On the domain of non-symmetric and, possibly, degenerate Ornstein–Uhlenbeck operators in separable Banach spaces
Davide Addona
Università degli Studi di Parma, ItalyGianluca Cappa
LUISS “Carlo Guidi”, Roma, ItalySimone Ferrari
Università del Salento, Lecce, Italy
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Abstract
Let be a separable Banach space and let be its topological dual. Let be a linear, bounded, non-negative, and symmetric operator and let be the infinitesimal generator of a strongly continuous semigroup of contractions on . We consider the abstract Wiener space , where is a centered non-degenerate Gaussian measure on with covariance operator defined, at least formally, as and is the Cameron–Martin space associated to .
Let be the reproducing kernel Hilbert space associated with with inner product . We assume that the operator extends to a bounded linear operator which satisfies , where denotes the identity operator on . Let and be the first and second order Fréchet derivative operators. We denote by and the closure in of the operators and , respectively, defined on smooth cylindrical functions, and by and , respectively, their domains in . Furthermore, we denote by the closure of the operator in defined on smooth cylindrical functions, and by the domain of in . We characterize the domain of the operator , associated to the bilinear form in . More precisely, we prove that coincides, up to an equivalent renorming, with a subspace of . We stress that we are able to treat the case when 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