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Generalized Ornstein--Uhlenbeck Semigroups in weighted $L^p$-spaces on Riemannian Manifolds (2107.03301v1)

Published 5 Jul 2021 in math.AP, math.DG, and math.FA

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 $P{\nabla}=\nabla{\dagger}\nabla u+\nabla_{(d\phi){\sharp}}u-\nabla_{X}u+Vu$, where $\nabla{\dagger}$ is the formal adjoint of $\nabla$, $(d\phi){\sharp}$ is the vector field corresponding to $d\phi$ via $g$, $X$ is a smooth real vector field on $M$, and $V$ is a self-adjoint locally integrable section of the bundle $\textrm{End }\mathcal{E}$. We show that (the negative of) the maximal realization $-H_{p,\max}$ of $P{\nabla}$ generates an analytic quasi-contractive semigroup in $Lp_{\mu}(\mathcal{E})$, $1<p<\infty$, where $d\mu=e{-\phi}d\nu_{g}$, 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, $P{d}=\Delta u+(d\phi){\sharp}u-Xu+Vu$, 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 $Lp_{\mu}(M)$ and, by imposing certain geometric conditions on $M$, we prove another semigroup generation result.

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