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

## TL;DR

This paper proves that the nonsymmetric Ornstein-Uhlenbeck operator is sectorial and analytic in certain weighted Gaussian measure spaces, extending the understanding of its functional-analytic properties.

## Contribution

It establishes the sectoriality and explicit analyticity sector of the Ornstein-Uhlenbeck operator in weighted Gaussian measure spaces, using Dirichlet form theory.

## Key findings

- The operator is sectorial in $L^p$ spaces for all $p
eq 1, ext{infinity}$.
- Explicit sector of analyticity is provided.
- Results rely on the theory of nonsymmetric Dirichlet forms.

## 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 $L^2(X,\nu_\infty)$.