Walking within growing domains: recurrence versus transience

TL;DR

This paper establishes a precise criterion to determine whether reflected Brownian motion and simple random walk in growing domains are recurrent or transient, based on the domains' growth rates.

Contribution

It provides a new sharp criterion linking the growth rate of domains to recurrence or transience for Brownian motion and random walk.

Findings

Recurrence and transience depend critically on the growth rate of the domain.

A sharp criterion is established for dimensions d>=3.

Results apply to both reflected Brownian motion and simple random walk.

Abstract

For normally reflected Brownian motion and for simple random walk on independently growing in time d-dimensional domains, d>=3, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.