Analyticity of non-symmetric Ornstein-Uhlenbeck semigroup with respect to a weighted Gaussian measure

Abstract

In this paper we show that the realization in $L^p(X,ν_\infty)$ of the nonsymmetric Ornstein-Uhlenbeck operator $L$ is sectorial for any $p\in(1,+\infty)$ and we provide an explicit sector of analyticity. Here $(X,μ_\infty,H_\infty)$ is an abstract Wiener space, i.e., $X$ is a separable Banach space, $μ_\infty$ is a centred non degenerate Gaussian measure on $X$ and $H_\infty$ is the associated Cameron-Martin space. Further, $ν_\infty$ is a weighted Gaussian measure, that is, $ν_\infty=e^{-U}μ_\infty$ where $U$ is a convex function which satisfies some minimal conditions. Our results strongly rely on the theory of nonsymmetric Dirichlet forms and on the divergence form of the realization of $L$ in $L^2(X,ν_\infty)$.