Stochastic differential equations on noncompact manifolds: moment stability and its topological consequences

TL;DR

This paper explores how the stability of stochastic differential equations on noncompact manifolds relates to the manifold's topology, providing conditions for stability and topological implications.

Contribution

It establishes necessary and sufficient conditions for SDE stability and links moment stability to the fundamental group's properties on manifolds.

Findings

Conditions for SDE moment stability in terms of coefficients

A vanishing result for the fundamental group based on geometric quantities

Connection between stochastic stability and topological properties of manifolds

Abstract

In this paper we discuss the stability of stochastic differential equations and the interplay between the moment stability of a SDE and the topology of the underlying manifold. Sufficient and necessary conditions are given for the moment stability of a SDE in terms of the coefficients. Finally we prove a vanishing result for the fundamental group of a complete Riemannian manifold in terms of purely geometrical quantities.