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- W4287214159 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 $A:D(A)subseteq Xrightarrow X$ 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 $Q_infty A^*:D(A^*)subseteq X^*rightarrow X$ extends to a bounded linear operator $Bin mathcal L(H)$ 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 Fr'echet derivative operators, we denote by $D_H$ and $D^2_H$ the closure in $L^2(X,mu_infty)$ of the operators $QD$ and $QD^2$ and by $W^{1,2}_H(X,mu_infty)$ and and $W^{2,2}_H(X,mu_infty)$ their domains in $L^2(X,mu_infty)$, respectively,. Furthermore, we denote by $D_{A_infty}$ the closure of the operator $Q_infty A^*D$ and by $W^{1,2}_{A_infty}(X,mu_infty)$ 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]_Hdmu_infty, qquad u,vin W^{1,2}_H(X,mu_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 $W^{2,2}_H(X,mu_infty)cap W^{1,2}_{A_infty}(X,mu_infty)$. We stress that we are able to treat the case when $L$ is degenerate and non-symmetric." @default.
- W4287214159 created "2022-07-25" @default.
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- W4287214159 date "2021-04-13" @default.
- W4287214159 modified "2023-09-28" @default.
- W4287214159 title "Regarding the domain of non-symmetric and, possibly, degenerate Ornstein--Uhlenbeck operators in separable Banach spaces" @default.
- W4287214159 hasPublicationYear "2021" @default.
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