Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers

TL;DR

This paper unifies results on heat semigroups and stochastic differential equations on open manifolds using weak uniform covers, providing insights into their behavior at infinity and non-explosion criteria.

Contribution

It introduces a unified approach employing weak uniform covers to analyze diffusion semigroups and stochastic flows at infinity, linking geometric and probabilistic properties.

Findings

Established non-explosion criteria using manifold covers.

Connected heat semigroup behavior with stochastic differential equations.

Reproduced known results on Brownian motion on manifolds with Ricci decay.

Abstract

A unified treatment is given of some results of H. Donnelly-P. Li and L. Schwartz concerning the behaviour of heat semigroups on open manifolds with given compactifications, on one hand, and the relationship with the behaviour at infinity of solutions of related stochastic differential equations on the other. A principal tool is the use of certain covers of the manifold: which also gives a non-explosion test. As a corollary we obtain known results about the behaviour of Brownian motions on a complete Riemannian manifold with Ricci curvature decaying at most quadratically in the distance function.