On the domain of non-symmetric and, possibly, degenerate Ornstein–Uhlenbeck operators in separable Banach spaces

Abstract

Let $X$ be a separable Banach space and let $X^$ be its topological dual. Let $Q:X^\rightarrow X$ be a linear, bounded, non-negative, and symmetric operator and let $A:D(A)\subseteq X\rightarrow X$ be the infinitesimal generator of a strongly continuous semigroup of contractions on $X$. We consider the abstract Wiener space $(X,\mu\_\infty,H\_\infty)$, where $\mu\_\infty$ is a centered non-degenerate Gaussian measure on $X$ with covariance operator defined, at least formally, as $$ Q\_\infty=\int\_0^{+\infty} e^{sA}Qe^{sA^\*},ds, $$ and $H\_\infty$ is the Cameron–Martin space associated to $\mu\_\infty$. Let $H$ be the reproducing kernel Hilbert space associated with $Q$ with inner product $\[\cdot,\cdot]H$. We assume that the operator $Q\infty A^:D(A^)\subseteq X^\rightarrow X$ extends to a bounded linear operator $B\in\mathcal{L}(H)$ which satisfies $B+B^=-\mathrm{Id}H$, where $\mathrm{Id}H$ denotes the identity operator on $H$. Let $D$ and $D^2$ be the first and second order Fréchet derivative operators. We denote by $D\_H$ and $(D\_H,D^2\_H)$ the closure in $L^2(X,\mu\infty)$ of the operators $QD$ and $(QD,QD^2)$, respectively, defined on smooth cylindrical functions, and by $W^{1,2}H(X,\mu\infty)$ and $W^{2,2}H(X,\mu\infty)$, respectively, their domains in $L^2(X,\mu\infty)$. Furthermore, we denote by $D\_{A\_\infty}$ the closure of the operator $Q\_\infty A^\*D$ in $L^2(X,\mu\_\infty)$ defined on smooth cylindrical functions, and by $W^{1,2}{A\infty}(X,\mu\_\infty)$ the domain of $D\_{A\_\infty}$ in $L^2(X,\mu\_\infty)$. We characterize the domain of the operator $L$, associated to the bilinear form $$ (u,v)\mapsto-\int\_{X}\[BD\_Hu,D\_Hv]H,d\mu\infty, \quad u,v\in W^{1,2}H(X,\mu\infty), $$ in $L^2(X,\mu\_\infty)$. More precisely, we prove that $D(L)$ coincides, up to an equivalent renorming, with a subspace of $W^{2,2}H(X,\mu\infty)\cap W^{1,2}{A\infty}(X,\mu\_\infty)$. We stress that we are able to treat the case when $L$ is degenerate and non-symmetric.