Generalized Ornstein--Uhlenbeck Semigroups in weighted $L^p$-spaces on Riemannian Manifolds

Abstract

Let $\mathcal{E}$ be a Hermitian vector bundle over a Riemannian manifold $M$ with metric $g$, let $\nabla$ be a metric covariant derivative on $\mathcal{E}$. We study the generalized Ornstein-Uhlenbeck differential expression $P^{\nabla }={\nabla }^{†}\nabla u+{\nabla }_{(d\varphi {)}^{♯}}u-{\nabla }_{X}u+Vu$, where ${\nabla }^{†}$ is the formal adjoint of $\nabla$, $(d\varphi {)}^{♯}$ is the vector field corresponding to $d\varphi$ via $g$, $X$ is a smooth real vector field on $M$, and $V$ is a self-adjoint locally integrable section of the bundle $\text{End }\mathcal{E}$. We show that (the negative of) the maximal realization $-H_{p,\max }$ of $P^{\nabla }$ generates an analytic quasi-contractive semigroup in $L_{\mu }^p(\mathcal{E})$, $1<p<\infty$, where $d\mu =e^{-\varphi }d{\nu }_g$, with ${\nu }_g$ being the volume measure. Additionally, we describe a Feynman-Kac representation for the semigroup generated by $-H_{p,\max }$. For the Ornstein-Uhlenbeck differential expression acting on functions, that is, $P^d=\Delta u+(d\varphi {)}^{♯}u-Xu+Vu$, where $\Delta$ is the (non-negative) scalar Laplacian on $M$ and $V$ is a locally integrable real-valued function, we consider another way of realizing $P^d$ as an operator in $L_{\mu }^p(M)$ and, by imposing certain geometric conditions on $M$, we prove another semigroup generation result.