\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 79, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/79\hfil Magnetic Benard problem]
{Ergodicity of the two-dimensional magnetic Benard problem}

\author[K. Yamazaki \hfil EJDE-2016/79\hfilneg]
{Kazuo Yamazaki}

\address{Kazuo Yamazaki \newline
Department of Mathematics and Statistics,
Washington State University, Pullman,
WA 99164-3113, USA}
\email{kyamazaki@math.wsu.edu, Phone 509-335-9812}

\thanks{Submitted December 3, 2015. Published March 16, 2016.}
\subjclass[2010]{35Q35, 37H99, 60H15}
\keywords{B\'enard problem; ergodicity; invariant measure; irreducibility;
\hfill\break\indent  Krylov-Bogoliubov theorem; strong Feller}

\begin{abstract}
 We study the two-dimensional magnetic B\'enard problem with noise,
 white in time. We prove the well-posedness including the path-wise
 uniqueness of the generalized solution, and the existence of the unique
 invariant, and consequently ergodic, measure under random perturbation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

The B\'enard problem is concerned  with the motion of a horizontal layer
 of viscous fluid heated from below and the magnetic B\'enard problem
 similarly with the electrically conducting viscous fluid. Both problems
 have attracted much attention in the past (cf. \cite{18, 38} for the
 B\'enard problem and \cite{19, 20, 33, 39, 40} for the magnetic B\'enard problem).
 In particular, systems of stochastic partial differential equations for
 these problems have been studied intensively (cf. \cite{2, 9, 10} for the
 B\'enard problem and \cite{3} for the magnetic B\'enard problem). For such systems,
 the existence of a unique invariant measure, if proven, describes the statistical
 equilibrium to which the system approaches. In this manuscript, we show the
 existence and uniqueness of the invariant measure, which is consequently ergodic,
 for the magnetic B\'enard problem.

\section{Preliminaries and statement of results}

We consider a spatial domain $D = (0,L) \times (0,1)$ with $L > 0$.
 We denote $\partial_t \triangleq \frac{\partial}{\partial t}$,
 $\partial_i \triangleq \frac{\partial}{\partial x_i}$, $i= 1, 2$
 and let $(e_1, e_2)$ represent the standard basis in $\mathbb{R}^2$.
We let $u(x,t) = (u_1, u_2)(x,t), b(x,t) = (b_1, b_2)(x,t), p(x,t), \theta(x,t)$
be the velocity and magnetic vector fields, pressure and temperature scalar
fields respectively. We furthermore denote by $\nu_1, \nu_2, \kappa$ the
kinematic viscosity, magnetic diffusivity, thermal diffusivity respectively
and $s = H^2\nu_1\nu_2$ where $H$ is the Hartman number. Finally we let
$n^{j}(x,t), j= 1, 2$, be the Gaussian random fields, white noise in time,
to be elaborated more below. With these notations, we consider
\begin{gather}
\partial_t u + (u\cdot\nabla) u - \nu_1 \Delta u + \nabla ( p + \frac{s}{2}
| b |^2) - s (b\cdot\nabla) b = \theta e_2 + n^1, \label{e1a}\\
\partial_t b + (u\cdot\nabla) b - (b\cdot\nabla) u - \nu_2 \Delta b = n^2, \label{e1b}\\
\partial_t \theta + (u\cdot\nabla) \theta - \kappa \Delta \theta - u_2 = 0, \label{e1c}
\end{gather}
equipped with the boundary and initial conditions
\begin{gather} \label{e2}
\begin{gathered}
u, p,  b, \theta, \partial_1 u,\partial_1 b,  \partial_1 \theta,
\text{ periodic in the $x_1$-direction with period } L,\\
u, b_2, \partial_2 b_1, \theta  = 0 \quad\text{on } \{x_2 = 0\} \cup \{x_2 = 1\},
\end{gathered}	\\
 \label{e3}
(u,b,\theta)(x,0) = (u_0, b_0, \theta_0)(x), 	
\end{gather}
where $x_i$ is the $i$-th coordinate of $x$ (cf. \cite[section 2.1.4 pg. 386]{3}).
For simplicity, hereafter we assume $s = 1$, write $\int f$ for
$\int_{D} f(x) dx$ and when an equality or an inequality consists of a constant
 $c$ that is of no significance, we write $A \approx B, A \lesssim B$ instead
of $A = cB, A \leq cB$ respectively. For convenience, hereafter we frequently
denote the solution by $y = (u,b,\theta)$. We now set up the standard functional
setting and subsequently state our main results. We let
\begin{align*}
H_1 &= H_2\triangleq \big\{ u \in (L^2(D))^2: \nabla \cdot u = 0, \;
 u_1 |_{\{ x_1 = 0 \} } = u_1 |_{\{x_1 = L\}},\\
& \quad\quad u_2|_{\{x_2 = 0 \}} = u_2 |_{\{x_2 = 1\}} = 0 \big\},\\
H_3&= (L^2(D))^2,
\end{align*}
and set $\mathbb{H} \triangleq H_1 \times H_2 \times H_3$, endowed with
its norm for $y^{j} = (u^{j}, b^{j}, \theta^{j})$,
\[
(y^1, y^2) = \int y^1 \cdot y^2, \quad | y^{j} |^2 = (y^{j}, y^{j}),
\quad j = 1, 2.
\]
We define
\begin{gather*}
V_1 = \{ u \in H_1 \cap (H^1(D))^2 : u|_{\{x_2 = 0\}} = u |_{\{x_2 = 1\}} = 0,
u \text{ is L-periodic in } x_1\},\\
V_2 = H_2 \cap (H^1(D))^2,\\
V_3 = \{\theta \in H^1(D): \theta |_{\{x_2 =0 \}} = \theta|_{\{x_2 = 1\}} = 0, \theta \text{ is L-periodic in } x_1\}.
\end{gather*}
We let $\mathbb{V} \triangleq V_1 \times V_2 \times V_3$, and $A_1$ be the Stokes
operator defined by its bilinear form
\[
a_1(u^1, u^2) = \nu_1 \sum_{i=1}^2 \int \nabla u_i^1 \cdot \nabla u_i^2,  \quad
a_1(u^{j}, u^{j}) = \| u^{j}  \|_{V_1}^2, \quad u^1, u^2 \in V_1\,.
\]
Similarly $A_2$ is defined by its bilinear form
\[
a_2(b^1, b^2) = \nu_2 \sum_{i=1}^2\int \nabla b_i^1 \cdot \nabla b_i^2, \quad
a_2(b^{j}, b^{j}) = \| b^{j} \|_{V_2}^2, \quad b^1, b^2 \in V_2\,.
\]
$A_3$ is generated by the Dirichlet form
\[
a_3(\theta^1, \theta^2) = \kappa \int \nabla \theta^1 \cdot \nabla \theta^2,
\quad a_3(\theta^{j}, \theta^{j}) = \| \theta^{j} \|_{V_3}^2, \quad
\theta^1, \theta^2 \in V_3\,.
\]
Finally $A$ be the operator defined by
\[
a(y^1, y^2) =  \sum_{i=1}^2 \int \nu_1\nabla u_i^1 \cdot \nabla u_i^2
+ \nu_2  \nabla b_i^1\cdot \nabla b_i^2
+ \kappa\nabla \theta^1 \cdot \nabla \theta^2, \quad
a(y^{j}, y^{j}) = \| y^{j} \|_{V}^2,
\]
$j = 1, 2$. We denote by
$V_1^{l} = D(A_1^{1/2}),  V_2^{l} = D(A_2^{1/2}), V_3^{l}
= D(A_3^{1/2})$ and $\mathbb{V}^{l} \triangleq D(A^{1/2}), l \in \mathbb{R}$
(cf. \cite[Chapter II Section 2.1]{38}). We also define bilinear continuous
mappings $B_i, i = 1, \dots, 5$ to satisfy
\begin{gather*}
\langle B_1(u^1, u^2), u^3\rangle = \int (u^1 \cdot\nabla)u^2 \cdot u^3, \quad
\langle B_2(b^1, b^2), u^3\rangle = \int (b^1 \cdot\nabla) b^2 \cdot u^3, \\
\langle B_3(u^1, b^2)\cdot b^3\rangle = \int (u^1 \cdot\nabla) b^2 \cdot b^3,\quad
\langle B_4(b^1, u^2), b^3\rangle = \int (b^1 \cdot\nabla) u^2 \cdot b^3, \\
\langle B_5(u^1, \theta^2), \theta^3\rangle
 = \int (u^1 \cdot\nabla) \theta^2 \theta^3.
\end{gather*}
For $y = (u, b, \theta)$, denoting
\[
B(y,y) \triangleq
\begin{pmatrix}
B_1(u,u) - B_2(b,b)\\
B_3(u,b) - B_4(b,u)\\
B_5(u,\theta)
\end{pmatrix},
\quad
Ry \triangleq
\begin{pmatrix}
\theta e_2\\
0\\
u_2
\end{pmatrix},
\quad
n \triangleq
\begin{pmatrix}
n^1\\
n^2\\
0
\end{pmatrix},
\]
we rewrite \eqref{e1a}--\eqref{e1c} as
\begin{equation} \label{e4}
\partial_t y + Ay + B(y,y) = Ry + n.
\end{equation}
We remark on a distinctive feature of the B\'enard and magnetic B\'enard
problems in comparison to the Navier-Stokes equations (NSE) and the
magnetohydrodynamics (MHD) system. Considering the case e.g. $n \equiv 0$
for the MHD system which is \eqref{e4} at $R \equiv 0, \theta \equiv 0$,
we see that there exists $\alpha' > 0$, for example
$\alpha' = \min \{\nu_1, \nu_2\} > 0$, such that
\[
(Ay, y) + (Ry, y) \geq \alpha' \| y \|_{\dot{H}^1}^2;
\]
i.e. $A + R$ is coercive on $\dot{H}^1(D)$, while in the case of the B\'enard
and the magnetic B\'enard problems, such a property is valid only if
$\min \{\nu_1, \nu_2, \kappa\} > 0$ is sufficiently large
(see \cite[Chapter III Section 3.5]{38}).

Concerning $n^{j}(t), j = 1, 2$, we let
$\{g_i^{j}, \lambda_i^{j}\}_{i=1}^{\infty}$, $j= 1, 2, 3$ be the eigenvectors
and their corresponding eigenvalues of $A_{j}$ respectively. Then we define
 $w^{j}(t) \triangleq \sum_{i=1}^{\infty} \beta_i^{j}(t) g_i^{j}$,
\begin{equation} \label{e5}
G^{j}w^{j}(t) \triangleq \sum_{i=1}^{\infty} \sigma_i^{j} \beta_i^{j}(t) g_i^{j},
\quad j = 1, 2
\end{equation}
where $\{\sigma_i^{j}\}_{i=1}^{\infty}, j=  1, 2$ are noise intensities,
$\{\beta_i^{j}\}_{i=1}^{\infty}$, $j = 1, 2$ are families of independent
standard one-dimensional Brownian motions defined for all  $t$,
on $(\Omega, \mathcal{F}, \mathbb{P})$ with expectation with respect to
$\mathbb{P}$ denoted by $E$ and filtration of
$\mathcal{F}_t = \sigma \{w(s) - w(\tau): \tau \leq s \leq t \}$ where
$w \triangleq (w^1, w^2, 0)$. We note that $\{g_i^{j}\}_{i=1}^{\infty}$, $j = 1, 2$
 forms an orthonormal basis for $H_{j}$, $j= 1,2,3$ respectively.

Now letting $G \triangleq (G^1, G^2, 0), G \in L(\mathbb{H})$,
we define our solution.

\begin{definition} \label{def2.1} \rm
The stochastic process $y = (u,b, \theta)$ is a generalized solution over
time interval $[t_0, T]$ of \eqref{e4} if $\mathbb{P}$ a.e.
 $\omega \in \Omega$,
\begin{enumerate}
\item
\begin{equation} \label{e6}
y(\cdot, \omega) \in C([t_0, T]; \mathbb{H})
\cap L^2(t_0, T; D(A_1^{1/4}) \times D(A_2^{1/4}) \times V_3),
\end{equation}

\item $y = (u,b,\theta)$ satisfies for all
$\phi = (\phi^1, \phi^2, \phi^3) \in D(A)$,
\begin{equation} \label{e7}
\begin{aligned}
& \langle y(t), \phi\rangle - \langle y(t_0), \phi\rangle
+ \int_{t_0}^{t} \langle Ay, \phi\rangle ds \\
&= -\int_{t_0}^{t} \langle B(y, y) + Ry, \phi\rangle ds
+ \langle Gw(t)-  Gw(t_0), \phi\rangle,
\end{aligned}
\end{equation}

\item $y = (u,b,\theta)$ is progressively measurable.
\end{enumerate}
\end{definition}

Let us now recall that irreducibility in $\mathbb{V}^{l}$ implies that
the transition function
$P(t, x, \Gamma) \triangleq \mathbb{P} (\{y(t; x) \in \Gamma\})$ satisfies
\[
P(t, x, \Gamma) > 0 \quad \forall   t > t_0,\; x \in \mathbb{V}^{l}
\]
for all  non-empty open set $\Gamma \subset \mathbb{V}^{l}$.
Moreover, their associated Markovian semigroup $\{P_t\}$ is defined by
\begin{equation} \label{e8}
(P_t \psi) (x) \triangleq E[\psi(y(t; x))]
\end{equation}
is Feller if $P_t(C_{b}(\mathbb{V}^{l})) \subset C_{b}(\mathbb{V}^{l})$ and
strong Feller if $P_t(B_{b}(\mathbb{V}^{l})) \subset C_{b}(\mathbb{V}^{l})$
for all $t \in (t_0,T]$ where $B_{b}(\mathbb{V}^{l}), C_{b}(\mathbb{V}^{l})$
are the sets of all real Borel functions that are bounded, continuous and
bounded respectively. We define a dual semigroup $P_t^{\ast}$ in $Pr(\mathbb{H})$,
the space of probability measures on $\mathbb{H}$, by
\[
\int_{\mathbb{H}} \psi d(P_t^{\ast} \mu) = \int_{\mathbb{H}} P_t \psi d\mu.
\]
If $P_t^{\ast} \mu = \mu$, then $\mu \in Pr(\mathbb{H})$ is called an
invariant measure for the dynamical system $(Pr(\mathbb{H}), P_t^{\ast})$.
We now state our first result.

\begin{theorem} \label{thm2.1}
Suppose that for both $j = 1, 2$, we have
\begin{gather} \label{e9}
\sigma_i^{j} \neq 0 \quad \forall \hspace{1mm} i= 1, 2, \dots;\\
\label{e10}
\sigma_i^{j} \leq \frac{C}{i^{\frac{1}{4} + 2\gamma_0}} \text{ for some } \gamma_0 > 0
\end{gather}
for $i$ large enough.
Then for all $y_{t_0} \triangleq (u, b, \theta) (t_0) \in \mathbb{H}$,
there exists a unique process $y$ that solves \eqref{e7} as defined in
 Definition \ref{def2.1} such that in addition to the regularity of \eqref{e6},
\begin{gather*}
 u(\cdot, \omega) \in L^2(t_0, T; D(A_1^{\min\{\frac{1}{4}
 + \gamma, \frac{1}{2}\}})) \cap L^4(t_0, T; D(A_1^{1/4})),\\
 b(\cdot, \omega) \in L^2(t_0, T; D(A_2^{\min\{\frac{1}{4}
 + \gamma, \frac{1}{2}\}})) \cap L^4(t_0, T; D(A_2^{1/4}))
\end{gather*}
for all $\gamma < \gamma_0$, $\mathbb{P}$-a.e. $\omega \in \Omega$. Moreover,
it is a Markov process. Finally, there exists an invariant measure for \eqref{e4}.
\end{theorem}

To prove the uniqueness of the invariant measure, it suffices to prove the
irreducibility and strong Feller property (\cite{8} and e.g. \cite[Theorem 4.2.1]{6}).
Under a slightly more restrictive condition, this is possible.

\begin{theorem} \label{thm2.2}
Let $l \in \mathbb{Z}, l \geq 2$. Suppose that for both
$j = 1, 2$, $\{\sigma_i^{j}\}_{i=1}^{\infty}$  satisfy \eqref{e9} and
\begin{equation} \label{e11}
\frac{c}{i^{\frac{l+1}{2}}} \leq \sigma_i^{j}  < \frac{C}{i^{1/2}}
\end{equation}
for $i's$ large enough. Then for all
$y_0\triangleq (u_0, b_0, \theta_0) \in \mathbb{V}^{l}$,
there exists a unique process $y$ that solves \eqref{e7} as defined in
 Definition \ref{def2.1} at $t_0 = 0$ such that for $\mathbb{P}$-a.e. $\omega \in \Omega$,
\[
y \in C([0,T]; \mathbb{V}^{l});
\]
moreover, it is a Markov process. Finally, there exists a unique invariant
measure $\mu$ of \eqref{e7} at $t_0 = 0$, supported in $\mathbb{V}^{l}$
such that
$\lim_{t\to\infty} P(t, x, \Gamma) = \mu(\Gamma)$
for all $x \in \mathbb{V}^{l}, \Gamma \in \mathcal{B} (\mathbb{V}^{l})$.
\end{theorem}

\begin{remark} \label{rmk2.1} \rm
(1) As a consequence of uniqueness, the invariant measure claimed in
Theorem \ref{thm2.2} is ergodic (cf. \cite[Theorem 3.2.6]{6}).
To the best of the author's knowledge, both Theorems \ref{thm2.1}
and \ref{thm2.2} represent  new results on the magnetic B\'enard problem.
Moreover, although the existence
 of an invariant measure for the B\'enard problem is shown in \cite{10}, not its
uniqueness. As the magnetic B\'enard problem when $b \equiv 0$ becomes
the B\'enard problem, Theorem \ref{thm2.2} proves the uniqueness of the invariant
 measure obtained in \cite{10} for the B\'enard problem as well.

(2) The proofs of Theorems \ref{thm2.1} and \ref{thm2.2}
 follow the approach of \cite{10, 12} closely.
The proof is also inspired by the previous work on ergodicity of the NSE,
Burgers' equation (see \cite{5, 11, 14, 16, 17});
 we also refer to \cite{1, 22, 24, 25, 26, 31, 32, 34}
as closely related important work. We remark however that in \cite{10, 14},
the path-wise uniqueness of the generalized solution was missing for a
technical reason; in fact, in \cite[Remark after Theorem 3.1]{14},
this uniqueness was stated as an open problem. We follow the approach
of \cite{13} to obtain such a path-wise uniqueness.

(3) For the generalization to $l$ non-integer and $l = 1$ cases,
we refer to the \cite[Section 6]{12};
we chose not to pursue this direction of research for simplicity of presentation.
\end{remark}

\section{Proof of Theorem \ref{thm2.1}: Existence}

We consider two Ornstein-Uhlenbeck processes: for $\alpha \geq 0$, $j = 1, 2$,
defined also for $t < 0$,
\begin{equation} \label{e12}
dz_{\alpha}^{j} + A_{j} z_{\alpha}^1(t) dt
= -\alpha z_{\alpha}^{j} (t) dt + dG^{j} w^{j},
\end{equation}
for which their solutions are of the form
\begin{equation} \label{e13}
z_{\alpha}^{j}(t) = \int_{-\infty}^{t} e^{(-A_{j} - \alpha)(t-s)} G^{j}dw^{j}(s)
\end{equation}
(cf. \cite{7}) and we collect their properties in the Appendix.
By \eqref{e10} we see that there exists  $\gamma_0 > 0$ such that
\[
\sum_{i=1}^{\infty} \frac{(\sigma_i^{j})^2}{(\lambda_i^{j})
^{\frac{1}{2} - 2 \gamma_0}}  < \infty
\]
because $\lambda_i^{j} \sim i$ as $i \to \infty$. Thus, by Lemma \ref{lem5.2} (1)
we see that $\mathbb{P}$-a.s.
\begin{equation} \label{e14}
z_{\alpha}^{j} \in C([t_0, T]; D(A_{j}^{\gamma_0 + \frac{1}{4}})),
\end{equation}
and by Lemma \ref{lem5.1} (2)
\begin{equation} \label{e15}
G^{j}w^{j} \in C([t_0, T]; D(A_{j}^{\gamma_0 - \frac{1}{4}})).
\end{equation}
Now we define
\begin{equation} \label{e16}
\overline{u} \triangleq u - z_{\alpha}^1, \quad
\overline{b} \triangleq b - z_{\alpha}^2
\end{equation}
so that by \eqref{e4} and \eqref{e12},
\begin{gather}
\begin{aligned}
&\partial_t \overline{u} + A_1 \overline{u} + B_1(\overline{u}, \overline{u}) + B_1(\overline{u}, z_{\alpha}^1) + B_1(z_{\alpha}^1, \overline{u}) - B_2(\overline{b}, \overline{b}) - B_2(\overline{b}, z_{\alpha}^2) - B_2(z_{\alpha}^2, \overline{b})\\
&= - B_1(z_{\alpha}^1, z_{\alpha}^1) + B_2(z_{\alpha}^2, z_{\alpha}^2)
+ \theta e_2 + \alpha z_{\alpha}^1,
\end{aligned} \label{e17a} \\
\begin{aligned}
& \partial_t \overline{b} + A_2 \overline{b} + B_3(\overline{u}, \overline{b})
 + B_3(\overline{u}, z_{\alpha}^2) + B_3(z_{\alpha}^1, \overline{b})
 - B_4(\overline{b}, \overline{u}) - B_4(\overline{b}, z_{\alpha}^1)
 - B_4(z_{\alpha}^2, \overline{u}) \\
&= -B_3(z_{\alpha}^1, z_{\alpha}^2) + B_4(z_{\alpha}^2, z_{\alpha}^1)
+ \alpha z_{\alpha}^2,
\end{aligned} \label{e17b}\\
\partial_t \theta + A_3 \theta + B_5 (\overline{u}, \theta)
= -B_5(z_{\alpha}^1, \theta) + \overline{u}_2 + z_{\alpha,2}^1,
\label{e17c}
\end{gather}
where $z_{\alpha,2}^1$ is the second component of $z_{\alpha}^1$.

We now fix $\omega \in \Omega$ and take $L^2$-inner products of
 \eqref{e17a}--\eqref{e17c}  with $(\overline{u}, \overline{b}, \theta)$.
Their addition yields
\begin{equation} \label{e18}
\begin{aligned}
& \frac{1}{2} \partial_t | (\overline{u}, \overline{b}, \theta) |^2
 + \| (\overline{u}, \overline{b}, \theta) \|_{\mathbb{V}}^2\\
&= \langle B_1(\overline{u}, \overline{u}),z_{\alpha}^1\rangle
 - \langle B_2(\overline{b}, \overline{u}), z_{\alpha}^2\rangle
 + \langle B_1(z_{\alpha}^1, \overline{u}), z_{\alpha}^1\rangle
 - \langle B_2(z_{\alpha}^2, \overline{u}), z_{\alpha}^2\rangle\\
&\quad+ \langle \theta e_2, \overline{u}\rangle
 + \alpha \langle  z_{\alpha}^1, \overline{u}\rangle
 + \langle B_3(\overline{u}, \overline{b}), z_{\alpha}^2\rangle
 - \langle B_4(\overline{b}, \overline{b}), z_{\alpha}^1\rangle \\
&\quad + \langle B_3(z_{\alpha}^1, \overline{b}), z_{\alpha}^2\rangle
 - \langle B_4(z_{\alpha}^2, \overline{b}), z_{\alpha}^1\rangle
 + \alpha \langle z_{\alpha}^2, \overline{b}\rangle
 + \langle \overline{u}, e_2\theta\rangle
 + \langle z_{\alpha}^1, e_2\theta\rangle  \\
&\leq \| \overline{u} \|_{L^4} \| \overline{u} \|_{V_1} \| z_{\alpha}^1 \|_{L^4}
 + \| \overline{b} \|_{L^4} \| \overline{u} \|_{V_1} \| z_{\alpha}^2 \|_{L^4}
 + \| z_{\alpha}^1 \|_{L^4}^2 \| \overline{u} \|_{V_1}
 + \| z_{\alpha}^2 \|_{L^4}^2 \| \overline{u} \|_{V_1}\\
&\quad  + | \theta | | \overline{u} |
 + \alpha | z_{\alpha}^1 | | \overline{u} |
 + \| \overline{u} \|_{L^4} \| \overline{b} \|_{V_2} \| z_{\alpha}^2 \|_{L^4}
 + \| \overline{b} \|_{L^4}\| \overline{b} \|_{V_2} \| z_{\alpha}^1 \|_{L^4} \\
&\quad + \| z_{\alpha}^1 \|_{L^4} \| \overline{b} \|_{V_2}
 \| z_{\alpha}^2 \|_{L^4}
 + \| z_{\alpha}^2 \|_{L^4} \| \overline{b} \|_{V_2} \| z_{\alpha}^1 \|_{L^4}
 + \alpha | z_{\alpha}^2 | | \overline{b} | + | \overline{u} | | \theta |
 + | z_{\alpha}^1 | | \theta |  \\
&\lesssim | \overline{u} |^{1/2} \| \overline{u} \|_{V_1}^{3/2}
  \| z_{\alpha}^1 \|_{L^4}
 + | \overline{b} |^{1/2} \| \overline{b} \|_{V_2}^{1/2}
  \| \overline{u} \|_{V_1} \| z_{\alpha}^2 \|_{L^4}
 + \| z_{\alpha}^1 \|_{L^4}^2 \| \overline{u} \|_{V_1}\\
&\quad + \| z_{\alpha}^2 \|_{L^4}^2 \| \overline{u} \|_{V_1}
 + | \theta | | \overline{u} | + | z_{\alpha}^1 | | \overline{u} |
 + | \overline{u} |^{1/2} \| \overline{u} \|_{V_1}^{1/2}
 \| \overline{b} \|_{V_2} \| z_{\alpha}^2 \|_{L^4} \\
&\quad + | \overline{b} |^{1/2} \| \overline{b} \|_{V_2}^{3/2}
 \| z_{\alpha}^1 \|_{L^4}
 + \| z_{\alpha}^1 \|_{L^4} \| \overline{b} \|_{V_2}
 \| z_{\alpha}^2 \|_{L^4} + | z_{\alpha}^2 | | \overline{b} |
 + | \overline{u} | | \theta | + | z_{\alpha}^1 | | \theta | \\
&\leq \frac{1}{2}  \| (\overline{u}, \overline{b}, \theta ) \|_{\mathbb{V}}^2
 + c | (\overline{u}, \overline{b}, \theta) |^2 (1+ \| (z_{\alpha}^1, z_{\alpha}^2 )
  \|_{L^4}^4) + c\| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4
\end{aligned}
\end{equation}
where we used integration by parts, H\"older's inequalities,
Gagliardo-Nirenberg inequalities of e.g.
$\| u \|_{L^4(D)} \lesssim | u |^{1/2} \| u \|_{V_1}^{1/2}$
(cf.  \cite[Lemma 6.2]{28}, and more generally \cite{30}),
 and Young's inequalities. We also relied on important cancelations such as
\begin{gather*}
- \langle B_2(\overline{b}, \overline{b}), \overline{u}\rangle
 - \langle B_4(\overline{b}, \overline{u}), \overline{b}\rangle = 0,\\
- \langle B_2(z_{\alpha}^2, \overline{b}), \overline{u}\rangle
 - \langle B_4(z_{\alpha}^2, \overline{u}), \overline{b}\rangle = 0.
\end{gather*}
After subtracting
$\frac{1}{2}  \| (\overline{u}, \overline{b}, \theta ) \|_{\mathbb{V}}^2$
from both sides of \eqref{e18}, we obtain by the Sobolev embedding of
$D(A_{j}^{1/4}) \hookrightarrow L^4(D)$ and \eqref{e14} that for all
$  t \in [t_0, T]$
\begin{equation} \label{e19}
| (\overline{u}, \overline{b}, \theta) (t) |^2
+ \int_{t_0}^{T}\| (\overline{u}, \overline{b},
\theta) \|_{\mathbb{V}}^2 ds \lesssim | (\overline{u}, \overline{b},
\theta)(t_0)|^2 + 1 \lesssim 1.
\end{equation}

\begin{proposition}  \label{prop3.1}
Under the hypothesis of Theorem \ref{thm2.1},
$\alpha \geq 0$, on $[t_0, T]$ such that $(u,b,\theta)(t_0) \in \mathbb{H}$,
there exists a unique solution $(\overline{u}, \overline{b}, \theta)$
to \eqref{e17a}--\eqref{e17c} such that
$(\overline{u}(t_0), \overline{b}(t_0), \theta(t_0))
 = (u(t_0) - z_{\alpha}^1(t_0), b(t_0) - z_{\alpha}^2(t_0), \theta(t_0))$ and
\begin{equation} \label{e20}
(\overline{u}, \overline{b}, \theta) \in C([t_0, T]; \mathbb{H})
\cap L^2(t_0, T; \mathbb{V}) \quad \mathbb{P}-\text{a.e. } \omega \in \Omega.
\end{equation}
Moreover, the solution is independent of $\alpha$.
\end{proposition}

\begin{proof}
With the \emph{a priori} estimates of \eqref{e19}, a standard Galerkin approximation
scheme shows the existence of the solution (see \cite{4, 28, 38}). We note that
due to \eqref{e16} the regularity of $u$ and $b$ are obtained from
$\overline{u}, z_{\alpha}^1$ and $\overline{b}, z_{\alpha}^2$ respectively.
This is the reason why in Definition \ref{def2.1}, we require only
$u \in L^2 (t_0, T; D(A_1^{1/4})), b \in L^2(t_0, T; D(A_2^{1/4}))$
instead of $L^2(t_0, T; V_1), L^2(t_0, T; V_2)$.

We now prove the uniqueness. Suppose there exists
$(\overline{u}^1, \overline{b}^1, \theta^1), (\overline{u}^2,
 \overline{b}^2, \theta^2)$ that are both solutions. Then defining
\begin{equation} \label{e21}
\delta \overline{u} \triangleq \overline{u}^1 - \overline{u}^2, \quad
\delta \overline{b} \triangleq \overline{b}^1 - \overline{b}^2, \quad
\delta \theta \triangleq \theta^1 - \theta^2,
\end{equation}
we see that from \eqref{e17a}--\eqref{e17c},
\begin{gather}
\begin{aligned}
\partial_t \delta \overline{u} + A_1 \delta \overline{u}
&=- B_1 (\delta \overline{u}, \overline{u}^1)
 - B_1(\overline{u}^2, \delta \overline{u})
 - B_1(\delta \overline{u}, z_{\alpha}^1)
 - B_1(z_{\alpha}^1, \delta \overline{u})  \\
&\quad + B_2(\delta \overline{b}, \overline{b}^1)
 + B_2(\overline{b}^2, \delta \overline{b})
 + B_2(\delta \overline{b}, z_{\alpha}^2)
 + B_2(z_{\alpha}^2, \delta \overline{b}) + \delta \theta e_2 ,
\end{aligned} \label{e22a}\\
\begin{aligned}
\partial_t \delta \overline{b} + A_2 \delta \overline{b}
& = -B_3 (\delta \overline{u}, \overline{b}^1)
 - B_3(\overline{u}^2, \delta \overline{b})
 - B_3(\delta \overline{u}, z_{\alpha}^2)
 - B_3(z_{\alpha}^1, \delta \overline{b})  \\
&\quad + B_4(\delta \overline{b}, \overline{u}^1)
 + B_4(\overline{b}^2, \delta \overline{u})
 + B_4(\delta \overline{b}, z_{\alpha}^1)
 + B_4(z_{\alpha}^2, \delta \overline{u}) ,
\end{aligned} \label{e22b}\\
 \partial_t \delta \theta + A_3 \delta \theta
= - B_5(\delta \overline{u}, \theta^1) - B_5(\overline{u}^2, \delta \theta)
 - B_5(z_{\alpha}^1,\delta \theta) + \delta \overline{u}_2, \label{e22c}
\end{gather}
with $\delta \overline{u}_2 = \overline{u}_2^1 - \overline{u}_2^2$,
 $\overline{u}_2^{j}$ being the second component of $\overline{u}^{j}$, $j = 1, 2$.

Taking $L^2$-inner products on \eqref{e22a}--\eqref{e22c}
 with $(\delta \overline{u}, \delta \overline{b}, \delta\theta$) respectively
 we obtain similarly to \eqref{e18},
\begin{align*}
& \frac{1}{2} \partial_t | (\delta \overline{u}, \delta \overline{b},
 \delta \theta) |^2 + \| (\delta \overline{u},
 \delta \overline{b}, \delta \theta) \|_{\mathbb{V}}^2 \\
&= -\int (\delta \overline{u}\cdot\nabla) \overline{u}^1 \cdot \delta \overline{u}
 + \int (\delta \overline{u} \cdot\nabla) \delta \overline{u} \cdot z_{\alpha}^1
 + \int (\delta \overline{b}\cdot\nabla) \overline{b}^1 \cdot \delta \overline{u}
 - \int (\delta \overline{b}\cdot\nabla) \delta \overline{u} \cdot z_{\alpha}^2 \\
&\quad + \int \delta \theta e_2\cdot \delta \overline{u}
 - \int (\delta \overline{u}\cdot\nabla) \overline{b}^1 \cdot \delta \overline{b}
 + \int (\delta \overline{u}\cdot\nabla)  \delta \overline{b}\cdot z_{\alpha}^2
 + \int (\delta \overline{b}\cdot\nabla) \overline{u}^1 \cdot \delta \overline{b}\\
&\quad - \int (\delta \overline{b}\cdot\nabla) \delta \overline{b}\cdot z_{\alpha}^1
 - \int (\delta \overline{u} \cdot\nabla) \theta^1 \delta \theta
 + \int \delta \overline{u}\cdot \delta \theta e_2 \\
&\leq \| \delta \overline{u} \|_{L^4}^2 \| \overline{u}^1 \|_{V_1}
 + \| \delta \overline{u} \|_{L^4} \| \delta \overline{u}
 \|_{V_1} \| z_{\alpha}^1 \|_{L^4}
 + \| \delta \overline{b} \|_{L^4} \| \overline{b}^1 \|_{V_2}
 \| \delta \overline{u} \|_{L^4}   \\
&\quad + \| \delta \overline{b} \|_{L^4} \| \delta \overline{u}
 \|_{V_1} \| z_{\alpha}^2 \|_{L^4}+ | \delta \theta | | \delta \overline{u} |
 + \| \delta \overline{u} \|_{L^4} \| \overline{b}^1\|_{V_2} \|
 \delta \overline{b} \|_{L^4}
 + \| \delta \overline{u} \|_{L^4} \| \delta \overline{b} \|_{V_2}
 \| z_{\alpha}^2 \|_{L^4}  \\
&\quad + \| \delta \overline{b} \|_{L^4}^2 \| \overline{u}^1 \|_{V_1}
 + \| \delta \overline{b} \|_{L^4} \| \delta \overline{b} \|_{V_2}
 \| z_{\alpha}^1 \|_{L^4}+ \| \delta \overline{u} \|_{L^4}
 \| \theta^1 \|_{V_3} \| \delta \theta \|_{L^4}
 + | \delta \overline{u} | | \delta \theta |\\
&\leq \frac{1}{2} \| (\delta \overline{u}, \delta \overline{b},
 \delta \theta) \|_{\mathbb{V}}^2
 + c | (\delta \overline{u}, \delta \overline{b},
 \delta \theta) |^2 (\| (\overline{u}^1,
 \overline{b}^1, \theta^1) \|_{\mathbb{V}}^2
 +  \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4  + 1)
\end{align*}
where we use the crucial cancelations
\begin{gather*}
\int (\overline{b}^2 \cdot\nabla) \delta \overline{b} \cdot \delta \overline{u}
+ \int (\overline{b}^2\cdot\nabla) \delta \overline{u} \cdot \delta \overline{b} = 0,\\
\int (z_{\alpha}^2\cdot\nabla) \delta \overline{b} \cdot \delta \overline{u}
+ \int (z_{\alpha}^2\cdot\nabla) \delta \overline{u} \cdot \delta \overline{b} = 0.
\end{gather*}
Subtracting $\frac{1}{2} \| (\delta \overline{u}, \delta \overline{b},
\delta \theta) \|_{\mathbb{V}}^2$ from both sides, we obtain
\begin{equation} \label{e23}
\begin{aligned}
&\partial_t | (\delta \overline{u}, \delta \overline{b},
 \delta \theta) |^2 + \| (\delta \overline{u}, \delta \overline{b},
 \delta \theta) \|_{\mathbb{V}}^2 \\
&\lesssim  | (\delta \overline{u}, \delta \overline{b},
 \delta \theta) |^2 (\| (\overline{u}^1, \overline{b}^1,
 \theta^1) \|_{\mathbb{V}}^2 +  \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4  + 1).
\end{aligned}
\end{equation}
Gronwall's inequality with \eqref{e14} and \eqref{e19} implies uniqueness.

Concerning the Markov property, it suffices to show that for $y = (u, b, \theta)$,
\[
E[\psi(y(t; t_0, y(t_0))) | \mathcal{F}_{s}]
= P_{t-s} (\psi) (y(s; t_0, y(t_0)))
\]
for all $t_0 < s < t, y(t_0) \in \mathbb{H}, \psi \in C_{b}(\mathbb{H})$.
This result is classical and can be found in \cite[Theorem 9.14]{7}.

We have already shown that the solution $(\overline{u}, \overline{b}, \theta)$
to \eqref{e17a}--\eqref{e17c} is unique for all  $\alpha\geq 0$.
For a fixed $\alpha \geq 0$, we let
$(\overline{u}_{\alpha}, \overline{b}_{\alpha},\theta)$
be the unique solution to \eqref{e17a}--\eqref{e17c}. Then by \eqref{e16},
$u_{\alpha} = \overline{u}_{\alpha} + z_{\alpha}^1, b_{\alpha}
= \overline{b}_{\alpha} + z_{\alpha}^2$ and for $\alpha \geq 0$,
 we have $(\overline{u}_{\alpha}, \overline{b}_{\alpha}, \theta)
\in L^2(0, T; \mathbb{V})$. Moreover, for $j = 1, 2$, denoting by $z^{j}$,
the solution to \eqref{e12} at $\alpha = 0$, we see that they satisfy
\begin{align*}
\partial_t (z_{\alpha}^{j} - z^{j}) = -A_{j} (z_{\alpha}^{j} - z^{j})
- \alpha z_{\alpha}^{j}
\end{align*}
so that
\[
(z_{\alpha}^{j} - z^{j}) (t) = e^{-(t - t_0)A_{j}} (z_{\alpha}^{j}
- z^{j})(t_0) - \int_{t_0}^{t} e^{-(t-s) A_{j}} \alpha z_{\alpha}^{j}(s) ds
\]
for all $\alpha \geq 0$ and it is well-known (cf. \cite{35}) that
 $(z_{\alpha}^{j} - z^{j})(t) \in L^2(t_0, T; V_{j})$.
Thus, $u_{\alpha} - z^1 \in L^2(t_0, T; V_1), b_{\alpha} - z^2
 \in L^2(t_0, T; V_2)$ and therefore for all $\alpha \geq 0$, we obtain
a unique solution $(u_{\alpha}, b_{\alpha})$ independent of $\alpha$;
it is also clear that $\theta$ is independent of $\alpha$.
\end{proof}

\begin{proposition} \label{prop3.2}
Under the hypothesis of Theorem \ref{thm2.1}, suppose that for some
$\gamma \in (0, \frac{1}{2}) \cap (0, 2 \gamma_0]$, where
 $\gamma_0 \in (0, \frac{3}{4}], y_{t_0} \triangleq y(t_0) \in D(A^{\gamma})$.
Then $\mathbb{P}$-a.e. $\omega \in \Omega$,
\[
y \in C([t_0, T] ; D(A^{\min\{\gamma, \frac{1}{4} + \gamma_0\}}))
\cap L^2([t_0, T]; D(A^{\min\{\frac{1}{2} + \gamma, \frac{1}{4} + \gamma_0\}})).
\]
\end{proposition}

\begin{proof}
For both $j = 1 ,2$, $z_{\alpha}^{j} \in C([t_0, T]; D(A_{j}^{\gamma_0
+ \frac{1}{4}}))$ by \eqref{e14} and hence due to \eqref{e16}, it suffices to
show that
\[
(\overline{u},\overline{b},\theta) \in C([t_0, T] ; D(A^{\min\{\gamma, \frac{1}{4}
+ \gamma_0\}})) \cap L^2([t_0, T]; D(A^{\min\{\frac{1}{2} + \gamma, \frac{1}{4}
+ \gamma_0\}})).
\]
We first remark that $| A_1^{k} \cdot | \approx | A_2^{k} \cdot |$.
We take $L^2$-inner products on \eqref{e17a}--\eqref{e17c} with
$(A_1^{2\gamma} \overline{u}, A_2^{2\gamma} \overline{b}, A_3^{2\gamma} \theta)$
to obtain
\begin{align*} %e24
& \frac{1}{2} \partial_t | (A_1^{\gamma} \overline{u}, A_2^{\gamma} \overline{b},
 A_3^{\gamma} \theta) |^2 + | (A_1^{\frac{1}{2} + \gamma} \overline{u},
 A_2^{\frac{1}{2} + \gamma} \overline{b}, A_3^{\frac{1}{2}  +\gamma} \theta) |^2 \\
&= -\int [(\overline{u}\cdot\nabla) \overline{u} + (\overline{u}\cdot\nabla)
 z_{\alpha}^1 + (z_{\alpha}^1 \cdot\nabla) \overline{u}
 - (\overline{b}\cdot\nabla) \overline{b} - (\overline{b}\cdot\nabla) z_{\alpha}^2
 - (z_{\alpha}^2\cdot\nabla) \overline{b}] \cdot A_1^{2\gamma} \overline{u} \\
&\quad - \int [(z_{\alpha}^1 \cdot\nabla) z_{\alpha}^1
 - (z_{\alpha}^2\cdot\nabla) z_{\alpha}^2 - \theta e_2
 - \alpha z_{\alpha}^1 ] \cdot A_1^{2\gamma} \overline{u} \\
&\quad - \int [(\overline{u}\cdot\nabla) \overline{b}
 + (\overline{u}\cdot\nabla) z_{\alpha}^2
 + (z_{\alpha}^1 \cdot\nabla) \overline{b}
 - (\overline{b}\cdot\nabla) \overline{u} -(\overline{b}\cdot\nabla) z_{\alpha}^1
 - (z_{\alpha}^2\cdot\nabla) \overline{u}] \cdot A_2^{2\gamma} \overline{b}  \\
& - \int [(z_{\alpha}^1 \cdot\nabla) z_{\alpha}^2
 - (z_{\alpha}^2\cdot\nabla) z_{\alpha}^1
 - \alpha z_{\alpha}^2]\cdot A_2^{2\gamma} \overline{b} \\
& - \int [(\overline{u}\cdot\nabla) \theta + (z_{\alpha}^1\cdot\nabla) \theta
 + \overline{u}_2 +  z_{\alpha,2}^1] A_3^{2\gamma} \theta.
\end{align*}
We bound e.g.
\[
\int (\overline{u}\cdot\nabla) \overline{u} \cdot A_1^{2\gamma} \overline{u}
\leq | A_1^{-(\frac{1}{2}  -\gamma)} [(\overline{u}\cdot\nabla)
\overline{u}] | | A_1^{\frac{1}{2} + \gamma} \overline{u} |
\lesssim | A_1^{\frac{1}{4} + \frac{\gamma}{2}} \overline{u} |^2
 | A_1^{\frac{1}{2} + \gamma} \overline{u} |
\]
by H\"older's inequality and Lemma \ref{lem5.3}. We compute similarly on other
nonlinear terms to obtain
\begin{equation} \label{e25}
\begin{aligned}
& \frac{1}{2} \partial_t | (A_1^{\gamma} \overline{u}, A_2^{\gamma} \overline{b},
  A_3^{\gamma} \theta ) |^2 + | (A_1^{\frac{1}{2}
 + \gamma} \overline{u}, A_2^{\frac{1}{2}
 + \gamma} \overline{b}, A_3^{\frac{1}{2} + \gamma} \theta ) |^2 \\
&\lesssim (| A_1^{\frac{1}{4} + \frac{\gamma}{2}} \overline{u} |^2
 + | A_1^{\frac{1}{4} + \frac{\gamma}{2}} \overline{u} | | A_1^{\frac{1}{4}
 + \frac{\gamma}{2}} z_{\alpha}^1 | + | A_1^{\frac{1}{4}
 + \frac{\gamma}{2}} \overline{b} |^2 + | A_1^{\frac{1}{4}
 + \frac{\gamma}{2}} \overline{b} | | A_1^{\frac{1}{4}
 + \frac{\gamma}{2}} z_{\alpha}^2 |) | A_1^{\frac{1}{2}
 + \gamma} \overline{u} |\\
&\quad + ( | A_1^{\frac{1}{4} + \frac{\gamma}{2}} z_{\alpha}^1 |^2
 + | A_1^{\frac{1}{4} +\frac{\gamma}{2}} z_{\alpha}^2 |^2
 + | A_1^{\gamma - \frac{1}{2}} \theta | + | A_1^{\gamma - \frac{1}{2}}
  z_{\alpha}^1 |) | A_1^{\frac{1}{2} + \gamma} \overline{u} | \\
&\quad + (| A_2^{\frac{1}{4} + \frac{\gamma}{2} } \overline{u} |
 | A_2^{\frac{1}{4} + \frac{\gamma}{2}} \overline{b}| + | A_2^{\frac{1}{4}
 + \frac{\gamma}{2}} \overline{u} | | A_2^{\frac{1}{4}
 +  \frac{\gamma}{2}} z_{\alpha}^2 | + | A_2^{\frac{1}{4}
 + \frac{\gamma}{2}} z_{\alpha}^1 | | A_2^{\frac{1}{4}
 + \frac{\gamma}{2}} \overline{b} |) | A_2^{\frac{1}{2} + \gamma} \overline{b} | \\
&\quad + (| A_2^{\frac{1}{4} + \frac{\gamma}{2}} z_{\alpha}^1 | | A_2^{\frac{1}{4}
 + \frac{\gamma}{2}} z_{\alpha}^2 | + | A_2^{\gamma - \frac{1}{2}} z_{\alpha}^2 |)
 | A_2^{\frac{1}{2} + \gamma} \overline{b} | \\
&\quad + (| A_3^{\frac{1}{4} + \frac{\gamma}{2}} \overline{u} | | A_3^{\frac{1}{4}
 + \frac{\gamma}{2}} \theta | + | A_3^{\frac{1}{4}
 + \frac{\gamma}{2}} z_{\alpha}^1 | | A_3^{\frac{1}{4}
 + \frac{\gamma}{2}} \theta | + | A_3^{\gamma - \frac{1}{2}} \overline{u} |
 + | A_3^{\gamma - \frac{1}{2}} z_{\alpha}^1 |) | A_3^{\frac{1}{2}
 + \gamma} \theta |.
\end{aligned}
\end{equation}

Now $z_{\alpha}^{j} \in C([t_0, T]; D(A_{j}^{\gamma_0 + \frac{1}{4}}))$
by \eqref{e14} and $\gamma \in (0, 2\gamma_0]$ by hypothesis so that
$\frac{1}{4} + \frac{\gamma}{2} \leq  \gamma_0 + \frac{1}{4}$. Thus,
\begin{align*} %e26
& \frac{1}{2} \partial_t | (A_1^{\gamma} \overline{u}, A_2^{\gamma}
 \overline{b}, A_3^{\gamma} \theta ) |^2
 + | (A_1^{\frac{1}{2} + \gamma} \overline{u}, A_2^{\frac{1}{2}
 + \gamma} \overline{b}, A_3^{\frac{1}{2} + \gamma} \theta ) |^2 \\
&\lesssim (1+ | A_1^{1/2} \overline{u} | | A_1^{\gamma} \overline{u} |
 + | A_1^{1/2} \overline{b}|| A_1^{\gamma} \overline{b} |
 + | A_1^{ \frac{1}{2}} \theta | | A_1^{\gamma} \theta |)| A_1^{\frac{1}{2}
 + \gamma} \overline{u} | \\
&\quad + (1+ | A_2^{1/2} \overline{u} | | A_2^{\gamma} \overline{u} |
 + | A_2^{1/2} \overline{b} || A_2^{\gamma} \overline{b} |) | A_2^{\frac{1}{2}
 + \gamma} \overline{b} | \\
&\quad + (1+ | A_3^{1/2} \overline{u} | | A_3^{\gamma}  \overline{u} |
 + | A_3^{1/2} \theta || A_3^{\gamma} \theta |) | A_3^{\frac{1}{2}
 + \gamma} \theta |  \\
&\leq \frac{1}{2} | (A_1^{\frac{1}{2} + \gamma} \overline{u}, A_2^{\frac{1}{2}
 + \gamma} \overline{b}, A_3^{\frac{1}{2} + \gamma} \theta ) |^2
 + c(1 + | (A_1^{\gamma} \overline{u}, A_2^{\gamma} \overline{b},
 A_3^{\gamma} \theta ) |^2)(1 + \| (\overline{u}, \overline{b}, \theta) \|_{V}^2)
\end{align*}
where we used that $\gamma - \frac{1}{2} \leq \frac{1}{4} + \frac{\gamma}{2}$,
 Gagliardo-Nirenberg and and Young's inequalities. Hence, by Gronwall's inequality
we obtain
\begin{align*}
& | (A_1^{\gamma}\overline{u}, A_2^{\gamma}\overline{b},
A_3^{\gamma} \theta ) (t) |^2 + \int_{t_0}^{t} | (A_1^{\frac{1}{2}
+ \gamma} \overline{u}, A_2^{\frac{1}{2} + \gamma} \overline{b}, A_3^{\frac{1}{2}
+ \gamma} \theta ) |^2 ds \\
&\lesssim | (A_1^{\gamma}\overline{u}, A_2^{\gamma}\overline{b},
A_3^{\gamma} \theta ) (t_0) |^2 e^{\int_{t_0}^{t} 1+ \| (\overline{u}, \overline{b},
\theta) \|_{\mathbb{V}}^2 ds}.
\end{align*}

The computation above may be repeated for $\gamma$ replaced by
$\min\{\gamma, \frac{1}{4} + \gamma_0\}$; therefore,
\begin{align*}
& | A^{\min\{\gamma, \frac{1}{4} + \gamma_0\}}(\overline{u}, \overline{b}, \theta)
 (t) |^2 + \int_{t_0}^{t} | (A^{\frac{1}{2} + \min\{\gamma, \frac{1}{4}
+ \gamma_0\}} (\overline{u},  \overline{b}, \theta) |^2 ds \\
&\lesssim | A^{\min\{\gamma, \frac{1}{4} + \gamma_0\}}(\overline{u}, \overline{b},
 \theta) (t_0) |^2 e^{\int_{t_0}^{t} 1+ \| (\overline{u}, \overline{b}, \theta)
\|_{\mathbb{V}}^2 ds}.
\end{align*}

Finally, as $(\overline{u}, \overline{b}, \theta)(t_0)
= (u - z_{\alpha}^1, b - z_{\alpha}^2, \theta)(t_0)$ due to \eqref{e16}
where $(u, b, \theta)(t_0) \in D(A^{\gamma})$ by hypothesis and
$z_{\alpha}^{j} \in C([t_0, T]; D(A_{j}^{\gamma_0 + \frac{1}{4}}))$
 by \eqref{e14}, we see that

\[
| A^{\min\{\gamma, \frac{1}{4} + \gamma_0\}}(\overline{u}, \overline{b},
 \theta) (t_0) |^2< \infty.
\]
Thus, by \eqref{e20}, the proof is complete.
\end{proof}

\begin{proposition}[Dissipativity] \label{prop3.3}
 Let $y(t, t_0, (0, 0, \theta(t_0)), \omega)$  be the solution
$y = (u, b, \theta)(\cdot, \omega)$ at time $t$ that has values
 $(u, b, \theta)(t_0) = (0, 0, \theta(t_0)) \in \mathbb{H}$. Suppose that
$-1 \leq \theta(t_0) \leq 1$ a.e. $x \in D$. Then under the hypothesis of
 Theorem \ref{thm2.1}, there exists a random variable
$r(\omega) \hspace{1mm}  \mathbb{P}$-a.s. finite such that for
$\gamma \in (0, \frac{1}{2}) \cap (0, 2\gamma_0]$
\begin{equation} \label{e27}
\sup_{-\infty < t_0 \leq 0} | A^{\gamma} y(0, t_0, y(t_0), \omega) | \leq r(\omega).
\end{equation}
\end{proposition}

\begin{proof}
We take $L^2$-inner products on \eqref{e17a}, \eqref{e17b} with
 $(\overline{u},\overline{b})$ to obtain similarly to \eqref{e18},
applying H\"older's, Gagliardo-Nirenberg and Young's inequalities,
\begin{equation} \label{e28}
\begin{aligned}
& \frac{1}{2} \partial_t | (\overline{u}, \overline{b}) |^2
 + \| \overline{u} \|_{V_1}^2 + \| \overline{b} \|_{V_2}^2  \\
&\leq \frac{1}{2}(\| \overline{u} \|_{V_1}^2 + \| \overline{b} \|_{V_2}^2)
 + \frac{\epsilon}{2} | (\overline{u}, \overline{b}) |^2
 + c(| \overline{u} |^2 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 \\
&\quad + | \overline{b} |^2 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4
 +\| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4
 + | (\theta, z_{\alpha}^1, z_{\alpha}^2) |^2)
\end{aligned}
\end{equation}
for $\epsilon > 0$ arbitrary small. We may choose $\epsilon > 0$ sufficiently
small so that by Poincare's inequality,
\[
\epsilon | (\overline{u}, \overline{b}) |^2
+ \frac{1}{2}(\| \overline{u} \|_{V_1}^2 + \| \overline{b} \|_{V_2}^2)
 \leq \| \overline{u} \|_{V_1}^2 + \| \overline{b} \|_{V_2}^2.
\]	
Thus, subtracting $\frac{1}{2}(\| \overline{u} \|_{V_1}^2
+ \| \overline{b} \|_{V_2}^2)$ from both sides of \eqref{e28}, we may obtain for
some constant $c_0 > 0$,
\begin{equation} \label{e29}
\begin{aligned}
& | (\overline{u}, \overline{b})(t) |^2
 - | (\overline{u}, \overline{b})(t_0) |^2 e^{\int_{t_0}^{t} - \epsilon
 + c_0 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 ds}\\
&\leq c\int_{t_0}^{t} (\| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4
 + | (\theta, z_{\alpha}^1, z_{\alpha}^2)|^2) e^{\int_{s}^{t} - \epsilon
 + c_0 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 dr} ds.
\end{aligned}	
\end{equation}	
Now $z_{\alpha}^{j}(t)$ is an ergodic process with values in $(L^4(D))^2$; thus,
\[
\lim_{t_0 \to -\infty} \frac{1}{-t_0} \int_{t_0}^{0} \| z_{\alpha}^{j} \|_{L^4}^4 ds = E[\| z_{\alpha}^{j}(0) \|_{L^4}^4]
\]
while by Sobolev embedding of $D(A_{j}^{1/4}) \hookrightarrow L^4(D)$ and
Lemma \ref{lem5.2} (2), for all  $s$, we know
\[
\lim_{\alpha \to \infty} E[\| z_{\alpha}^{j}(s) \|_{L^4}^4]
= 0
\]
so that for $c_0 > 0$ fixed, we have
\[
-\epsilon + c_0 E[\| (z_{\alpha}^1, z_{\alpha}^2)(0) \|_{L^4}^4]
\leq - \frac{\epsilon}{2}
\]	
for $\alpha > 0$ sufficiently large. Thus,
\[
 \lim_{t_0 \to -\infty} \frac{1}{-t_0} \int_{t_0}^{0} -\epsilon
 + c_0 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 ds
\leq - \frac{\epsilon}{2}.
\]	
This implies that for given $\omega \in \Omega$ for some $\tau(\omega) < 0$
sufficiently small, we have
\begin{equation} \label{e30}
\sup_{t_0 < \tau(\omega)} \int_{t_0}^{0} - \epsilon
+ c_0 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 ds
\leq (-\frac{\epsilon}{4}) (-t_0) = (\frac{\epsilon}{4}) (t_0).
\end{equation}

Moreover, by \eqref{e14} and the Sobolev embedding of
$D(A_{j}^{\gamma_0 + \frac{1}{4}}) \hookrightarrow (L^4(D))^2$,
there exists a constant $c(\omega)$ such that
\begin{equation} \label{e31}
| \sup_{\tau (\omega) \leq t_0 \leq 0} \int_{t_0}^{0} -\epsilon
+ c_0 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 ds |  \leq c(\omega).
\end{equation}
Thus, by \eqref{e30} and \eqref{e31} we have the bound
\begin{align*}
\int_{t_0}^{0} -\epsilon + c_0 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 ds
&\leq \begin{cases}
(\frac{\epsilon}{4}) (t_0) & \text{if } t_0 < \tau (\omega), \\
c(\omega) & \text{if } \tau (\omega) \leq t_0,
\end{cases}\\
&\triangleq \overline{\epsilon}(t_0, \omega).
\end{align*}	
We observe that for a fixed $\tau(\omega)$,
$\lim_{t_0 \to -\infty} \overline{\epsilon} (t_0, \omega)  = -\infty$.
Besides, for $\tau \leq -1, \forall \hspace{1mm}  t \in [-1, 0]$, we also have
\begin{align*}
\int_{t_0}^{t} -\epsilon + c_0 \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 ds
\leq \overline{\epsilon} (t_0, \omega) + c(\omega).
\end{align*}
Under our hypothesis, $z_{\alpha}^{j}$ is a stationary process with values in
$(L^4(D))^2$; thus, for all $t < 0$,
\begin{equation} \label{e32}
\| z_{\alpha}^{j}(t, \omega) \|_{L^4} \leq c(\omega) (1+ | t |), \quad
| z_{\alpha}^{j}(t, \omega) | \leq c(\omega) (1+ | t |)
\end{equation}
for some $c(\omega) > 0$ taken larger than before if necessary.
Thus, for all $t \in [-1, 0], t_0 \leq t$, from \eqref{e29},
\begin{equation} \label{e33}
| (\overline{u}, \overline{b})(t) |^2
\leq  | (\overline{u}, \overline{b})(t_0) |^2 e^{\overline{\epsilon}(t_0, \omega)
+ c(\omega)} + \int_{t_0}^{t} e^{\overline{\epsilon} (t_0, \omega)
+ c(\omega)} c(\omega)(1+ | s |) ds \leq r_1(\omega)
\end{equation}	
for some $r_1(\omega)$ because of Proposition \ref{prop5.4}.

We go back to \eqref{e28}; subtracting
 $\frac{1}{2}(\| \overline{u} \|_{V_1}^2 + \| \overline{b} \|_{V_2}^2)$
from both sides of \eqref{e28} leads to
\begin{align*}
\partial_t | (\overline{u}, \overline{b}) |^2 + \| \overline{u} \|_{V_1}^2
+ \| \overline{b} \|_{V_2}^2
\lesssim | (\overline{u}, \overline{b}) |^2 (1+ \| (z_{\alpha}^1, z_{\alpha}^2)
\|_{L^4}^4) + \| (z_{\alpha}^1, z_{\alpha}^2) \|_{L^4}^4 +| \theta |^2.
\end{align*}
Integrating over $[-1, 0]$ leads to
\begin{equation} \label{e34}
\int_{-1}^{0} \| \overline{u} \|_{V_1}^2 + \| \overline{b} \|_{V_2}^2 ds
\leq r_2(\omega)
\end{equation}
for some $r_2(\omega)$ due to \eqref{e32}, \eqref{e33} and Proposition \ref{prop5.4}.
Next, taking $L^2$-inner products of \eqref{e17c} with $\theta$, applying Young's
inequality and integrating over $[-1, 0]$ give
\begin{equation} \label{e35}
\int_{-1}^{0} \| \theta \|_{V_3}^2 ds
\leq \frac{| \theta(-1) |^2}{2} + \int_{-1}^{0} | \theta |^2
+ \frac{| \overline{u} |^2}{2} + \frac{| z_{\alpha}^1 |^2}{2} ds \leq r_3(\omega)
\end{equation}
for some $r_3(\omega)$ due to Proposition \ref{prop5.4}, \eqref{e32} and \eqref{e33}.

Next, from \eqref{e25}, we compute
\begin{equation}
\begin{aligned} \label{e36}
& \frac{1}{2} \partial_t | (A_1^{\gamma} \overline{u}, A_2^{\gamma} \overline{b}) |^2
 + | (A_1^{\frac{1}{2} + \gamma} \overline{u}, A_2^{\frac{1}{2}
+ \gamma} \overline{b}) |^2 \\
&\leq | (A_1^{\frac{2}{2} + \gamma} \overline{u}, A_2^{\frac{1}{2} + \gamma}
 \overline{b})|^2  + c\Big( 1+ | (A_1^{\frac{1}{4} 
 + \frac{\gamma}{2}} \overline{u},
 A_2^{\frac{1}{4} + \frac{\gamma}{2}} \overline{b}) |^4 \\
&\quad + | (A_1^{\frac{1}{4}
+ \frac{\gamma}{2}} z_{\alpha}^1, A_2^{\frac{1}{4}
+\frac{\gamma}{2}} z_{\alpha}^2) |^4 
+ \| \theta \|_{V_3}^2\Big).
\end{aligned}
\end{equation}
Subtracting $ | (A_1^{\frac{2}{2} + \gamma} \overline{u}, A_2^{\frac{1}{2}
+ \gamma} \overline{b})|^2$ from both sides of (36) , using that
 $\frac{1}{4} + \frac{\gamma}{2} \leq \gamma_0 + \frac{1}{4}$ and
Gagliardo-Nirenberg and Gronwall's inequalities over $[-1, t], t \in [-1, 0]$
lead to
\begin{equation} \label{e37}
\begin{aligned}
&\sup_{t \in [-1, 0]} | (A_1^{\gamma} \overline{u}, A_2^{\gamma}
 \overline{b})(t) |^2 \\
&\leq | (A_1^{\gamma} \overline{u}, A_2^{\gamma} \overline{b})(-1) |^2
 e^{\int_{-1}^{0} \| (\overline{u}, \overline{b}) \|_{V_1 \times V_2}^2 ds} \\
&\quad + e^{\int_{-1}^{0} \| (\overline{u}, \overline{b}) \|_{V_1 \times V_2}^2 ds}
 \int_{-1}^{0} 1 + | (A_1^{\gamma_0 + \frac{1}{4}} z_{\alpha}^1,
  A_2^{\gamma_0 + \frac{1}{4}} z_{\alpha}^2) |^4 + \| \theta \|_{V_3}^2 ds
 \lesssim r_4(\omega)
\end{aligned}
\end{equation}
for some $r_4(\omega)$ due to \eqref{e34}, \eqref{e35} and \eqref{e14}.
Finally, we go back to computations of \eqref{e25} again to similarly obtain
\begin{align*}
& \frac{1}{2} \partial_t | A_3^{\gamma} \theta|^2 + | A_3^{\frac{1}{2}
+ \gamma} \theta  |^2 \\
&\leq | A_3^{\frac{1}{2} + \gamma} \theta |^2 + c( 1+ | (A_1^{\frac{1}{4}
+ \frac{\gamma}{2}} \overline{u}, A_3^{\frac{1}{4} + \frac{\gamma}{2}} \theta ) |^4
 + | A_1^{\frac{1}{4} + \frac{\gamma}{2}} z_{\alpha}^1 |^4) \\
&\leq | A_3^{\frac{1}{2} + \gamma} \theta |^2 + c( 1+ \| \overline{u}
 \|_{V_1}^2 | A_1^{\gamma} \overline{u} |^2
 + \| \theta \|_{V_3}^2 | A_3^{\gamma} \theta |^2 + | A_1^{\frac{1}{4}
 + \frac{\gamma}{2}} z_{\alpha}^1 |^4).
\end{align*}
Subtracting $| A_3^{\frac{1}{2} + \gamma} \theta |^2$ from both sides and
applying Gronwall's inequality give
\begin{align*}
| A_3^{\gamma} \theta(0) |^2
&\lesssim | A_3^{\gamma} \theta (-1) |^2 e^{\int_{-1}^{0} \| \theta \|_{V_3}^2 ds}\\
&+ e^{\int_{-1}^{0} \| \theta \|_{V_3}^2 ds}\int_{-1}^{0}
 (1+ \| \overline{u} \|_{V_1}^2 | A_1^{\gamma} \overline{u} |^2
+  | A_1^{\frac{1}{4} + \frac{\gamma}{2}} z_{\alpha}^1 |^4) ds.
\end{align*}
Thus, by \eqref{e34}, \eqref{e35}, \eqref{e37} and \eqref{e14}, for all
$t_0 \leq 0$, we obtain
\begin{equation} \label{e38}
| A_3^{\gamma} \theta (0, t_0, y(t_0), \omega) |^2 \leq r_5(\omega)
\end{equation}
for some $r_5(\omega)$. With \eqref{e37} and \eqref{e38} we now conclude
the proof by defining $r(\omega) \triangleq r_4(\omega) + r_5(\omega)$. 	
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
We now prove the path-wise uniqueness of the generalized solution
 $y$ to \eqref{e4}. The issue here is that we do not know
if $(u, b)(\cdot, \omega)$ belongs to
$L^2(t_0, T; D(A_1^{1/2}) \times D(A_2^{1/2}))$,
in contrast to the deterministic case, because
$z_{\alpha}^{j} \in C([t_0, T]; D(A_{j}^{\gamma_0 + \frac{1}{4}})), \gamma_0 > 0$
in \eqref{e14}, although from Proposition \ref{prop3.1}, we have
$(\overline{u}, \overline{b})(\cdot,\omega) \in L^2(t_0, T; V_1 \times V_2)$
for $\mathbb{P}$-a.e. $\omega \in \Omega$. The key observation from \cite{13}
is that along with the same type of classical results in \cite{37}, we only need
$(u, b)(\cdot, \omega) \in L^4(t_0, T; D(A_1^{1/4}) \times D(A_2^{1/4}))$
to prove the path-wise uniqueness and this follows from the interpolation
inequality of, e.g. in the case of $u$,
\begin{align*}
\| \overline{u} \|_{L^4(t_0, T; D(A_1^{1/4}))}^4
\lesssim \| \overline{u} \|_{L^{\infty}([t_0, T]; H_1)}^2
\| \overline{u} \|_{L^2(t_0, T; V_1)}^2  \lesssim 1
\end{align*}
and similarly for $b$. Therefore, by \eqref{e14} and \eqref{e16}, we obtain
$(u, b)(\cdot, \omega)$ belongs to $L^4(t_0, T; D(A_1^{1/4}) \times D(A_2^{1/4}))$
$\mathbb{P}$-a.e. $\omega \in \Omega$.

As in the proof of Proposition \ref{prop3.1}, we may consider
$(u^1, b^1, \theta^1), (u^2, b^2, \theta^2)$ that both solve
\eqref{e4} so that defining
$\delta u \triangleq u^1 - u^2, \delta b \triangleq b^1 -b^2$,
$\delta \theta \triangleq \theta^1 - \theta^2$, we obtain
\begin{gather*}
\partial_t \delta u + A_1 \delta u + (\delta u \cdot\nabla) u^1
- (u^2 \cdot \nabla) \delta u - (\delta b \cdot\nabla) b^1 - (b^2 \cdot\nabla)
\delta b = \delta \theta e_2,\\
\partial_t \delta b + A_2 \delta b + (\delta u \cdot\nabla) b^1
- (u^2 \cdot\nabla) \delta b - (\delta b \cdot\nabla) u^1
- (b^2 \cdot\nabla) \delta u = 0 ,\\
\partial_t \delta \theta + A_3 \delta \theta + (\delta u \cdot\nabla) \theta^1
 + (u^2 \cdot\nabla) \delta \theta = \delta u_2.
\end{gather*}
Therefore, taking $L^2$-inner products with $(\delta u, \delta b, \delta \theta)$,
a computation very similar to the proof of Proposition \ref{prop3.1} shows that
\begin{align*}
& \frac{1}{2} \partial_t | (\delta u, \delta b, \delta \theta) |^2
 + \| (\delta u, \delta b, \delta \theta) \|_{\mathbb{V}}^2 \\
&\leq \frac{1}{2} \| (\delta u, \delta b, \delta \theta) \|_{\mathbb{V}}^2
 + c | (\delta u, \delta b, \delta \theta) |^2(1+ \| (u^1, b^1) \|_{D(A_1^{1/4})
\times D(A_2^{1/4})}^4 + | \theta^1 |^2 \| \theta^1 \|_{V_3}^2).
\end{align*}
Thus, Gronwall's inequality implies  $(\delta u, \delta b, \delta \theta) = 0$.

Next, we prove the existence of an invariant measure.
With $r(\omega)$ from \eqref{e27} we let
$\Omega_{N} \triangleq \{\omega \in \Omega: r(\omega) \leq N\}$
so that $\Omega_{N} \nearrow \Omega$
$(N\to\infty)$,
$\cup_{N} \Omega_{N} = \Omega, \lim_{N\to\infty} \mathbb{P} (\Omega_{N}) = 1$
and hence for all  $\epsilon > 0$ fixed, there exists
$N_{\epsilon} > 0$ sufficiently large so that
$1 - \epsilon < \mathbb{P}(\Omega_{N_{\epsilon}})$. On the other hand,
by Proposition \ref{prop3.3}
\[
\mathbb{P} (\Omega_{N_{\epsilon}})
< \mathbb{P} (\{| y(0, t_0, y(t_0), \omega) |_{D(A^{\gamma})}
\leq N_{\epsilon} \})
\]
where $y(0, t_0, y(t_0), \omega)$ is a solution to \eqref{e4} at time $0$
with value $y(t_0)$ at time $t_0$. Therefore,
\[
1- \epsilon < \mathbb{P} (\{| y(0, t_0, y(t_0), \omega) |_{D(A^{\gamma})}
 \leq N_{\epsilon} \}).
\]
By the compact embedding of $D(A^{\gamma}) \hookrightarrow \mathbb{H}$
for all $\gamma > 0$ (cf. \cite[Theorem 16.1]{29}), this implies that for all
$ \epsilon > 0$, there exists a compact set $K_{\epsilon} \subset \mathbb{H}$
such that $1 - \epsilon < \mathbb{P} (y(0, t_0, y(t_0), \omega) \in K_{\epsilon})$.

Next, we let $\nu_{\tau} \triangleq \mathcal{L}(y(0, -\tau, y(-\tau), \omega))$
so that $\nu_{\tau} = P_{\tau}^{\ast} \delta_{y(0)} = \mathcal{L} (y_0(\tau, \omega))
 = \mathcal{L}(y_{-\tau}(0,\omega)), \tau \geq 0$ and define
\begin{equation} \label{e39}
\mu_{T} \triangleq \frac{1}{T} \int_0^{T} \nu_{\tau} d\tau
\end{equation}
so that we know $\{\mu_{T}\}_{T \geq 0}$ is tight.
By Prokhorov's theorem \cite{36}, it is weakly convergent.
Thus, the existence of an invariant measure now follows from
the Krylov-Bogoliubov theorem \cite{27}, specifically
\cite[Corollary 11.8]{7}.
\end{proof}

\section{Proof of Theorem \ref{thm2.2}: Uniqueness}

We return to \eqref{e4}, denote by $z^{j}, j= 1, 2$, the solution to \eqref{e12}
at $\alpha = 0$, but for $t \geq 0$ with the additional condition that $z^{j}(0) = 0$,
 and let $\overline{u} \triangleq u - z^1, \overline{b} \triangleq b - z^2$
and hence similarly to \eqref{e17a}-\eqref{e17c} we obtain
\begin{equation} \label{e40}
\begin{gathered}
\begin{aligned}
\partial_t \overline{u} + A_1 \overline{u}
& = - B_1(\overline{u}, \overline{u}) - B_1(\overline{u}, z^1)
- B_1(z^1, \overline{u}) - B_1(z^1, z^1) \\
&\quad + B_2(\overline{b}, \overline{b}) + B_2(\overline{b}, z^2) + B_2(z^2,
 \overline{b}) + B_2(z^2, z^2) + \theta e_2,
\end{aligned}\\
\begin{aligned}
\partial_t \overline{b} + A_2 \overline{b}
&= - B_3(\overline{u}, \overline{b}) - B_3(\overline{u},z^2) - B_3(z^1,
 \overline{b}) - B_3(z^1, z^2) \\
&\quad + B_4(\overline{b}, \overline{u}) + B_4(\overline{b}, z^1)
 + B_4(z^2, \overline{u}) + B_4(z^2, z^1),
\end{aligned}\\
\partial_t\theta + A_3 \theta = -B_5(\overline{u}, \theta)
 - B_5(z^1, \theta) + \overline{u}_2 + z^1_2,
\end{gathered}
\end{equation}
where $z_2^1$ is the second component of $z^1$

\begin{proposition} \label{prop4.1}
For $l \in \mathbb{Z}$, $l \geq 2$, suppose that for both $j = 1, 2$,
$\sigma_i^{j} <\frac{C}{i^{1/2}}$. Then for all
$y_0 \triangleq y(0) \in \mathbb{V}^{l}$, there exists a unique solution
$y = (u,b,\theta)$ to
\begin{equation} \label{e41}
y(t) + \int_0^{t} Ay + B(y,y) ds = y_0 + \int_0^{t} Ry ds + Gw(t)
\end{equation}
such that $y \in C([0,T]; \mathbb{V}^{l})$ $\mathbb{P}$-a.s., and
it is a Markov process, satisfying the Feller property in $\mathbb{V}^{l}$.
\end{proposition}

\begin{proof}
From our hypothesis that $\sigma_i^{j} <\frac{C}{i^{1/2}}$, we obtain for
 both $j = 1, 2$,
\begin{equation} \label{e42}
E[| A_{j}^{1/2} z^{j}(t) |^2] < \infty
\end{equation}
(see \cite{12}). Moreover, from the higher regularity of the initial data,
taking $L^2$-inner products of \eqref{e40} with
$(A_1^{l} \overline{u}, A_2^{l} \overline{b}, A_3^{l} \theta)$ respectively
and using the Banach algebra property of $H^{s}, s > 1$ in two-dimensional
space, we  compute
\begin{align*}
&\frac{1}{2} \partial_t | (A_1^{1/2} \overline{u}, A_2^{1/2}
 \overline{b}, A_3^{1/2} \theta ) |^2 + | (A_1^{\frac{l+1}{2}}
\overline{u}, A_2^{\frac{l+1}{2}} \overline{b}, A_3^{\frac{l+1}{2}} \theta) |^2\\
&= -\int A_1^{\frac{l-1}{2}} [B_1(\overline{u}, \overline{u})
 +B_1 (\overline{u}, z^1) + B_1(z^1, \overline{u}) + B_1(z^1, z^2)\\
& \quad - B_2 (\overline{b}, \overline{b}) - B_2(\overline{b}, z^2)
 - B_2(z^2, \overline{b}) - B_2(z^2, z^2)] \cdot A_1^{\frac{l+1}{2}} \overline{u}
 + \int A_1^{1/2} \theta e_2 A_1^{1/2} \overline{u}\\
&\quad- \int A_2^{\frac{l-1}{2}} [B_3(\overline{u}, \overline{b})
 + B_3(\overline{u}, z^2) + B_3(z^1, \overline{b}) + B_3(z^1, z^2)\\
& \quad - B_4(\overline{b}, \overline{u}) - B_4(\overline{b}, z^1)
 - B_4(z^2, \overline{u}) - B_4(z^2, z^1)] \cdot A_2^{\frac{l+1}{2}} \overline{b}\\
&\quad - \int A_3^{\frac{l-1}{2}} [B_5(\overline{u}, \theta)
 + B_5 (z^1, \theta)] A_3^{\frac{l+1}{2}} \theta + \int [A_3^{1/2} \overline{u}_2
 + A_3^{1/2} z^1_2] A_3^{1/2} \theta \\
&\leq \frac{1}{2} | (A_1^{\frac{l+1}{2}} \overline{u}, A_2^{\frac{l+1}{2}}
 \overline{b}, A_3^{\frac{l+1}{2}} \theta ) |^2\\
&\quad + c (\| \overline{u} \|_{V_1^{l}}^4
 + \| (\overline{u}, z^1) \| _{ V_1^{l}}^4
 + \| (z^1, z^2) \| _{ V_1^{l}}^4+ \| \overline{b} \| _{V_1^{l}}^4
 + \| (\overline{b}, z^2) \| _{V_1^{l}}^4 +  \| z^2 \| _{V_1^{l}}^4 \\
&\quad + \| (\theta, \overline{u} ) \|_{V_1^{l}}^2
 + \| (\overline{u}, \overline{b}) \| _{V_2^{l}}^4
 + \|  (\overline{u}, z^2) \| _{V_2^{l}}^4
 + \| (z^1, \overline{b}) \| _{V_2^{l}}^4 + \| (z^1, z^2) \| _{V_2^{l}}^4\\
&\quad + \| (\overline{u},\theta) \| _{V_3^{l}}^4
 + \| (z^1, \theta ) \| _{ V_3^{l}}^4 + \| (\overline{u}, z^1) \|_{V_3^{l}}^2).
\end{align*}	
Subtracting $\frac{1}{2} | (A_1^{\frac{l+1}{2}} \overline{u},
A_2^{\frac{l+1}{2}} \overline{b}, A_3^{\frac{l+1}{2}} \theta ) |^2$
from both sides, and using the fact that we know from the basic estimate
$\mathbb{P}$-almost surely,
\[
\sup_{t \in [0,T]}| (\overline{u}, \overline{b}, \theta) |^2
+ \int_0^{T} \| (\overline{u}, \overline{b}, \theta ) \|_{\mathbb{V}}^2 dt
\lesssim 1,
\]	
inductively we obtain
\begin{equation} \label{e43}
\sup_{t \in [0,T]} \| (\overline{u}, \overline{b}, \theta) \|_{\mathbb{V}^{l}}^2
 + \int_0^{T} \| (\overline{u}, \overline{b}, \theta) \|_{\mathbb{V}^{l+1}}^2 dt
\lesssim 1.
\end{equation}	
With such \emph{a priori} estimates, a standard Galerkin approximation scheme
proves the existence of the unique solution
\[
\overline{u}, \overline{b}, \theta \in C([0,T]; \mathbb{V}^{l})
\cap L^2(0, T; \mathbb{V}^{l+1})
\]
and hence
\begin{align*}
u \in C([0,T]; V_1^{l}), \quad b  \in C([0,T]; V_2^{l})
\end{align*}	
as $z_{j} \in C([0,T]; V_{j}^{l}), j = 1, 2$ by \eqref{e42}. 		
\end{proof}

\subsection{Irreducibility}

Let $B(v, \rho)$ be a ball of radius $\rho > 0$ centered at
$v \in \mathbb{V}^{l}$. By denseness of $\mathbb{V}^{l+2}$ in $\mathbb{V}^{l}$,
 we find $y_{\tau} \in \mathbb{V}^{l+2}$ such that
$\| v- y_{\tau} \|_{\mathbb{V}^{l}}<\rho/2$. Thus,
\begin{equation} \label{e44}
P (\tau, x, B(v,\rho)) \geq P(\tau, x, B(y_{\tau}, \frac{\rho}{2}))
\end{equation}
where $P(t, x, \Gamma) = \mathbb{P} (\{y(t, x) \in \Gamma \})$.

\begin{proposition} \label{prop4.2}
For $l \in \mathbb{Z}, l \geq 2$, suppose that for both $j = 1, 2$,
 $\sigma_i^{j} <\frac{C}{i^{1/2}}$. Then, given any
$\tau \in (0, T]$, $x = (x_1, x_2, x_3) = (u_0, b_0, \theta_0)
\in \mathbb{V}^{l}, y_{\tau} \in \mathbb{V}^{l+2}$, there exists
\begin{gather*}
\hat{y} \in C([0,\tau]; \mathbb{V}^{l}) \cap C((0, \tau]; \mathbb{V}^{l+2}),\\
\hat{w}_{G} \in \operatorname{Lip}([0,\tau]; \mathbb{V}^{l}),
\end{gather*}
such that $\hat{y} = (\hat{u}, \hat{b}, \hat{\theta})$ is a solution
to \eqref{e41} with
$\hat{w}_{G} = (G^1\hat{w}^1, G^2\hat{w}^2, 0)$,
$\hat{y}(0) = x$, $\hat{y}(\tau) = y_{\tau}$.
\end{proposition}

\begin{proof}
Given $x \in \mathbb{V}^{l}$, we consider \eqref{e41} with $G^1w^1 = G^2w^2 = 0$.
Then we know there exists a unique
$\hat{y} \in C([0,T]; \mathbb{V}^{l}) \cap L^2(0, T; \mathbb{V}^{l+1})$.
Since $\hat{y} \in L^2(0, T; \mathbb{V}^{l+1})$, in particular, there exists
$\tau_1 \in (0, \tau)$ arbitrary close to $0$ such that
$\hat{y}(\tau_1) \in \mathbb{V}^{l+1}$. Restarting from $\tau_1$, we find
\[
\hat{y} \in C([\tau_1, T]; \mathbb{V}^{l+1}) \cap L^2(\tau_1, T; \mathbb{V}^{l+2}).
\]	
Now we take $\tau_2 \in (\tau_1, \tau)$ arbitrary close to $\tau_1$ so that
$\hat{y}(\tau_2) \in \mathbb{V}^{l+2}$. We let
\begin{equation} \label{e45}
\hat{y}(t) = \hat{y}(\tau_2) + \frac{t - \tau_2}{\tau - \tau_2} [y_{\tau}
- \hat{y}(\tau_2)]
\end{equation}
for $t \in [\tau_2, \tau]$; we observe that $\hat{y}(t) |_{t = \tau} = y_{\tau}$.
We set
\[
\hat{\xi} \triangleq  \partial_t\hat{y} + A \hat{y} + B(\hat{y}, \hat{y}) - R \hat{y};
\]	
it follows that
\[
\hat{\xi} \in \mathbb{V}^{l+2} + C([\tau_2, \tau]; \mathbb{V}^{l})
+ C([\tau_2, \tau]; \mathbb{V}^{l+1}) + C([\tau_2, \tau]; \mathbb{V}^{l+2}).
\]	
Now integrating $\hat{\xi}$ over $[\tau_2, t]$, we see that
\begin{equation} \label{e46}
\hat{w}_{G}(t) \triangleq \int_{\tau_2}^{t} \hat{\xi}(s) ds
\in \operatorname{Lip} ([\tau_2, \tau]; \mathbb{V}^{l}).
\end{equation}
\end{proof}

Next, with $\omega \in \Omega$ fixed we prove the following proposition:

\begin{proposition} \label{prop4.3}
For $l \in \mathbb{Z}, l \geq 2$, suppose
$w_{G}, \hat{w}_G{} \in C^{\gamma} ([0,T];
\mathbb{V}^{l+1 - \frac{\epsilon}{2}}), \gamma
> \frac{1}{2} - \frac{\epsilon}{4}, x \in \mathbb{V}^{l}$, for both
$j = 1, 2$, $\sigma_i^{j} < \frac{c}{i^{1/2}}$ and $y, \hat{y}$ are the
corresponding solutions to \eqref{e41} with $w_{G}, \hat{w}_{G}$ respectively. Then
\[
\| y - \hat{y} \|_{C([0,T]; \mathbb{V}^{l})}
\lesssim \| w_{G} - \hat{w}_{G} \|_{C^{\gamma} ([0,T]; V_1^{l+1 - \frac{\epsilon}{2}}
 \times V_2^{l+ 1 - \frac{\epsilon}{2}})}.
\]
\end{proposition}

\begin{proof}
By \cite[Theorem 5.2]{15}, we obtain for $j = 1, 2$,
\begin{align*}
z^{j}(t) = e^{-tA_{j}} G^{j} w^{j}(t)
+ \int_0^{t} A_{j} e^{-(t-s) A_{j}} (G^{j} w^{j}(t) - G^{j} w^{j}(s)) ds.
\end{align*}
Thus, with $z = (z^1, z^2), \hat{z} = (\hat{z}^1, \hat{z}^2)$ where for $j = 1, 2$,
\begin{align*}
d\hat{z}_{j} + A_{j} \hat{z}^{j} dt = G^{j} d\hat{w}^{j}, \quad \hat{z}^{j}(0) = 0,
\end{align*}
so that similarly applying \cite[Theorem 5.2]{15},  for $j = 1, 2$, we obtain
\[
\hat{z}_{j}(t) = e^{-tA_{j}} G^{j}\hat{w}^{j}(t)
+ \int_0^{t} A_{j} e^{-(t-s) A_{j}} (G^{j}\hat{w}^{j}(t) - G^{j} \hat{w}^{j}(s)) ds
\]
and hence it follows that
\begin{equation} \label{e47}
\| z - \hat{z} \|_{C([0,T]; V_1^{l} \times V_2^{l})}
\lesssim\| w_{G} - \hat{w}_{G} \|_{C^{\gamma}([0,T]; V_1^{l+ 1 - \frac{\epsilon}{2}}
\times V_2^{l+ 1 - \frac{\epsilon}{2}})}.
\end{equation}

Now by hypothesis we have $y$ and $\hat{y}$ as the corresponding solutions
 with $w_{G}, \hat{w}_{G}$ respectively so that defining
\[
\overline{u} \triangleq u - z^1, \quad \overline{b} \triangleq b - z^2, \quad \overline{\hat{u}} \triangleq \hat{u} - \hat{z}^1, \quad \overline{\hat{b}} \triangleq \hat{b} - \hat{z}^2
\]
and furthermore
\[
\overline{U} \triangleq \overline{u} - \overline{\hat{u}}, \quad
\overline{B} \triangleq \overline{b} - \overline{\hat{b}}, \quad
\Theta \triangleq \theta - \hat{\theta}, \quad
\overline{z}^1 \triangleq z^1 - \hat{z}^1, \quad
\overline{z}^2 \triangleq z^2 - \hat{z}^2,
\]
we obtain
\begin{gather*}
\begin{aligned}
&\partial_t \overline{U} + A_1 \overline{U} + B_1 (\overline{u} + z^1,
  \overline{U} + \overline{z}^1) + B_1(\overline{U} + \overline{z}_1,
 \overline{\hat{u}} + \hat{z}^1)\\
& - B_2 (\overline{b} + z^2, \overline{B} + \overline{z}^2) - B_2(\overline{B}
+ \overline{z}^2, \overline{\hat{b}} + \hat{z}^2) = \Theta e_2,
\end{aligned}\\
\begin{aligned}
&\partial_t \overline{B} + A_2 \overline{B}
+ B_3(\overline{u} + z^1, \overline{B} + \overline{z}^2)
+ B_3(\overline{U} + \overline{z}^1,  \overline{\hat{b}} + \hat{z}^2)\\
&-B_4(\overline{b} + z^2, \overline{U} + \overline{z}^1)
 -B_4(\overline{B} + \overline{z}^2, \overline{\hat{u}} + \hat{z}^1),
\end{aligned}\\
\partial_t \Theta + A_3 \Theta + B_5(\overline{u} + z^1, \Theta)
+ B_5(\overline{U} + \overline{z}^1, \hat{\theta}) - (\overline{U}
 + \overline{z}^1)_2,
\end{gather*}
where $(\overline{U} + \overline{z}^1)_2$ is the second component of
$\overline{U} + \overline{z}^1$. 	
Taking $L^2$-inner products with
$(A_1^{l} \overline{U}, A_2^{l} \overline{B}, A_3^{l} \Theta)$ respectively gives
\begin{align*}
& \frac{1}{2} \partial_t | (A_1^{1/2} \overline{U}, A_2^{1/2}
 \overline{B},
 A_3^{1/2} \Theta) |^2  + | (A_1^{\frac{l+1}{2}} \overline{U},
  A_2^{\frac{l+1}{2}} \overline{B}, A_3^{\frac{l+1}{2}} \Theta ) |^2\\
&= -\int A_1^{\frac{l-1}{2}} \text{div} [(\overline{u} + z^1)
 \otimes (\overline{U} + \overline{z}_1) + (\overline{U}
 + \overline{z}_1)\otimes (\overline{\hat{u}} + \hat{z}^1)\\
&\quad - (\overline{b}+ z^2) \otimes (\overline{B} + \overline{z}^2)
 - (\overline{B} + \overline{z}^2) \otimes (\overline{\hat{b}}
 + \hat{z}^2)]\cdot  A_1^{\frac{l+1}{2}} \overline{U}
 + \int A_1^{1/2} \Theta e_2 \cdot A_1^{1/2} \overline{U}\\
&\quad - \int A_2^{\frac{l-1}{2}} \text{div} [(\overline{u} + z^1)
 \otimes (\overline{B} + \overline{z}^2) + (\overline{U} + \overline{z}^1)
 \otimes (\overline{\hat{b}} + \hat{z}^2) \\
&\quad -(\overline{b} + z^2) \otimes (\overline{U} + \overline{z}^1)
 - (\overline{B} + \overline{z}^2) \otimes (\overline{\hat{u}} + \hat{z}^1)]\cdot
   A_2^{\frac{l+1}{2}} \overline{B}\\
&\quad - \int A_3^{\frac{l-1}{2}} \text{div} [(\overline{u} + z^1) \Theta
 + (\overline{U} + \overline{z}^1) \hat{\theta}] A_3^{\frac{l+1}{2}} \Theta \\
&\quad- \int A_3^{1/2} \overline{U}\cdot A_3^{1/2} \Theta e_2
 - \int A_3^{1/2} \overline{z}^1 \cdot A_3^{1/2} \Theta e_2 \\
&\leq \frac{1}{2} | (A_1^{\frac{l+1}{2}} \overline{U}, A_2^{\frac{l+1}{2}}
 \overline{B}, A_3^{\frac{l+1}{2}} \Theta) |^2\\
&\quad+ c (1+ \| (\overline{u}, z^1, \overline{\hat{u}}, \hat{z}^1,
 \overline{b}, z^2, \overline{\hat{b}}, \hat{z}^2, \hat{\theta})
 \|_{\mathbb{V}^{l}}^2)\| (\overline{U}, \overline{B},
 \Theta) \|_{\mathbb{V}^{l}}^2\\
&\quad + c (1+ \| (\overline{u}, z^1, \overline{\hat{u}}, \hat{z}^1,
 \overline{b}, z^2, \overline{\hat{b}}, \hat{z}^2, \hat{\theta})
 \|_{\mathbb{V}^{l}}^2) \| (\overline{z}_1, \overline{z}_2) \|_{V_1^{l}
 \times V_2^{l}}^2
\end{align*}	
by Young's inequalities and Banach algebra property. Therefore, after
subtracting $\frac{1}{2} | (A_1^{\frac{l+1}{2}} \overline{U}, A_2^{\frac{l+1}{2}}
 \overline{B}, A_3^{\frac{l+1}{2}} \Theta) |^2$ from both sides, by
 Proposition \ref{prop4.1}, \eqref{e42} and that
$\overline{U}(0) = \overline{B}(0) = \Theta(0) = 0$, we obtain
\begin{equation} \label{e48}
\| (\overline{U}, \overline{B}, \Theta) \|_{C([0,T]; \mathbb{V}^{l})}
\lesssim \| (\overline{z}_1, \overline{z}_2) \|_{C([0,T]; V_1^{l}
\times V_2^{l})}.
\end{equation}	
Thus,
\begin{align*}
\| y - \hat{y} \|_{C([0,T]; \mathbb{V}^{l})}
&\lesssim \| (\overline{U}, \overline{B}, \Theta) \|_{C([0,T]; \mathbb{V}^{l})}
 + \| (\overline{z}_1, \overline{z}_2) \|_{C([0,T]; V_1^{l} \times V_2^{l})} \\
&\lesssim \| (\overline{z}_1, \overline{z}_2) \|_{C([0,T]; V_1^{l} \times V_2^{l})}\\
&\lesssim \| w_{G} - \hat{w}_{G} \|_{C^{\gamma} ([0,T];
 V_1^{l+ 1 - \frac{\epsilon}{2}} \times V_2^{l + 1 - \frac{\epsilon}{2}})}
\end{align*}
by \eqref{e47} and \eqref{e48}. 	
\end{proof}

We now complete the proof that the Markovian semigroup $P_t$ is irreducible
in $\mathbb{V}^{l}$:

\begin{proposition} \label{prop4.4}
For $l \in \mathbb{Z}, l \geq 2$, suppose that for both $j = 1, 2$,
 $\{\sigma_i^{j}\}_{i=1}^{\infty}$ satisfies \eqref{e9},
 $\sigma_i^{j} < \frac{C}{i^{1/2}}$,
$x, v \in \mathbb{V}^{l}$, $\tau \in (0, T]$, $\rho > 0$. Then
\[
P(\tau, x, B(v, \rho)) > 0.
\]
\end{proposition}

\begin{proof}
We denote by $y(\cdot, w_{G}; x)$ the solution to \eqref{e41} emanating
from $x$ as initial data with $w_{G} = Gw = (G^1w^1, G^2w^2, 0)^{T}$ and obtain
\[
\| y(\cdot, w_{G}; x) - \hat{y}(\cdot, \hat{w}_{G}; x) \|_{C([0,\tau];
\mathbb{V}^{l})}
\leq c_0 \| w_{G} - \hat{w}_{G} \|_{W^{s,p} (0, \tau;
V_1^{l - 1 + \frac{\epsilon}{2}} \times V_2^{l - 1 + \frac{\epsilon}{2}})}
\]
for some $c_0 \geq 0$ by Proposition \ref{prop4.3} if
$\gamma > \frac{1}{2} - \frac{\epsilon}{4}$ and the one-dimensional
 Sobolev embedding of $W^{s,p} (0,\tau) \hookrightarrow C^{\gamma} ([0,\tau])$
for all  $\gamma \leq s - \frac{1}{p}$. Thus,
\begin{equation} \label{e49}
\begin{aligned}
&\mathbb{P} (\{ \| w_{G} - \hat{w}_{G} \|_{W^{s,p} (0, \tau; V_1^{l - 1 + \frac{\epsilon}{2}} \times V_2^{l - 1 + \frac{\epsilon}{2}})}< \frac{\rho}{2c_0}\}) \\
&\leq  \mathbb{P} (\{ \| y(\cdot, w_{G}; x) - \hat{y}(\cdot, \hat{w}_{G}; x) \|_{C([0,\tau]; \mathbb{V}^{l})} <  \frac{\rho}{2} \} ).
\end{aligned}
\end{equation}
It is shown in \cite{12} that
\begin{equation} \label{e50}
\mathbb{P} (\{ \| w_{G} - \hat{w}_{G} \|_{W^{s,p} (0, \tau;
 V_1^{l - 1 + \frac{\epsilon}{2}} \times V_2^{l - 1
+ \frac{\epsilon}{2}})} < \frac{\rho}{2c_0} \} > 0
\end{equation}
because the law of $Gw$ is a full measure on
$W^{s,p} (0, T; V_1^{l - 1 + \frac{\epsilon}{2}}
\times V_2^{l - 1 + \frac{\epsilon}{2}})$. Thus, applying
\eqref{e50} to \eqref{e49} allows us to conclude that
\begin{align*}
0 < \mathbb{P}( \{\| y(\tau, w_{G}; x) - \hat{y}(\tau, \hat{w}_{G}; x)
\|_{\mathbb{V}^{l}} < \frac{\rho}{2} \}).
\end{align*}
By Proposition \ref{prop4.2}, we see that $0 < P(\tau, x, B(y_{\tau}, \frac{\rho}{2}))$.
By \eqref{e44}, the proof is complete.
\end{proof}

\subsection{Strong Feller property}

We let $H_{n}^{j} = \text{span} \{g_1^{j}, \dots, g_{n}^{j}\}$ where
we recall that $\{g_i^{j}\}_{i=1}^{\infty}$ are the complete orthonormal
systems of eigenfunctions of $A_{j}, j = 1, 2, 3$ and
$\pi_{n}^{j}: H_{j} \to H_{n}^{j}, \pi_{n}: \mathbb{H}
\to \prod_{j=1}^3 H_{n}^{j} \triangleq H_{n}$ be the projection operator
 defined by $\pi_{n}^{j} x = \sum_{i=1}^{n} \langle x, g_i^{j}\rangle g_i^{j}$.
For a fixed $R> 0$, we define a cutoff function
$\Psi_{R} \in C^{\infty} (\mathbb{R})$ that satisfies
\[
\Psi_{R}(t) = \begin{cases}
1 & t \in [-R, R],\\
0 & t \in \mathbb{R} \setminus (-R-1, R+1).
\end{cases}
\] 	
We consider $y_{n}^{(R)} \triangleq (u_{n}^{(R)}, b_{n}^{(R)}, \theta_{n}^{(R)})$,
for fixed $n \in \mathbb{N}$, that solves
\begin{gather}
\begin{aligned}
&du_{n}^{(R)} + [A_1 u_{n}^{(R)} + \Psi_{R} (\| y_{n}^{R} \|_{\mathbb{V}^{l}}^2)
 \pi_{n}^1 B_1 (u_{n}^{(R)}, u_{n}^{(R)})\\
& -\Psi_{R} (\| y_{n}^{(R)}  \|_{\mathbb{V}^{l}}^2) \pi_{n}^1
 B_2(b_{n}^{(R)}, b_{n}^{(R)})] dt\\
&= \pi_{n}^1 \theta_{n}^{(R)} e_2 dt + \pi_{n}^1 G^1 dw^1,
\end{aligned} \nonumber \\
\begin{aligned}
&db_{n}^{(R)} + [A_2 b_{n}^{(R)}
+ \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^2 B_3(u_{n}^{(R)},
b_{n}^{(R)})\\
& - \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^2 B_4 (b_{n}^{(R)},
 u_{n}^{(R)}) ] dt \\
&= \pi_{n}^2 G^2 dw^2,
\end{aligned} \nonumber\\
d\theta_{n}^{(R)} + [A_3 \theta_{n}^{(R)}
+ \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^3 B_5(u_{n}^{(R)},
\theta_{n}^{(R)})] dt
= \pi_{n}^3 u_{n,2}^{(R)} dt,
 \nonumber \\
(u_{n}^{(R)}, b_{n}^{(R)}, \theta_{n}^{(R)})(0)
= (\pi_{n}^1x_1, \pi_{n}^2x_2, \pi_{n}^3x_3) = \pi_{n} x, \label{e51}
\end{gather}
where $u_{n,2}^{(R)}$ is the second component of $u_{n}^{(R)}$,
and its associated Markovian semigroup $P_{t,n}^{(R)}$.
We prove the following proposition:

\begin{proposition} \label{prop4.5}
Under the hypothesis of Theorem \ref{thm2.2}, for all  $t > 0$ and all $R > 0$,
there exists a constant $L = L(R, t) \geq 0$ such that
\[
| P_{t,n}^{(R)} \psi(x) -P_{t,n}^{(R)} \psi(v) |
\leq L \| x - v \|_{\mathbb{V}^{l}}
\]
for all  $n \in \mathbb{N}$, $x, v \in H_{n}$, $\psi \in C_{b}(H_{n})$ such that
$\| \psi \|_{C_{b}} \triangleq \sup_{x \in H_{n}} | \psi(x) | \leq 1$.
\end{proposition}

\begin{proof}
Firstly, $P_{t,n}^{(R)}$ is Feller and hence
$P_{t,n}^{(R)}(C_{b}(\mathbb{V}^{l})) \subset C_{b}(\mathbb{V}^{l})$;
moreover, by mean value theorem,
\[
| P_{t,n}^{(R)} \psi(x) - P_{t,n}^{(R)} \psi(v) |
\leq \sup_{k, h \in H_{n}, \| h \|_{\mathbb{V}^{l}} \leq 1}
| D P_{t,n}^{(R)} \psi(k) \cdot h | \| x - v \|_{\mathbb{V}^{l}}
\]	
where $DP_{t,n}^{(R)} \psi(k) \cdot h$ denotes the derivative of the
mapping $v\mapsto P_{t,n}^{(R)}\psi(v)$ at the point $k$ in the direction $h$.
We rewrite \eqref{e51} similarly to \eqref{e4} as   	
\[
dy_{n}^{(R)} + [Ay_{n}^{(R)} + \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2)
\pi_{n} B(y_{n}^{(R)}, y_{n}^{(R)})] dt = \pi_{n} Ry_{n}^{(R)}dt + \pi_{n} Gdw.
\]
Thus, by Elworthy's formula (see e.g. \cite[pg. 267]{7}), for all  $h \in H_{n}$
with an $n$-dimensional standard Wiener process $\beta_{n}$,
\begin{align*}
[DP_{t,n}^{(R)} \psi(x) ] \cdot h = \frac{1}{t} E[\psi(y_{n}^{(R)} (t; x))
\int_0^{t} \langle (\pi_{n} GG^{\ast} \pi_{n})^{-\frac{1}{2}} [Dy_{n}^{(R)} (s; x)]
\cdot h, d\beta_{n} (s)\rangle].
\end{align*}	
By Burkholder-Davis-Gundy inequality (e.g. \cite{23}),
\begin{equation} \label{e52}
| [DP_{t,n}^{(R)} \psi(x)] \cdot h  |
\leq \frac{c}{t} \| \psi \|_{C_{b}}
E\Big[\Big( \int_0^{t} | (\pi_{n} G G^{\ast} \pi_{n})^{-\frac{1}{2}}
[Dy_{n}^{(R)} (s; x)] \cdot h |^2 ds \Big)^{1/2}\Big]
\end{equation}	
where we used that $\psi \in C_{b}(H_{n})$. Due to \eqref{e9} and \eqref{e11}
we have for a constant $c$ independent of $n$,
\begin{equation} \label{e53}
| (\pi_{n} GG^{\ast} \pi_{n})^{-\frac{1}{2}} v |
\leq c \| v \|_{\mathbb{V}^{l+1}} \quad \forall \hspace{1mm}  y \in H_{n}.	
\end{equation} 	
We let
\[
\overline{u}_{n}^{(R)} \triangleq u_{n}^{(R)} - \pi_{n}^1 z^1, \quad
\overline{b}_{n}^{(R)} \triangleq b_{n}^{(R)} - \pi_{n}^2 z^2, \quad
\overline{y}_{n}^{(R)} \triangleq (\overline{u}_{n}^{(R)},
\overline{b}_{n}^{(R)}, \theta_{n}^{(R)}),
\]	
where $z^{j}, j= 1,2$ are solutions to \eqref{e12} with
 $\alpha = 0$, $t \geq 0$, $z^{j} (0) = 0$ and also denote
\begin{align*}
Y_{n}^{(R)} (t)
&\triangleq [D\overline{y}_{n}^{(R)} (t; x) ] \cdot h\\
&=
\begin{pmatrix}
D_{x_1} \overline{u}_{n}^{(R)} \cdot h & D_{x_1} \overline{b}_{n}^{(R)} \cdot h
 & D_{x_1} \theta_{n}^{(R)} \cdot h\\
D_{x_2} \overline{u}_{n}^{(R)} \cdot h & D_{x_2} \overline{b}_{n}^{(R)} \cdot h
 & D_{x_2} \theta_{n}^{(R)} \cdot h\\
D_{x_3} \overline{u}_{n}^{(R)} \cdot h & D_{x_3} \overline{b}_{n}^{(R)} \cdot h
  & D_{x_3} \theta_{n}^{(R)} \cdot h
\end{pmatrix} \triangleq
\begin{pmatrix}
\eta_1 & \eta_2 & \eta_3 \\
\eta_4 & \eta_5 & \eta_{6} \\
\eta_{7} & \eta_{8} & \eta_{9}
\end{pmatrix}
\end{align*}	
where $D \triangleq (D_{x_1}, D_{x_2}, D_{x_3})$ which is a derivative
 of the mapping $x \mapsto y_{n}^{(R)} (t; x)$.
Thus, $\partial_t Y_{n}^{(R)} = \partial_t [D \overline{y}_{n}^{(R)} (t; x) ]
\cdot h$ and it can be checked that
\begin{gather*}
\begin{aligned}
&\partial_t \eta_1 + A_1 \eta_1 - \pi_{n}^1 \eta_3 e_2\\
&= 2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \langle A^{1/2} y_{n}^{(R)},
 A^{1/2} D_{x_1} y_{n}^{(R)} \cdot h\rangle \pi_{n}^1 [-B_1 (u_{n}^{(R)},
  u_{n}^{(R)}) + B_2(b_{n}^{(R)}, b_{n}^{(R)})]\\
&\quad + \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^1 [-B_1(\eta_1,
 u_{n}^{(R)} ) - B_1(u_{n}^{(R)}, \eta_1) + B_2 (\eta_2, b_{n}^{(R)})
 + B_2(b_{n}^{(R)}, \eta_2)],
\end{aligned}\\
\begin{aligned}
&\partial_t \eta_2 + A_2 \eta_2\\
&= 2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \langle A^{1/2} y_{n}^{(R)},
 A^{1/2} D_{x_1} y_{n}^{(R)} \cdot h\rangle \pi_{n}^2 [-B_3 (u_{n}^{(R)},
 b_{n}^{(R)}) + B_4(b_{n}^{(R)}, u_{n}^{(R)})]\\
&\quad + \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^2
 [-B_3(\eta_1, b_{n}^{(R)}) - B_3(u_{n}^{(R)}, \eta_2) + B_4 (\eta_2, u_{n}^{(R)})
 + B_4(b_{n}^{(R)}, \eta_1)] ,
\end{aligned}\\
\begin{aligned}
\partial_t \eta_3 + A_3 \eta_3
&= -2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \langle A^{1/2} y_{n}^{(R)},
 A^{1/2} D_{x_1} y_{n}^{(R)} \cdot h\rangle \pi_{n}^3 B_5 (u_{n}^{(R)},
 \theta_{n}^{(R)})\\
&\quad - \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^3
  [B_5(\eta_1, \theta_{n}^{(R)}) + B_5(u_{n}^{(R)}, \eta_3)]
 + \pi_{n}^3 \eta_{1,2},
\end{aligned}
\end{gather*}	
 that
\begin{gather*}
\begin{aligned}
&\partial_t \eta_4 + A_1 \eta_4 - \pi_{n}^1 \eta_{6} e_2\\
&= 2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \langle A^{1/2} y_{n}^{(R)},
 A^{1/2} D_{x_2} y_{n}^{(R)} \cdot h\rangle \pi_{n}^1
 [-B_1 (u_{n}^{(R)}, u_{n}^{(R)}) + B_2(b_{n}^{(R)}, b_{n}^{(R)})]\\
&\quad + \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^1
 [-B_1(\eta_4, u_{n}^{(R)} ) - B_1(u_{n}^{(R)}, \eta_4)
 + B_2 (\eta_5, b_{n}^{(R)}) + B_2(b_{n}^{(R)}, \eta_5)],
\end{aligned}\\
\begin{aligned}
&\partial_t \eta_5 + A_2 \eta_5\\
&= 2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2)
 \langle A^{1/2} y_{n}^{(R)}, A^{1/2} D_{x_2} y_{n}^{(R)}
 \cdot h\rangle \pi_{n}^2 [-B_3 (u_{n}^{(R)}, b_{n}^{(R)})
 + B_4(b_{n}^{(R)}, u_{n}^{(R)})]\\
&\quad + \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^2
 [-B_3(\eta_4, b_{n}^{(R)}) - B_3(u_{n}^{(R)}, \eta_5)
 + B_4 (\eta_5, u_{n}^{(R)}) + B_4(b_{n}^{(R)}, \eta_4)],
\end{aligned}\\
\begin{aligned}
\partial_t \eta_{6} + A_3 \eta_{6}
&= -2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \langle A^{1/2} y_{n}^{(R)},
  A^{1/2} D_{x_2} y_{n}^{(R)} \cdot h\rangle \pi_{n}^3 B_5 (u_{n}^{(R)},
 \theta_{n}^{(R)})\\
&\quad - \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^3
 [B_5(\eta_4, \theta_{n}^{(R)}) + B_5(u_{n}^{(R)}, \eta_{6})]
 + \pi_{n}^3 \eta_{4,2},
\end{aligned}
\end{gather*}
and that
\begin{gather*}
\begin{aligned}
&\partial_t \eta_{7} + A_1 \eta_{7} - \pi_{n}^1 \eta_{9} e_2\\
&= 2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \langle A^{1/2} y_{n}^{(R)},
  A^{1/2} D_{x_3} y_{n}^{(R)} \cdot h\rangle \pi_{n}^1 [-B_1 (u_{n}^{(R)},
  u_{n}^{(R)}) + B_2(b_{n}^{(R)}, b_{n}^{(R)})]\\
&\quad + \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^1 [-B_1(\eta_{7},
 u_{n}^{(R)} ) - B_1(u_{n}^{(R)}, \eta_{7}) + B_2 (\eta_{8}, b_{n}^{(R)})
  + B_2(b_{n}^{(R)}, \eta_{8})],
\end{aligned}\\
\begin{aligned}
&\partial_t \eta_{8} + A_2 \eta_{8}\\
& = 2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \langle A^{1/2} y_{n}^{(R)},
 A^{1/2} D_{x_3} y_{n}^{(R)} \cdot h\rangle \pi_{n}^2 [-B_3 (u_{n}^{(R)},
 b_{n}^{(R)}) + B_4(b_{n}^{(R)}, u_{n}^{(R)})]\\
&\quad + \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^2
 [-B_3(\eta_{7}, b_{n}^{(R)}) - B_3(u_{n}^{(R)}, \eta_{8})
 + B_4 (\eta_{8}, u_{n}^{(R)}) + B_4(b_{n}^{(R)}, \eta_{7})],
\end{aligned}\\
\begin{aligned}
\partial_t \eta_{9} + A_3 \eta_{9}
&= -2\Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \langle A^{1/2} y_{n}^{(R)},
A^{1/2} D_{x_3} y_{n}^{(R)} \cdot h\rangle \pi_{n}^3 B_5 (u_{n}^{(R)},
\theta_{n}^{(R)})\\
&\quad - \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) \pi_{n}^3
  [B_5(\eta_{7}, \theta_{n}^{(R)}) + B_5(u_{n}^{(R)}, \eta_{9})]
 + \pi_{n}^3 \eta_{7,2},
\end{aligned}
\end{gather*}
where $\eta_{1,2}, \eta_{4,2}, \eta_{7,2}$ are the second components of
$\eta_1, \eta_4, \eta_{7}$ respectively.

We take $L^2$-inner products with
\[
(A_1^{l} \eta_1, A_2^{l} \eta_2, A_3^{l} \eta_3, A_1^{l}
\eta_4, A_2^{l} \eta_5, A_3^{l} \eta_{6}, A_1^{l} \eta_{7}, A_2^{l} \eta_{8},
 A_3^{l} \eta_{9})
\]
 respectively to estimate for example,
\begin{align*}
&\frac{1}{2} \partial_t | A_1^{1/2} \eta_1 |^2 + | A_1^{\frac{l+1}{2}} \eta_1 |^2\\
&\lesssim \Psi_{R}'(\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) | A^{1/2} y_{n}^{(R)} |
  | A^{1/2} Y_{n}^{(R)} | (| A_1^{1/2} u_{n}^{(R)} |^2
 + | A_1^{1/2} b_{n}^{(R)} |^2) | A_1^{\frac{l+1}{2}} \eta_1 | \\
&\quad+ \Psi_{R} (\| y_{n}^{(R)} \|_{\mathbb{V}^{l}}^2) ( | A_1^{1/2} \eta_1 |
 | A_1^{1/2} u_{n}^{(R)} | + | A_1^{1/2} \eta_2 | | A_1^{1/2} b_{n}^{(R)} |)
 | A_1^{\frac{l+1}{2}} \eta_1 | \\
&\quad + | A_1^{1/2} \eta_3 | | A_1^{1/2} \eta_1 | \\
&\lesssim c(R) (| A^{1/2} Y_{n}^{(R)} | + | A_1^{1/2} \eta_1 |
 + | A_1^{1/2} \eta_2 |) | A_1^{\frac{l+1}{2}} \eta_1 | + | A_1^{1/2} \eta_3 |
  | A_1^{1/2} \eta_1 | \\
&\leq  \frac{1}{2} | A_1^{\frac{l+1}{2}} \eta_1 |^2 + c(R) | A^{1/2} Y_{n}^{(R)} |^2
\end{align*}
where we used H\"older's and Young's inequalities. Subtracting
$\frac{1}{2} | A_1^{\frac{l+1}{2}} \eta_1 |^2$ from both sides gives
\[
\partial_t | A_1^{1/2} \eta_1 |^2 + | A_1^{\frac{l+1}{2}} \eta_1 |^2
\leq c(R) | A^{1/2} Y_{n}^{(R)} |^2.
\]	
Similar computations on the other eight equations show that in sum
\begin{equation} \label{e54}
\partial_t | A^{1/2} Y_{n}^{(R)} |^2 + | A^{\frac{l+1}{2}} Y_{n}^{(R)} |^2
\leq c(R) | A^{1/2} Y_{n}^{(R)} |^2.
\end{equation}
Thus,
\[
\| Y_{n}^{(R)} (t) \|_{\mathbb{V}^{l}}^2 \leq \| h\|_{\mathbb{V}^{l}}^2 e^{c(R)t}
\]	
and hence integrating \eqref{e54} gives
\[
\int_0^{t} \| Y_{n}^{(R)}(s) \|_{\mathbb{V}^{l+1}}^2 ds
\leq c(R,t)e^{c(R) t} \| h \|_{\mathbb{V}^{l}}^2.
\]	
This implies
\[
E[\int_0^{t} \| Y_{n}^{(R)}(s) \|_{\mathbb{V}^{l+1}}^2 ds]
\leq c(R, t) \| h \|_{\mathbb{V}^{l}}^2
\]
so that
\begin{equation} \label{e55}
E[\int_0^{t} \|[ D y_{n}^{(R)} (s; x) ] \cdot h \|_{\mathbb{V}^{l+1}}^2 ds]
= E[\int_0^{t} \| Y_{n}^{(R)} (s) \|_{\mathbb{V}^{l+1}}^2 ds] \leq c(R, t) \| h \|_{\mathbb{V}^{l}}^2.
\end{equation}
Therefore,
\begin{align*}
& \sup_{k, h \in H_{n}, \| h \|_{\mathbb{V}^{l}} \leq 1}
 | [D P_{t,n}^{(R)} \psi(k)] \cdot h | \\
&\leq  \sup_{k, h \in H_{n}, \| h \|_{\mathbb{V}^{l}} \leq 1} \frac{c}{t}
\| \psi \|_{C_{b}} E[(\int_0^{t} | (\pi_{n} G G^{\ast} \pi_{n})^{-\frac{1}{2}}
[Dy_{n}^{(R)} (s; k)] \cdot h |^2 ds )^{1/2}]\\
&\leq  \sup_{k, h \in H_{n}, \| h \|_{\mathbb{V}^{l}} \leq 1}  \frac{c}{t}
\| \psi \|_{C_{b}} E[(\int_0^{t} \| [D y_{n}^{(R)} (s; k)] \cdot
h \|_{\mathbb{V}^{l+1}}^2 ds)^{1/2}]\\
&\leq  \sup_{k, h \in H_{n}, \| h \|_{\mathbb{V}^{l}} \leq 1}
(\frac{\| \psi \|_{C_{b}}}{t}) c(R,t) \| h \|_{\mathbb{V}^{l}}
\leq \frac{C(R,t)}{t}
\end{align*}
by \eqref{e52}, \eqref{e53}, H\"older's inequality and \eqref{e55}.
\end{proof}

We now consider
\begin{gather*}
\begin{aligned}
&du^{(R)} + [A_1 u^{(R)} + \Psi_{R} (\| y^{R} \|_{\mathbb{V}^{l}}^2) B_1 (u^{(R)},
 u^{(R)}) -\Psi_{R} (\| y^{(R)} \|_{\mathbb{V}^{l}}^2) B_2(b^{(R)}, b^{(R)})] dt\\
&= \theta^{(R)} e_2 dt + G^1 dw^1,
\end{aligned}\\
\begin{aligned}
&db^{(R)} + [A_2 b^{(R)} + \Psi_{R} (\| y^{(R)} \|_{\mathbb{V}^{l}}^2)
B_3(u^{(R)}, b^{(R)}) - \Psi_{R} (\| y^{(R)} \|_{\mathbb{V}^{l}}^2)
B_4 (b^{(R)}, u^{(R)}) ] dt \\
&= G^2 dw^2,
\end{aligned}\\
d\theta^{(R)} + [A_3 \theta^{(R)} + \Psi_{R} (\| y^{(R)} \|_{\mathbb{V}^{l}}^2)
B_5(u^{(R)}, \theta^{(R)})] dt
= u_2^{(R)} dt,\\
(u^{(R)}, b^{(R)}, \theta^{(R)})(0) = (x_1, x_2, x_3) = x,
\end{gather*}
with $P_t^{(R)}$ as its associated Markovian semigroup, pass the limit and
obtain the following proposition:

\begin{proposition} \label{prop4.6}
Under the hypothesis of Theorem \ref{thm2.2}, for all  $t > 0$ and all $R> 0$,
there exists a constant $L = L(R, t) > 0$ such that
\[
| P_t^{(R)}\psi(x) - P_t^{(R)} \psi(v) | \leq L \| x - v \|_{\mathbb{V}^{l}}
\]	
for all  $x, v \in \mathbb{V}^{l}$, $\psi \in C_{b}(\mathbb{V}^{l})$ such that
$\| \psi \|_{C_{b}} \leq 1$. Moreover, $P_t^{(R)}\psi$ is Lipschitz
continuous for all  $\psi \in B_{b}(\mathbb{V}^{l})$. 	
\end{proposition}

\begin{proof}
The \emph{a priori} estimates in Proposition \ref{prop4.1} are 
available for the Galerkin approximation system \eqref{e51};
hence, by the well-known compact embedding results (see \cite{4, 28, 38}),
and passing to a subsequence if necessary and relabeling it, we obtain
\[
y_{n}^{(R)} \to y^{(R)} \quad \text{a.e. } t, \; \mathbb{P}\text{-a.s. }
 (n\to\infty).
\]	
We let $\psi \in C_{b}(\mathbb{V}^{l})$ so that $\psi \in C_{b}(H_{n})$ and
 because $y_{n}^{(R)} \in H_{n}$, we obtain
\[
\psi(y_{n}^{(R)} (\cdot; x)) \to \psi(y^{(R)} (\cdot; x)) \quad
\text{in } L^1(0,T) \quad (n\to\infty)
\]
by dominated convergence theorem. Taking expectation, using again that
$\psi$ is continuous  and bounded, due to dominated convergence theorem,
we obtain
\[
E\Big[\int_0^{T} | \psi(y_{n}^{(R)} (\cdot; x)) - \psi(y^{(R)} (\cdot; x)) |
dt\Big] \to 0 \quad (n\to\infty)
\]	
and hence by Fubini's theorem
\[
\int_0^{T} E[ | \psi (y_{n}^{(R)}(\cdot, x)) - \psi(y^{(R)} (\cdot; x)) |] dt \to 0
\quad (n\to\infty).
\]	
Taking a subsequence again, and relabeling it we obtain for a.e. $t \in [0,T]$,
\begin{equation} \label{e56}
E[| \psi(y_{n}^{(R)} (\cdot; x) ) - \psi(y^{(R)} (\cdot; x)) |] \to 0 \quad
(n\to\infty).
\end{equation}

By \eqref{e8}, we now see that for all  $x, v \in \mathbb{V}^{l}$,
$\psi\in C_{b}(\mathbb{V}^{l})$ such that $\| \psi \|_{C_{b}} \leq 1$,
there exists $L = L(R,t) \geq 0$ such that
\begin{align*}
&| P_t^{(R)} \psi(x) - P_t^{(R)} \psi(v) |\\
&\leq  | P_t^{(R)} \psi(x) - P_{t,n}^{(R)} \psi(x) |
 + | P_{t,n}^{(R)} \psi(x) - P_{t,n}^{(R)} \psi(v) |
 + | P_{t,n}^{(R)} \psi(v) - P_t^{(R)} \psi(v) | \\
&\leq  E[ | \psi(y^{(R)} (\cdot; x)) - \psi(y_{n}^{(R)} (\cdot; x)) |]
 + L \| x - v \|_{\mathbb{V}^{l}}\\
&\quad + E[| \psi(y^{(R)} (\cdot; v)) - \psi(y_{n}^{(R)} (\cdot; v)) |]\\
&\to  L \| x - v \|_{\mathbb{V}^{l}}
\end{align*}
by Proposition \ref{prop4.5} and \eqref{e56}. Since the trajectories of $y^{(R)}(t;x)$
in $\mathbb{V}^{l}$ are continuous, this holds for all  $t \in [0,T]$.
Considering the total variation of $\mathbb{V}^{l}$, this holds for all
$\psi \in B_{b}(\mathbb{V}^{l})$.
\end{proof}

The following proposition is now an immediate consequence of the above result.

\begin{proposition} \label{prop4.7}
Under the hypothesis of Theorem \ref{thm2.2}, for all  $t > 0$,
\[
\lim_{R\to\infty} \| P(t, x, \cdot) - P^{(R)} (t, x, \cdot) \|_{\rm Var} = 0
\]
uniformly with respect to $x$ in bounded sets of $\mathbb{V}^{l}$.
\end{proposition}

\begin{proof}[Proof of Theorem \ref{thm2.2}]
We now complete the proof of the strong Feller property.
By Proposition \ref{prop4.7} with $x, v$ in some ball of $ \in \mathbb{V}^{l}$,
for all $\epsilon > 0$ fixed, $t > 0$, there exists $R_{\epsilon}  > 0$ such that
$R \geq R_{\epsilon}$ implies
\[
\| P(t, x, \cdot) - P^{(R)} (t, x, \cdot) \|_{\rm Var}
+ \| P(t, v, \cdot) - P^{(R)} (t, v, \cdot) \|_{\rm Var} < 2 \epsilon.
\]
Thus, for $x, v$ such that $\| x - v \|_{\mathbb{V}^{l}} < \frac{\epsilon}{L}$ where
$L$ is that of Proposition \ref{prop4.6},
\begin{align*}
& \| P(t, x, \cdot) - P(t, v, \cdot) \|_{\rm Var} \\
&\leq  \| P(t, x, \cdot) - P^{(R)} (t, x, \cdot) \|_{\rm Var}
 + \| P^{(R)} (t, x, \cdot) - P^{(R)}(t, v, \cdot) \|_{\rm Var}\\
&\quad + \| P^{(R)} (t, v, \cdot) - P(t, v, \cdot) \|_{\rm Var} \\
&\leq 2 \epsilon + L \| x - v \|_{\mathbb{V}^{l}}\leq 3\epsilon
\end{align*}
by Proposition \ref{prop4.6}. Thus,
$P_t: B_{b}(\mathbb{V}^{l}) \to C_{b}(\mathbb{V}^{l})$ for all $t \in (0, T]$;
i.e. $\{P_t\}$ is strong Feller.
\end{proof}

\section{Appendix}

\begin{lemma}[\cite{10}]  \label{lem5.1}
Let $\{g_i^{j}, \lambda_i^{j}\}_{i=1}^{\infty}$ be the
eigenvectors and their corresponding eigenvalues of $A_{j}$, $j = 1, 2$. Then
\begin{enumerate}
\item for $f = \sum_{i=1}^{\infty} f_i(t) g_i^{j}(x)$, $j = 1, 2$,
$\mathbb{P}$-a.s.,
\[
f \in D(A_{j}^{k}) \text{ if and only if }
\sum_{i=1}^{\infty} (f_i(t))^2 (\lambda_i^{j})^{2k} < \infty;
\]	

\item in particular, for $G^{j}w^{j}(t) = \sum_{i=1}^{\infty} \sigma_i^{j}
\beta_i^{j}(t) g_i^{j}$, $j = 1, 2$, from \eqref{e5}, we have $\mathbb{P}$-a.s.,
\[
G^{j}w^{j}(t) \in D(A_{j}^{k}) \text{ if and only if }
\sum_{i=1}^{\infty} (\sigma_i^{j})^2(\lambda_i^{j})^{2k} < \infty.
\]
\end{enumerate}
	\end{lemma}

\begin{lemma}[\cite{10}] \label{lem5.2}
 For $z_{\alpha}^{j}(t) = \int_{-\infty}^{t} e^{(-A_{j}- \alpha)(t-s)} dG^{j}w^{j}(s)$
 in \eqref{e13}, where \\
$G^{j}w^{j}(s) = \sum_{i=1}^{\infty} \sigma_i^{j} \beta_i^{j}
(s) g_i^{j}$, $j = 1, 2$ from \eqref{e5}: we have
$\mathbb{P}$-a.s.,
\begin{gather*}
z_{\alpha}^{j} \in D(A_{j}^{k}) \text{ if and only if }
\sum_{i=1}^{\infty} \frac{(\sigma_i^{j})^2 (\lambda_i^{j})^{2k}}{2(\lambda_i^{j}
+ \alpha)} < \infty;
\\
\lim_{\alpha \to \infty} E[| A_{j}^{1/4} z_{\alpha}^{j} |^4] = 0.
\end{gather*}
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.2]{21}}] \label{lem5.3}
In an $n$-dimensional bounded domain $D$, let $P$ be a continuous projector
from $(L^{r}(D))^{n}$ to the closure of
$\{u \in (C_0^{\infty}(D))^{n}: \nabla\cdot u = 0\}$ in
$(L^{r}(D))^{n}$ and $A = - P \Delta$. Then for
$0 \leq \delta  < \frac{1}{2} + \frac{n(1 - \frac{1}{r})}{2}$
\[
\| A^{-\delta} P (u\cdot\nabla) v\|_{L^{r}}
\lesssim_{\delta, \theta, \rho, r} \| A^{\theta}  u\|_{L^{r}}
 \| A^{\rho} v \|_{L^{r}}
\]
if $\delta + \theta + \rho \geq \frac{n}{2r} + \frac{1}{2}$, $\theta  >0$,
$\rho > 0$, $\rho + \delta > 1/2$.
\end{lemma}

For the next result, see  \cite[Proposition 6.1]{10} and
\cite[Chapter III, Section 3.5.2]{38}.

\begin{proposition} \label{prop5.4}
 Suppose $-1 \leq \theta(x, t_0) \leq 1$ a.e. $x \in D$.
Then $-1 \leq \theta(x,t) \leq 1$ a.e. $x \in D$, for all $t \geq t_0$,
$\mathbb{P}$-almost surely.
\end{proposition}


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