Volume growth and escape rate of Brownian motion on a complete Riemannian manifold

TL;DR

This paper establishes an upper escape rate function for Brownian motion on complete Riemannian manifolds, linking it to the manifold's volume growth and using reflection techniques to estimate crossing times.

Contribution

It introduces a new method to relate Brownian escape rates to volume growth, employing reflection of Brownian motions for precise tail probability estimates.

Findings

Derived an effective upper escape rate function based on volume growth.

Estimated small tail probabilities of crossing times using reflection techniques.

Provided insights into the relationship between geometric properties and stochastic behavior.

Abstract

We give an effective upper escape rate function for Brownian motion on a complete Riemannian manifold in terms of the volume growth of the manifold. An important step in the work is estimating the small tail probability of the crossing time between two concentric geodesic spheres by reflecting Brownian motions on the larger geodesic ball.