Stochastic Properties of the Laplacian on Riemannian Submersions

TL;DR

This paper investigates the stochastic properties of the Laplacian on Riemannian submersions, establishing conditions under which the total space inherits properties like Feller, parabolicity, or stochastic completeness from the base manifold.

Contribution

It proves that Riemannian submersions with compact minimal fibers transfer stochastic properties from the base to the total space under certain conditions.

Findings

Immersed submanifolds with bounded mean curvature of Cartan-Hadamard manifolds are Feller.

The total space of a Riemannian submersion is Feller, parabolic, or stochastically complete if and only if the base manifold has the same property.

Abstract

Based on ideas of Pigolla and Setti \cite{PS} we prove that immersed submanifolds with bounded mean curvature of Cartan-Hadamard manifolds are Feller. We also consider Riemannian submersions $\pi \colon M \to N$ with compact minimal fibers, and based on various criteria for parabolicity and stochastic completeness, see \cite{Grygor'yan}, we prove that $M$ is Feller, parabolic or stochastically complete if and only if the base $N$ is Feller, parabolic or stochastically complete respectively.