Transcience/recurrence for normally reflected Brownian motion in unbounded domains

TL;DR

This paper investigates whether reflected Brownian motion in certain unbounded, horn-shaped domains is transient or recurrent, focusing on domains defined by a function controlling the boundary shape.

Contribution

It characterizes the transience and recurrence behavior of reflected Brownian motion in a broad class of unbounded domains, including horn-shaped and slab domains.

Findings

Provides criteria for transience vs. recurrence based on domain geometry.

Extends understanding of reflected Brownian motion in complex unbounded domains.

Includes analysis of domains with varying boundary functions.

Abstract

Let $D\subsetneq R^d$ be an unbounded domain and let $B(t)$ be a Brownian motion in $D$ with normal reflection at the boundary. We study the transcience/recurrence dichotomy, focusing mainly on domains of the form $D=\{(x,z)\in R^{l+m}:|z|<H(|x|)\}$, where $d=l+m$ and $H$ is a sufficiently regular function. This class of domains includes various horn-shaped domains and generalized slab domains.