Recurrence and transience for normally reflected Brownian motion in warped product manifolds

TL;DR

This paper provides an integral test to determine when normally reflected Brownian motion in specific unbounded warped product manifolds is recurrent or transient, extending previous results and applying to hyperbolic space domains.

Contribution

It introduces a new integral criterion for recurrence and transience of reflected Brownian motion in warped product manifolds, generalizing prior flat space results.

Findings

Established an exact cutoff criterion for recurrence vs. transience.

Extended classical results to warped product manifolds and hyperbolic space.

Unified understanding of Brownian motion behavior in various geometric settings.

Abstract

We establish an integral test describing the exact cut-off between recurrence and transience for normally reflected Brownian motion in certain unbounded domains in a class of warped product manifolds. Besides extending a previous result by R. Pinsky, who treated the case in which the ambient space is flat, our result recovers the classical test for the standard Brownian motion in model spaces. Moreover, it allows us to discuss the recurrence/transience dichotomy for certain generalized tube domains around totally geodesic submanifolds in hyperbolic space.