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

Abstract

Let ${{\mathcal {X}}}$ be a real separable Hilbert space. Let C be a linear, bounded, non-negative self-adjoint operator on ${{\mathcal {X}}}$ and let A be the infinitesimal generator of a strongly continuous semigroup in ${{\mathcal {X}}}$. Let $\{W(t)\}_{t\ge 0}$ be a ${{\mathcal {X}}}$-valued cylindrical Wiener process on a filtered (normal) probability space $(\Omega ,{\mathcal {F}},\{{\mathcal {F}}_t\}_{t\ge 0},{\mathbb {P}})$. Let $F:{{\text {Dom}}}(F)\subseteq {{\mathcal {X}}}\rightarrow {{\mathcal {X}}}$ be a smooth enough function. We are interested in the generalized mild solution ${\lbrace X(t,x)\rbrace _{t\ge 0}}$ of the semilinear stochastic partial differential equation

$$\begin{aligned} {\left\{ \begin{array}{ll} dX(t,x)=\big (AX(t,x)+F(X(t,x))\big )dt+ \sqrt{C}dW(t), &{} t>0;\\ X(0,x)=x\in {{\mathcal {X}}}. \end{array}\right. } \end{aligned}$$

We consider the transition semigroup defined by

$$\begin{aligned} P(t)\varphi (x):={{\mathbb {E}}}[\varphi (X(t,x))], \qquad \varphi \in B_b({{\mathcal {X}}}),\ t\ge 0,\ x\in {{\mathcal {X}}}. \end{aligned}$$

If ${{\mathcal {O}}}$ is an open set of ${{\mathcal {X}}}$, we consider the Dirichlet semigroup defined by

$$\begin{aligned} P^{{\mathcal {O}}}(t)\varphi (x):={\mathbb {E}}\left[ \varphi (X(t,x)){\mathbb {I}}_{\{\omega \in \Omega \; :\;\tau _x(\omega )> t\}}\right] ,\quad \varphi \in B_b({\mathcal {O}}),\; x\in {{\mathcal {O}}},\; t>0 \end{aligned}$$

where $\tau _x$ is the exit time defined by

$$\begin{aligned} \tau _x=\inf \{ s> 0\; : \; X(s,x)\in {{\mathcal {O}}}^c \}. \end{aligned}$$

We study the infinitesimal generator of P(t), $P^{{\mathcal {O}}}(t)$ in $L^2({{\mathcal {X}}},\nu )$, $L^2({{\mathcal {O}}},\nu )$ respectively, where $\nu$ is the unique invariant measure of P(t).