim \norm{u}_{L^r(\tilde{\Omega}, L^2(0,T; V))}
	\\
	&\lesssim \norm{u}_{L^{r\alpha}(\tilde{\Omega}, L^{\gamma}(0,T;V))}<\infty.
	\end{align*}
It remains to estimate the stochastic convolution involving the operator $G$.
The stochastic Strichartz estimate \eqref{eqn-Strichartz-213}, and \eqref{crescitaGEA} and \eqref{originalSmoothnessSolutions} yield
\begin{align*}
\left\Vert \int_0^{\cdot} e^{-\im (\cdot-\tau)A}G(u(\tau))\df \tilde{\newW}(\tau)\right\Vert_{L^r(\tilde{\Omega},Y_T^{\vartheta })}
&\lesssim \norm{G(u)}_{L^r(\tilde{\Omega}, L^2(0,T; \gamma(Y_2,\mathcal{D}(A^{\frac{\vartheta }{2}}))))}
\\
&\lesssim \norm{G(u)}_{L^r(\tilde{\Omega}, L^2(0,T; \gamma(Y_2,V)))}
\\
&= \mathbb{E} \left[ \left(\int_0^T \|G(u(\tau))\|^2_{\gamma(Y_2,V)}\,{\rm d}\tau \right)^{\frac r2} \right]
\\
&\lesssim_{r,T} 1 + \|u\|_{L^r(\tilde{\Omega}, L^2(0,T;V))}
\lesssim 1 + \|u\|_{L^{r\alpha}(\tilde{\Omega}, L^{\gamma}(0,T;V))} < \infty.
\end{align*}
Thus the mild equation \eqref{mild_form} holds $\tilde{\mathbb{P}}$-a.s. in $\mathcal{D}(A^{\frac{\vartheta }{2}})$ for each $t \in [0,T]$
 and therefore, thanks to the pathwise continuity of the deterministic and stochastic integrals, we get \eqref{more_regularity}.
\end{proof}




We are now ready to prove the pathwise uniqueness of the solution to \eqref{Problem}. We will need the following result which exploits the gain of regularity of solutions proved in Proposition \ref{mild_prop}.
\begin{Lemma}
\label{psi}
Fix $T>0$.
Let $\alpha>1$, $r>1$ and $\gamma \ge \max\{\alpha,2\}$.
Let $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{W},\tilde{\newW},\tilde{\mathbb{F}},u_i\right)$ be two solutions to
\eqref{Problem} with $u_i \in L^{\alpha r}(\tilde{\Omega},L^{\gamma}(0,T;V))$, for $i=1,2$.
Then the process $h$ defined by
\begin{equation*}
h(s):= \left[\|u_1(s)\|_{L^{\infty}}^{\alpha-1} +\|u_2(s)\|^{\alpha-1}_{L^{\infty}}\right], \qquad s \in [0,T],
\end{equation*}
belongs to $L^1(0,T)$, $\tilde{\mathbb{P}}$-a.s.
\end{Lemma}
\begin{proof}
Let us choose $2<p,q<\infty$ with $\frac 2p+\frac 2q=1$. Thanks to Proposition \ref{mild_prop},
$u_i \in L^p(0,T;\mathcal{D}(A_q^{\frac{\vartheta }{2}-\frac{2}{3p}}))$ $\tilde{\mathbb{P}}$-a.s. for each $\vartheta  \in \left(\frac{4}{3p},1\right)$,
 for $i=1,2$. Since $1-\frac{2}{3p}>\frac{4}{3p}$ we take $\vartheta  \in \left(1-\frac{2}{3p},1\right)$.
Thanks to Proposition \ref{B1}(ii), Definition \ref{def-sobolev spaces with boundary conditions} and Theorem \ref{dom_D_N},
which ensures that $\mathcal{D}(A_{B_q}^{\frac{\vartheta }{2}-\frac{2}{3p}}) \subset H^{\vartheta -\frac{4}{3p},q}(\mathcal{O})$,
for $B=D,N$),  and Proposition \ref{propB6}(i)-(ii), we infer  that $\mathcal{D}(A_q^{\frac{\vartheta }{2}-\frac{2}{3p}}))\embed L^{\infty}$.
In fact, this embeddings holds true
provided $\vartheta -\frac{4}{3p}-\frac2q>0\iff \vartheta  >1- \frac{2}{3p}$.
Let now $\alpha \in (1,3]$;
applying the H\"older inequality in time with $p>2\ge \alpha-1$ we see that the process $h$ satisfies
\begin{equation}
\label{esHold}
\|h\|_{L^1(0,T)} \lesssim \|u_1\|^{\alpha-1}_{L^p(0,T;\mathcal{D}(A_q^{\frac{\vartheta }{2}-\frac{2}{3p}}))}+\|u_2\|^{\alpha-1}_{L^p(0,T;\mathcal{D}(A_q^{\frac{\vartheta }{2}-\frac{2}{3p}}))}.
\end{equation}
If instead we take $\alpha \in (3, \infty)$, it is immediate to show that estimate \eqref{esHold} holds true with $p=\alpha-1$.

We thus conclude that $h\in L^1(0,T)$ $ \tilde{\mathbb{P}}$-a.s. as a consequence of Proposition \ref{mild_prop}.
\end{proof}


\begin{Theorem}
Let $\alpha>1$ $r>1$, $
\gamma \ge \max\{\alpha,4\}$, $s \ge \max\{\alpha r, 4\}$. Let the same Assumptions of Theorem \ref{existTheorem} be in force.
Then solutions of problem \eqref{Problem} are pathwise unique in $L^s(\tilde{\Omega},L^{\gamma}(0,T;V))$
i.e. given two solutions
$\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{W},\tilde{\newW},\tilde{\mathbb{F}},u_i\right)$ with
\begin{equation}
\label{reg_Sch}
u_i \in L^s(\tilde{\Omega},L^{\gamma}(0,T;V)),
\end{equation}
for $i=1,2$, we have
$\tilde{\mathbb{P}}\left(\|u_1(t)-u_2(t)\|_H=0 \ \text{for all} \ t \in [0,T]\right)=1$.


\end{Theorem}
\begin{proof}
Since the noise is not conservative we can not work pathwise as in \cite{BHW-2019}. We prove the uniqueness of the solution by means of a rather classical argument, see \cite{Sch1997}.

Take two solutions $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{W},\tilde{\newW},\tilde{\mathbb{F}},u_i\right)$ with $u_i \in L^s(\tilde{\Omega},L^{\gamma}(0,T;V))$,
for $i=1,2$ and define $v:= u_1-u_2$. This difference satisfies
\begin{equation*}
\begin{cases}
{\rm d}v(t)= -[\im Av(t)+\im\left(F(u_1(t))-F(u_2(t))\right)+\beta v(t)]{\rm d}t +
\\
\qquad \quad \ - \im B(v(t)) \circ {\rm d} \tilde{W}(t)-\im[G(u_1(t))-G(u_2(t))] {\rm d}\tilde{\newW}(t)
\\
v(0)=0.
\end{cases}
\end{equation*}
We use the It\^o formula to compute ${\rm d}\left( e^{-\int_0^t\psi(s)\, {\rm d}s}\|v(t)\|^2_H\right)$, by choosing $\psi$ as
\begin{equation}
\label{u0}
\psi(t):=2\left[\|u_1(t)\|_{L^{\infty}}^{\alpha-1} +\|u_2(t)\|^{\alpha-1}_{L^{\infty}}-\beta\right] +L_G , \qquad t \in [0,T],
\end{equation}
with $L_G$ the Lipschitz constant given in \eqref{Lipschitz_G}. Thanks to Lemma \ref{psi}, $\psi \in L^1(0,T)$, $\tilde{\mathbb{P}}$-a.s.. For $t \in [0,T]$, we have
\begin{align}
\label{u1}
{\rm d}\left( e^{-\int_0^t\psi(s)\, {\rm d}s}\|v(t)\|^2_H \right)= -\psi(t)e^{-\int_0^t\psi(s)\,{\rm d}s}\|v(t)\|^2_H\, {\rm d}t + e^{-\int_0^t\psi(s)\, {\rm d}s}\,{\rm d}\|v(t)\|^2_H,
\end{align}
where
\begin{align}
\label{u2}
{\rm d} \|v(t)\|_H^2
&=
\left[ 2\text{Re}\langle v(t), -\im F(u_1(t))+ \im F(u_2(t))\rangle -2 \beta \|v(t)\|^2_H+\|G(u_1(t))-G(u_2(t))\|^2_{\gamma(Y_2,H)} \right] \, {\rm d}t
\notag\\
&+ 2 \text{Re}\langle v(t), [G(u_1(t))-G(u_2(t))]\, {\rm d}\tilde{\newW}(t)\rangle.
\end{align}
By means of inequality \eqref{Lipschitz_G} we estimate
\begin{equation}
\label{u3}
\|G(u_1(t))-G(u_2(t))\|^2_{\gamma(Y_2,H)} \le L_G \|v(t)\|^2_H.
\end{equation}
The estimate
\begin{equation*}
|F(z_1)-F(z_2)| \lesssim \left(|z_1|^{\alpha-1}+|z_2|^{\alpha-1}\right) |z_1-z_2|, \qquad z_1, z_2 \in \mathbb{C},
\end{equation*}
and the H\"older inequality yield
\begin{align}
\label{u4}
\text{Re}\langle v(t), -\im F(u_1(t))+ \im F(u_2(t))\rangle
&\lesssim \|v(t)\|^2_H \left[\|u_1(t)\|_{L^{\infty}}^{\alpha-1} +\|u_2(t)\|^{\alpha-1}_{L^{\infty}}\right].
\end{align}
By means of \eqref{u2}, \eqref{u3} and \eqref{u4} we estimate \eqref{u1} as follows
\begin{align*}
{\rm d}\left( e^{-\int_0^t\psi(s)\, {\rm d}s}\|v(t)\|^2_H \right)
&\lesssim
-\psi(t)e^{-\int_0^t\psi(s)\,{\rm d}s}\|v(t)\|^2_H\, {\rm d}t
\\
&+ e^{-\int_0^t\psi(s)\, {\rm d}s} \left[ 2 \|v(t)\|^2_H \left[\|u_1(t)\|_{L^{\infty}}^{\alpha-1} +\|u_2(t)\|^{\alpha-1}_{L^{\infty}}\right]\, {\rm d}t-2 \beta \|v(t)\|^2_H\, {\rm d}t \right.
\\
&\left.\qquad +L_G\|v(t)\|^2_H\, {\rm d}t+ 2 \text{Re}\langle v(t), [G(u_1(t))-G(u_2(t))]\, {\rm d}\tilde{\newW}(t)\rangle \right]
\\
&= e^{-\int_0^t\psi(s)\, {\rm d}s}\|v(t)\|^2_H
\left[-\psi(t) +2 \left[\|u_1(t)\|_{L^{\infty}}^{\alpha-1} +\|u_2(t)\|^{\alpha-1}_{L^{\infty}}-\beta\right]+L_G\right]\, {\rm d}t
\\
& \qquad + 2 e^{-\int_0^t\psi(s)\, {\rm d}s}\text{Re}\langle v(t), [G(u_1(t))-G(u_2(t))]\, {\rm d}\tilde{\newW}(t)\rangle.
\end{align*}
Therefore, recalling \eqref{u0}, we obtain
\begin{equation*}
{\rm d}\left( e^{-\int_0^t\psi(s)\, {\rm d}s}\|v(t)\|^2_H \right)
\lesssim
2e^{-\int_0^t\psi(s)\, {\rm d}s} \text{Re}\langle v(t), [G(u_1(t))-G(u_2(t))]\, {\rm d}\tilde{\newW}(t)\rangle.
\end{equation*}
Integrating on both sides we get
\begin{equation}
\label{Schmalfuss}
e^{-\int_0^t\psi(s)\, {\rm d}s}\|v(t)\|^2_H
\lesssim
2\int_0^te^{-\int_0^r\psi(s)\, {\rm d}s} \text{Re}\langle v(r), [G(u_1(r))-G(u_2(r))]\, {\rm d}\tilde{\newW}(r)\rangle.
\end{equation}
Let us observe that the RHS of \eqref{Schmalfuss} is a square integrable martingale. Indeed, using inequality \eqref{Lipschitz_G} we get
\begin{align*}
\mathbb{E}& \left[ \left( \int_0^te^{-\int_0^r\psi(s)\, {\rm d}s} \text{Re}\langle v(r), [G(u_1(r))-G(u_2(r))]\, {\rm d}\tilde{\newW}(r)\rangle\right)^2\right]
\\
&\le \mathbb{E}\left[ \int_0^te^{-2\int_0^r\psi(s)\, {\rm d}s}\|v(r)\|^2_H\|G(u_1(r))-G(u_2(r))\|^2_{\gamma(Y_2,H)}\,{\rm d}r\right]
\\
& \lesssim L_G^2\mathbb{E}\left[ \int_0^t\|v(r)\|^4_H{\rm d}r\right] \lesssim \|v\|_{L^4(\tilde{\Omega};L^4(0,T;H))}
\end{align*}
which is finite thanks to \eqref{reg_Sch}. Therefore, by taking the expected value in both sides of \eqref{Schmalfuss}
 we get
\begin{equation*}
\mathbb{E} \left[e^{-\int_0^t\psi(s)\, {\rm d}s}\|v(t)\|^2_H \right]\le 0, \qquad \forall \ t \in [0,T].
\end{equation*}
Thus, in particular, for any $t \in \left[0,T\right]$
\begin{equation*}
e^{-\int_0^t\psi(s)\, {\rm d}s}\|v(t)\|^2_H=0, \qquad \tilde{\mathbb{P}}-a.s.
\end{equation*}
Therefore, if we take a sequence $\{t_k\}_{k=1}^{\infty}$, which is dense in $[0,T]$, we have
\begin{equation*}
\tilde{\mathbb{P}}\left( \|v(t_k)\|_H=0 \ \text{for all}\ k \in \mathbb{N}\right)=1.
\end{equation*}
Since by Proposition \ref{mild_prop} each path of the process belongs to $C([0,T];H)$, we get
\begin{equation*}
\tilde{\mathbb{P}}\left(u_1(t)=u_2(t) \ \text{for all} \ t \in [0,T]\right)=1
\end{equation*}
and this concludes the proof.
\end{proof}

Pathwise uniqueness and existence of martingale solutions
implies existence of a strong solution, see e.g. \cite{Ikeda1989}.






\section{Sequential weak  Feller property}\label{sec-weakCont}

Our aim is to prove the sequential weak  Feller property. This is one of the ingredients to prove existence of invariant measures
and relies on an argument of continuous dependence of the solution on the initial data.
 For the solution of problem \eqref{Problem},  we put in evidence the initial data $u_0 \in {V}$
by writing $u(\cdot;u_0)$.
Let us now recall the following fundamental definition. A function
$\phi:{V}\to\mathbb R$ is sequentially continuous w.r.t. the weak
topology  on ${V}$ (we write $\phi \in SC({V}_w)$) if $\phi(x_k)\to \phi(x)$ if $x_n\rightharpoonup x \text{ in } {V}$.
Using  the subindex  $SC_b({V}_w)$ we add the property of boundedness.
We recall that the following inclusions hold
\[
  C_b({V}_w) \subset SC_b({V}_w) \subset    C_b({V}) .
\]
Here  ${V}_w$  denotes ${V}$ equipped with the weak topology  and  ${V}_n$  denotes ${V}$ equipped with
the strong (norm)  topology.

Let us also notice that because ${V}$ is a separable space,
the weak Borel and the (strong) Borel $\sigma$
fields on it are equal, see, e.g., \cite{Edgar}, i.e.
$\mathcal{B}({V}_n)=\mathcal{B}({V}_w)$.

 Here we consider the unique solution $u$ of problem \eqref{Problem} on   the probability space
$(\Omega,\mathscr{F}}%{\mathcal{F},\mathbb{P})$.
We know from \cite[Corollary 23]{On2005}
that the transition function is jointly measurable, that is  for any Borelian subset $\Gamma$  of ${V}$
the map $V \times [0,\infty)\ni  (x,t)\mapsto \mathbb{P}\{u(t;x)\in\Gamma\}\in \mathbb{R}$ is measurable.
 We define the  family  of operators $P_t$, for any $t>0$,
\begin{equation}\label{eqn-Markov}
(P_t\phi)(x)= {\mathbb E}[\phi( u(t;x))], \;\; x\in V.
\end{equation}
If $\phi:V\to \mathbb R$ is a  bounded and Borel measurable  function,
then the same holds for $P_t\phi$.

We now prove the sequential weak Feller property, that is $P_t:SC_b({V}_w)\to SC_b({V}_w)$.
\begin{Prop}\label{Feller}
We consider the same assumptions as in Theorem \ref{existTheorem} i). For  any $t>0$, if
$\phi \in SC_b({V}_w)$ then $P_t\phi \in SC_b({V}_w)$.
\end{Prop}
\begin{proof} Let us choose and fix  $t>0$ and $\phi \in SC_b({V}_w)$.
We have to prove that, given a sequence $\left(x_k \right)_k \subset {V}$ which converges weakly in ${V}$ to
$x$,  the sequence $ P_t\phi(x_k)$ converges to
$P_t\phi(x) $.
We  fix $T>t$, and we will prove the result for any time  in the time interval $ [0,T]$.
\\
Set $u(\cdot,x_k)=u_k$.
For the sequence $(u_k)_k$ we have the same a priori estimates as in Section \ref{sec-exis},
that is this sequence  is tight in $Z_T$. Hence, following the proofs of that section we get that
there exist a subsequence $(u_{k_j})_j$,
a new stochastic basis $\left(\tilde{\Omega},\tilde{\mathscr{F}}%{\mathcal{F}},\tilde{\mathbb{P}},\tilde{\mathbb{F}}\right)$ on which are defined
$Z_T$-valued Borel measurable variables $ v$ and $(v_j)_j$ such that
for any $j$ the laws of $u_{k_j}$ and $v_j$
are the same and $v_j$ converges pathwise  to $v$
in $Z_T$. Moreover, $v$ coincides with the solution of
\eqref{Problem} with initial datum $x$.

In particular, for fixed $t$, $v_j(t;x_{k_j})$ weakly
converges in ${V}$ to   $v(t;x)$, $\tilde {\mathbb P}$-a.s.,
according to the fact that
$v_j \to  v$ in $C([0,T];{V}_w)$.
Hence, given any   sequential weak
continuous function $\phi:{V} \to\mathbb R$, we have that
$\phi(v_{j}(t;x_{k_j})) \to \phi(v(t;x))$ $\tilde {\mathbb P}$-almost surely.
Therefore, since  $\phi$ is also bounded,  by invoking the Lebesgue Dominated Convergence Theorem we
deduce that  $\tilde{\mathbb E}[\phi(v_{j}(t;x_{k_j}))]
 \to \tilde{\mathbb E}[\phi(v(t;x))]$.
Since $v$ has the same law as $u$ and   $v_j$ has the same
law as $u_{k_j}$, we infer that ${\mathbb E}[\phi(u_{k_j}(t;x_{k_j}))]
\to {\mathbb E}[\phi(u(t;x))]$.
Moreover, by the uniqueness the whole sequence converges. Bearing in mind the definition \eqref{eqn-Markov}, we conclude our proof.
\end{proof}


Now we consider the  Markov property.
\begin{Prop}
For every $\phi\in SC_b({V}_w )$,
$x\in {V}$  and $t,s>0$ we have
\begin{equation}\label{Markov-reg}
  {\mathbb E}\left[  \phi\left(  u(t+s;x)\right)  |{\mathcal F}_s\right]
  =
  \left(P_{t}\phi\right)  \left(  u(s;x)\right)  \qquad {\mathbb P}\text{-a.s.}
\end{equation}
\end{Prop}
\begin{proof}
The proof is classical when the solution is a continuous process
taking values in a separable Banach space, endowed with the strong topology;
see, e.g. \cite[Theorem 9.14 ]{DPZ1}.
Hence we highlight only the differences when dealing with the weak topology in ${V}$.

 For  $t>t_0\ge 0$ let us denote by $u(t;t_0,x)$
the solution of equation \eqref{Problem}  evaluated at time $t$ when the initial value is $u(t_0)=x$,
namely the evolution is on the time interval $[t_0,t]$. According to the previous notation used so far we have $u(t;x)=u(t;0,x)$.

By the pathwise uniqueness we know that
\begin{equation}u(t+s;0, x)=u(t+s;s,u(s;0,x)) \qquad \ a.s.
\label{eqn-question 01}
\end{equation}
Set $\eta=u(s;0,x)$ so that identity
 \eqref{Markov-reg} can be written as
\begin{equation}\label{markov}
{\mathbb E}\left[  \phi\left(u(t+s;s,\eta)\right)  |{\mathcal F}_{s}\right]
  =
  \left(P_{t}\phi\right)  \left( \eta\right) \qquad  \mathbb{P}\text{-a.s.}
\
\end{equation}
Hence it is enough to show that equality \eqref{markov} holds for an arbitrary integrable  $\mathcal F_s$-measurable random variable $\eta$.

Let us first suppose that  $\eta$ is a simple random variable of the form
$\sum_{j=1}^N x_j \mathbbm 1_{\Gamma_j}$ with $x_j \in {V}$ and a partition
$\Gamma_1,\ldots,\Gamma_N\subset \mathcal F_s$. Then   \eqref{markov}
is proved as usual  by noticing that $u(t+s;s,\eta)=\sum_{j=1}^N u(t+s;s,x_j)\mathbbm 1_{\Gamma_j}$:
\[\begin{split}
{\mathbb E}\left[  \phi\left(u(t+s;s,\eta)\right)  |{\mathcal F}_{s}\right]
&=\sum_{j=1}^N {\mathbb E}\left[ \mathbbm 1_{\Gamma_j} \phi\left(u(t+s;s,x_j)\right)  |{\mathcal F}_{s}\right]
\\&=\sum_{j=1}^N \mathbbm 1_{\Gamma_j}  {\mathbb E}\left[  \phi\left(u(t+s;s,x_j)\right)  |{\mathcal F}_{s}\right]
\\& =\sum_{j=1}^N \mathbbm 1_{\Gamma_j}  {\mathbb E}\left( \phi(u(t+s;s,x_j))\right)
\\&=\sum_{j=1}^N \mathbbm 1_{\Gamma_j}  (P_t\phi)(x_j)=(P_t\phi)(\eta).
\end{split}\]
We used that $ u(t+s;s,x_j)$  is independent of  ${\mathcal F}_{s}$ and  that $u(t+s;s,x_j)$ and $u(t;0,x_j)$ have the same law.


Otherwise, for general $\eta$
 there exists a sequence of simple random variables $\eta_n$  with $\lim_{n\to\infty}\eta_n=\eta$  in $L^1(\Omega)$ and $\mathbb{P}$-a.s.
(in the  strong topology of ${V}$, hence weak too); we checked before that
\begin{equation}\label{markov-step}
\mathbb E \left[  \phi\left(u(t+s;s,\eta_n)\right)  |{\mathcal F}_{s}\right]
  =
  \left(P_{t}\phi\right)  \left( \eta_n\right) \qquad  \mathbb P\text{-a.s.}
\
\end{equation}
By Proposition \ref{Feller} the RHS converges a.s. to $\left(P_{t}\phi\right)  (\eta)$ as $n \to \infty$. For the l.h.s.
 bearing in mind the proof of Proposition \ref{Feller} we have that $u(t+s;s,\eta_n)$
 has the same law of a random variable converging a.s. in $V$ to a random variable equivalent to $u(t+s;s,\eta)$;
 so, using that $\phi\in SC_b({V}_w )$ we can pass to the limit in the l.h.s. side of \eqref{markov-step} and get finally \eqref{markov}.
\end{proof}

Therefore we deduce  as usual that
 the family $\{P_t\}_{t\ge 0}$ is a Markov semigroup, namely $P_{t+s}=P_t P_s$ for any $s,t>0$.



\section{Existence of an invariant measure}\label{s-inv}

Given the  sequential weak Feller  Markov semigroup on the  separable Hilbert space $V$,
 we can define an invariant measure $\pi$ for equation \eqref{Problem} as a
Borel probability measure  on $V$ such that for any time $t\ge 0$
\begin{equation}\label{def-invmeas}
\int_{{V}} P_t \phi\ d\pi=\int_{{V}} \phi\ d\pi, \qquad \forall  \phi\in SC_b(V_w).
\end{equation}

Let us recall a result of Maslowski-Seidler \cite{Mas-Seid} about  the
existence of an invariant measure. This is a modification of
the Krylov-Bogoliubov technique, usually  presented in the setting of strong topologies, see, e.g.,  \cite{KB} and \cite{DPZ2}.

\begin{Theorem}\label{MSprop}
Assume that\\
i) the semigroup $\{P_t\}_{t\ge 0}$ is sequential weak Feller in ${V}$;\\
ii) for any $\varepsilon >0$ there exists $R_\varepsilon>0$ such that
\[
\sup_{T\ge 1} \frac 1T \int_0^T \mathbb P\left(\|u(t;0)\|_{{V}}>R_\varepsilon\right)dt<\varepsilon .
\]
Then there exists  at least one invariant measure for equation \eqref{Problem}.
\end{Theorem}

Hence we get our main result on invariant measures.
\begin{Theorem}\label{th:mis-inv-2d}
Under Assumptions \ref{spaceAssumptions},
\ref{F_def}, \ref{stochasticAssumptions},
if
\eqref{condizione-beta} is fulfilled then
then there exists at least one invariant measure  $\mu$ for
equation \eqref{Problem} and $\mu({V})=1$.
\end{Theorem}
\begin{proof}  The proof is based on Theorem \ref{MSprop}.
The sequential weak Feller property has been proved before in Proposition \ref{Feller}.
For the tightness
it is enough to recall \eqref{Ebound} and the Chebyshev inequality, so
\[
\mathbb P\left(\|u(t;0)\|_{{V}}>R \right)\le \frac{\mathbb{E} \|u(t;0)\|_{V}^2}{R^2}\le \frac C{R^2} \qquad \text{ for all }t\ge 0
\]
where the constant $C$ is independent of $t$.  Hence we have verified that the two assumptions of Theorem \ref{MSprop} are fulfilled.
\end{proof}



\section{Existence and uniqueness of the invariant measure with purely multiplicative noise}\label{sect-unique}

Assume that the coefficients characterizing the operator $G$  are such that $C_1=0$; this implies that $G(0)=0$, see \eqref{crescitaG}.
Hence the zero process is a solution of equation \eqref{Problem}, or equivalently $\delta_0$ is an invariant measure.
Let us prove that this is the unique invariant measure if
\[
\beta>\tfrac 12 \tilde C_1^2.
\]
This is our result
\begin{Theorem}
\label{cordelta0}
Let Assumptions \ref{spaceAssumptions} and \ref{F_def} be in force and assume that  Assumption \ref{stochasticAssumptions} holds with $C_1=0$, that is
\begin{equation}\label{crescitaG-0}
\|G(u)\|_{\gamma(\Yg,H)} \le \tilde C_1\|u\|_H\qquad \forall u \in H.
\end{equation}
If
\begin{equation}
\beta>\tfrac 12 \tilde C_1^2,
\end{equation}
then  there exists a unique invariant measure for equation \eqref{Problem} given by $\pi=\delta_0$.
\end{Theorem}

The proof is based on an auxiliary result
\begin{Lemma}
\label{lem1_delta0}
Under the assumptions of Theorem \ref{cordelta0},
 there exists a constant $\lambda>0$ such that, if $u$ is a solution to \eqref{Problem} with $u_0\in H$, then the process
 $\{e^{\lambda t}\|u(t)\|^2_H\}_{t \ge 0}$ is a non-negative continuous  supermartingale.
\end{Lemma}
\begin{proof} Let $u$ be the unique solution to equation \eqref{Problem} starting from $u_0 \in V$.
We apply the It\^o formula to the process $g(r, u(r))$ with $g(r,x):=e^{\lambda r}\|x\|^2_H$, for $r \in [s,t]$, since we know that the paths are in $  C([0,T];H)$.
We have
\begin{equation*}
{\rm d}\left(e^{\lambda t}\|u(t)\|_H^2\right)= \lambda e^{\lambda t} \|u(t)\|_H^2+e^{\lambda t}{\rm d}\|u(t)\|_H^2.
\end{equation*}
By the same computations done in the proof of Proposition \ref{MassEstimateGalerkinSolution}
we get
\[
d\|u(t)\|_H^2+2\beta\|u(t)\|_H^2 dt = \|G(u(t))\|^2_{\gamma(Y_2,H)} dt + 2 \operatorname{Re} \skpH{u(t)}{ -\im  G(u(t))  \df \textbf{W}(t)}.
\]
Hence, using \eqref{crescitaG-0}
\[
{\rm d}\left(e^{\lambda t}\|u(t)\|_H^2\right)
\le
 (\lambda -2\beta+\tilde C_1 ^2) e^{\lambda t} \|u(t)\|_H^2+ 2 \operatorname{Re} \skpH{u(t)}{ -\im  G(u(t))  \df \textbf{W}(t)}.
 \]
Taking the conditional expected value on both sides, we get
\begin{equation*}
\frac{{\rm d}}{{\rm d}t}\mathbb{E}[e^{\lambda t}  \|u(t)\|_H^2 |\mathcal{F}_s]
\le (\lambda -2\beta+\tilde C_1 ^2)\mathbb{E}[e^{\lambda t} \|u(t)\|_H^2|\mathcal{F}_s], \quad s <t,
\end{equation*}
so that
\begin{equation*}
\mathbb{E}[e^{\lambda t}  \|u(t)\|_H^2 |\mathcal{F}_s]
\le
e^{(\lambda-2\beta+\tilde C_1 ^2)(t-s)}e^{\lambda s} \|u(s)\|_H^2 ,\quad s <t.
\end{equation*}
If we now choose $\lambda>0$ such that $\lambda-2\beta+\tilde C_1 ^2<0$ we obtain
\begin{equation}
\label{eq4.}
\mathbb{E}[e^{\lambda t}\|u(t)\|_H^2 |\mathcal{F}_s] \le e^{\lambda s} \|u(s)\|_H^2  \qquad \forall  \ s<t.
\end{equation}
\end{proof}


 Let us now prove Theorem \ref{cordelta0}.
\begin{proof}[Proof of Theorem \ref{cordelta0}]
 For the (unique) solution of problem \eqref{Problem}, we put in evidence the initial datum $u_0 \in {V}$
by writing $u(\cdot;u_0)$.
Lemma \ref{lem1_delta0} yields
\begin{equation*}
\label{es1_delta0}
\mathbb{E} \|u(t;u_0)\|^2_H \le e^{-\lambda t} \|u_0\|^2_H, \qquad \forall \ t \ge 0.
\end{equation*}
Proceeding as in  \cite[Proof of Theorem 1.4]{BrzMasSei} one then shows by means of the Borel-Cantelli Lemma
that, for every $\bar \lambda \in (0, \lambda)$,
there exists a $\mathbb{P}$-a.s. finite function $t_0: \Omega \rightarrow [0, \infty]$ such that
\begin{equation*}
 \|u(t;u_0)\|^2_H \le e^{-\bar \lambda t} \|u_0\|^2_H, \qquad \forall \ t \ge t_0, \quad \mathbb{P}-a.s.
\end{equation*}
Hence
 \begin{equation*}
\|u(t;u_0)\|_H \rightarrow 0, \quad \text{as} \ t \rightarrow \infty, \quad \mathbb{P}-a.s.
\end{equation*}

Now take any function $\phi:V\to\mathbb R$ which is continuous with respect to the $H$-norm; write
$\phi \in C(V_H)$.  By the above we have
\[
\lim_{t\to+\infty} \phi(u(t;u_0))=\phi(0)
\]
for any initial data $u_0$. Moreover this function $\phi$ belongs to $SC_b(V_w)$, because
 the embedding
$V\subset H$ is compact, so that any sequence weakly convergent in $V$ is strongly convergent in $H$.

Now let $\pi$ be any invariant measure. Then we have
\[
\int_V P_t \phi\ d\pi=\int_V \phi\  d\pi \qquad \forall \phi \in SC_b(V_w), t\ge 0.
\]
Taking $\phi  \in C_b(V_H)$, by the Dominated Convergence Theorem the left hand side converges to $\phi(0)$
as $t \to +\infty$. This implies that
\[
\phi(0)=\int_V \phi\  d\pi \qquad \forall \phi \in C_b(V_H).
\]
We can conclude that $\pi=\delta_0$ if this equality holds for any  $\phi \in SC_b(V_w)$.
In fact, take $\phi(u)=e^{ i \skp{u}{h}_{V}}$ so to get that the integral defines the characteristic function and this is enough to determine the measure.

This can be shown
by approximation.
For any $u\in V$ and $\epsilon>0$  define $u_\epsilon=(I+\epsilon A)^{-1}u$ so that
 \begin{equation}\label{pointwise}
\lim_{\epsilon \to 0}\|u_\epsilon-u\|_V=0
\end{equation}
and
\[
\|u_\epsilon\|_V \lesssim_\epsilon \|u\|_H
\]
(here the constant $C(\epsilon)$ in this estimate is not bounded as $\epsilon \to 0$).\\
Now take any  $\phi \in SC_b(V_w)\subset C_b(V)$ and define
\[\phi_\epsilon(u)=\phi((I+\epsilon A)^{-1}u).\]
It is clear $\phi_\epsilon(u)\to \phi(u)$ for any $u \in V$.
Moreover  from the previous arguments we have that $\phi_\epsilon\in C_b(V_H)$ and therefore we can write the identity
\[
\int_V \phi_\epsilon\  d\pi =\phi(0).
\]
Now passing in the limit as $\epsilon \to 0$ in the l.h.s., by means of the  dominated convergence theorem
 and \eqref{pointwise} we get
 \[
\int_V \phi\  d\pi =\phi(0)\qquad \forall \phi \in SC_b(V_w).
\]
\end{proof}






