Analyticity of Nonsymmetric Ornstein-Uhlenbeck Semigroup with Respect to a Weighted Gaussian Measure

Abstract

In this paper we show that the realization in $\mathrm {L}^{p}(X,\nu _{\infty })$ of a nonsymmetric Ornstein-Uhlenbeck operator Lp 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 nondegenerate 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 L2 in $\mathrm {L}^{2}(X,\nu _{\infty })$ .