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

Abstract

In this paper we show that the realization in $L^p(X,{\nu }_{\infty })$ of the nonsymmetric Ornstein-Uhlenbeck operator $L$ is sectorial for any $p\in (1,+\infty )$ and we provide an explicit sector of analyticity. Here $(X,{\mu }_{\infty },H_{\infty })$ is an abstract Wiener space, i.e., $X$ is a separable Banach space, ${\mu }_{\infty }$ is a centred non degenerate Gaussian measure on $X$ and $H_{\infty }$ is the associated Cameron-Martin space. Further, ${\nu }_{\infty }$ is a weighted Gaussian measure, that is, ${\nu }_{\infty }=e^{-U}{\mu }_{\infty }$ 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 $L^2(X,{\nu }_{\infty })$.