Stochastic completeness and volume growth

TL;DR

This paper investigates the relationship between volume growth conditions and stochastic completeness in Riemannian manifolds, providing counterexamples that challenge previously suggested implications and their converses.

Contribution

It demonstrates that the proposed volume growth condition does not imply stochastic completeness and offers counterexamples to the converse implication.

Findings

The volume growth condition does not guarantee stochastic completeness.

Counterexamples show the converse implication also fails.

The results clarify limitations of volume growth criteria in stochastic analysis.

Abstract

It has been suggested in 1999 that a certain volume growth condition for geodesically complete Riemannian manifolds might imply that the manifold is stochastically complete. This is motivated by a large class of examples and by a known analogous criterion for recurrence of Brownian motion. We show that the suggested implication is not true in general. We also give counter-examples to a converse implication.