$L^2$-theory for transitions semigroups associated to dissipative systems

Let $\mathcal{X}$ be a real separable Hilbert space. Let $C$ be a linear, bounded and positive operator on $\mathcal{X}$ and let $A$ be the infinitesimal generator of a strongly continuous semigroup on $\mathcal{X}$. Let $\{W(t)\}_{t\geq 0}$ be a $\mathcal{X}$-valued cylindrical Wiener process on a filtered (normal) probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0},\mathbb{P})$. Let $F:D(F)\subseteq\mathcal{X}\rightarrow\mathcal{X}$ be a smooth enough function. Under suitable conditions on $A$, $C$ and $F$ the following semilinear stochastic partial differential equation \begin{gather*} \begin{cases} dX(t,x)=\big(AX(t,x)+F(X(t,x))\big)dt+ \sqrt{C}dW(t), & t>0;\\ X(0,x)=x\in \mathcal{X}, \end{cases} \end{gather*} has a unique generalized mild solution $\{X(t,x)\}_{t\geq 0}$. We consider the transition semigroup defined by \begin{align*} P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_b(\mathcal{X}),\ t\geq 0,\ x\in \mathcal{X}. \end{align*} If $\mathcal{O}$ is an open set of $\mathcal{X}$, we consider the stopped semigroup defined by \begin{equation*} P^{\mathcal{O}}(t)\varphi(x):=\mathbb{E}\left[\varphi(X(t,x))\mathbb{I}_{\{\omega\in\Omega\; :\;\tau_x(\omega)> t\}}\right],\quad \varphi\in B_b(\mathcal{O}),\; x\in\mathcal{O},\; t>0 \end{equation*} where $\tau_x$ is the stopping time defined by \begin{equation*} \tau_x=\inf\{ s> 0\; : \; X(s,x)\in \mathcal{O}^c \}. \end{equation*} We will study the infinitesimal generators of $P(t)$ and $P^{\mathcal{O}}(t)$ in $L^2(\mathcal{X},\nu)$ and $L^2(\mathcal{O},\nu)$ respectively, where $\nu$ is the unique invariant measure of $P(t)$. We will focus on investigating how these two semigroups are related to the operator formally defined by \begin{equation*} N\varphi(x):=\frac{1}{2}\mbox{Tr}[C\nabla^2\varphi(x)]+\langle Ax+F(x), \nabla\varphi(x) \rangle. \end{equation*}