The Feller property on Riemannian manifolds

TL;DR

This paper explores the Feller property on Riemannian manifolds, aiming to develop a comprehensive set of tools and criteria similar to those established for parabolicity and stochastic completeness.

Contribution

It advances the understanding of the Feller property by developing new analytical tools and criteria for its characterization on Riemannian manifolds.

Findings

Established new geometric conditions for the Feller property.

Developed comparison techniques for analyzing the Feller property.

Connected the Feller property with existing concepts like parabolicity and stochastic completeness.

Abstract

The asymptotic behavior of the heat kernel of a Riemannian manifold gives rise to the classical concepts of parabolicity, stochastic completeness (or conservative property) and Feller property (or $C^{0}$-diffusion property). Both parabolicity and stochastic completeness have been the subject of a systematic study which led to discovering not only sharp geometric conditions for their validity but also an incredible rich family of tools, techniques and equivalent concepts ranging from maximum principles at infinity, function theoretic tests (Khas'minskii criterion), comparison techniques etc... The present paper aims to move a number of steps forward in the development of a similar apparatus for the Feller property.