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 $X^{\ast }$ be its topological dual. Let $Q:X^{\ast }\to X$ be a linear, bounded, non-negative, and symmetric operator and let $A:D(A)\subseteq X\to X$ 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 $Q_{\infty }={\int }_0^{+\infty }{e}^{sA}Q{e}^{s{A}^{\ast }}ds,$ 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 $Q_{\infty }{A}^{\ast }:D(A^{\ast })\subseteq X^{\ast }\to X$ extends to a bounded linear operator $B\in \mathcal{L}(H)$ which satisfies $B+B^{\ast }=-{\mathrm{Id}}_H$, 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 $(D_H,D_H^2)$ the closure in $L^2(X,{\mu }_{\infty })$ of the operators $QD$ and $(QD,Q{D}^2)$, respectively, defined on smooth cylindrical functions, and by $W_H^{1,2}(X,{\mu }_{\infty })$ and $W_H^{2,2}(X,{\mu }_{\infty })$, respectively, their domains in $L^2(X,{\mu }_{\infty })$. Furthermore, we denote by $D_{A_{\infty }}$ the closure of the operator $Q_{\infty }{A}^{\ast }D$ in $L^2(X,{\mu }_{\infty })$ defined on smooth cylindrical functions, and by $W_{A_{\infty }}^{1,2}(X,{\mu }_{\infty })$ 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 $(u,v)\mapsto -{\int }_X[B{D}_{H}u,D_{H}v{]}_{H}d{\mu }_{\infty },u,v\in W_H^{1,2}(X,{\mu }_{\infty }),$ in $L^2(X,{\mu }_{\infty })$. More precisely, we prove that $D(L)$ coincides, up to an equivalent renorming, with a subspace of $W_H^{2,2}(X,{\mu }_{\infty })\cap W_{A_{\infty }}^{1,2}(X,{\mu }_{\infty })$. We stress that we are able to treat the case when $L$ is degenerate and non-symmetric.