Regarding the domain of non-symmetric and, possibly, degenerate Ornstein--Uhlenbeck operators in separable Banach spaces
D. Addona, G. Cappa, S. Ferrari
Abstract
Let $X$ be a separable Banach space and 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 $(X,μ_\infty,H_\infty)$ where $μ_\infty$ 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 $μ_\infty$. 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 $B+B^*=-{\rm Id}_H$, where ${\rm Id}_H$ 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 $D^2_H$ the closure in $L^2(X,μ_\infty)$ of the operators $QD$ and $QD^2$ and by $W^{1,2}_H(X,μ_\infty)$ and and $W^{2,2}_H(X,μ_\infty)$ their domains in $L^2(X,μ_\infty)$, respectively,. Furthermore, we denote by $D_{A_\infty}$ the closure of the operator $Q_\infty A^*D$ and by $W^{1,2}_{A_\infty}(X,μ_\infty)$ its domain in $L^2(X,μ_\infty)$. 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 $L^2(X,μ_\infty)$. More precisely, we prove that $D(L)$ coincides, up to an equivalent remorming, with a subspace of $W^{2,2}_H(X,μ_\infty)\cap W^{1,2}_{A_\infty}(X,μ_\infty)$. We stress that we are able to treat the case when $L$ is degenerate and non-symmetric.