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

### Davide Addona

Università degli Studi di Parma, Italy### Gianluca Cappa

LUISS “Carlo Guidi”, Roma, Italy### Simone Ferrari

Università del Salento, Lecce, Italy

## Abstract

Let $X$ be a separable Banach space and let $X_{∗}$ be its topological dual. Let $Q:X_{∗}→X$ be a linear, bounded, non-negative, and symmetric operator and let $A:D(A)⊆X→X$ be the infinitesimal generator of a strongly continuous semigroup of contractions on $X$. We consider the abstract Wiener space $(X,μ_{∞},H_{∞})$, where $μ_{∞}$ is a centered non-degenerate Gaussian measure on $X$ with covariance operator defined, at least formally, as $Q_{∞}=∫_{0}e_{sA}Qe_{sA_{∗}}ds,$ and $H_{∞}$ is the Cameron–Martin space associated to $μ_{∞}$.

Let $H$ be the reproducing kernel Hilbert space associated with $Q$ with inner product $[⋅,⋅]_{H}$. We assume that the operator $Q_{∞}A_{∗}:D(A_{∗})⊆X_{∗}→X$ extends to a bounded linear operator $B∈L(H)$ which satisfies $B+B_{∗}=−Id_{H}$, where $Id_{H}$ 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 $(D_{H},D_{H})$ the closure in $L_{2}(X,μ_{∞})$ of the operators $QD$ and $(QD,QD_{2})$, respectively, defined on smooth cylindrical functions, and by $W_{H}(X,μ_{∞})$ and $W_{H}(X,μ_{∞})$, respectively, their domains in $L_{2}(X,μ_{∞})$. Furthermore, we denote by $D_{A_{∞}}$ the closure of the operator $Q_{∞}A_{∗}D$ in $L_{2}(X,μ_{∞})$ defined on smooth cylindrical functions, and by $W_{A_{∞}}(X,μ_{∞})$ the domain of $D_{A_{∞}}$ in $L_{2}(X,μ_{∞})$. We characterize the domain of the operator $L$, associated to the bilinear form $(u,v)↦−∫_{X}[BD_{H}u,D_{H}v]_{H}dμ_{∞},u,v∈W_{H}(X,μ_{∞}),$ in $L_{2}(X,μ_{∞})$. More precisely, we prove that $D(L)$ coincides, up to an equivalent renorming, with a subspace of $W_{H}(X,μ_{∞})∩W_{A_{∞}}(X,μ_{∞})$. We stress that we are able to treat the case when $L$ is degenerate and non-symmetric.

## Cite this article

Davide Addona, Gianluca Cappa, Simone Ferrari, On the domain of non-symmetric and, possibly, degenerate Ornstein–Uhlenbeck operators in separable Banach spaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 33 (2022), no. 2, pp. 297–359

DOI 10.4171/RLM/972