Regarding 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 $Q:X^{\ast }\to X$ 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 centred non-degenerate Gaussian measure on $X$ with covariance operator defined, at least formally, as \begin{align*} Q_\infty=\int_0^{+\infty} e^{sA}Qe^{sA^*}ds, \end{align*} 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 $D_H^2$ the closure in $L^2(X,{\mu }_{\infty })$ of the operators $QD$ and $Q{D}^2$ and by $W_H^{1,2}(X,{\mu }_{\infty })$ and and $W_H^{2,2}(X,{\mu }_{\infty })$ their domains in $L^2(X,{\mu }_{\infty })$, respectively,. Furthermore, we denote by $D_{A_{\infty }}$ the closure of the operator $Q_{\infty }{A}^{\ast }D$ and by $W_{A_{\infty }}^{1,2}(X,{\mu }_{\infty })$ its domain in $L^2(X,{\mu }_{\infty })$. We characterize the domain of the operator $L$, associated to the bilinear form \begin{align*} (u,v)\mapsto-\int_{X}[BD_Hu,D_Hv]_Hdμ_\infty, \qquad u,v\in W^{1,2}_H(X,μ_\infty), \end{align*} in $L^2(X,{\mu }_{\infty })$. More precisely, we prove that $D(L)$ coincides, up to an equivalent remorming, 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.

References in corpus (2)

• Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes
• Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients

Cited by in corpus (1)

• Differentiability in infinite dimension and the Malliavin calculus