Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity

TL;DR

This paper investigates the behavior of multidimensional reflected Brownian motion in unbounded, generalized parabolic domains, identifying conditions for explosion or superdiffusive transience and developing new criteria for analysis.

Contribution

It introduces natural conditions for explosion and superdiffusivity in such domains, along with novel semimartingale criteria for studying these phenomena.

Findings

Explosion occurs if the domain narrows sufficiently fast at infinity.

Superdiffusive transience is established with a strong law of large numbers.

In planar domains, explosion occurs iff the domain's area is finite.

Abstract

For a multidimensional driftless diffusion in an unbounded, smooth, sub-linear generalized parabolic domain, with oblique reflection from the boundary, we give natural conditions under which either explosion occurs, if the domain narrows sufficiently fast at infinity, or else there is superdiffusive transience, which we quantify with a strong law of large numbers. For example, in the case of a planar domain, explosion occurs if and only if the area of the domain is finite. We develop and apply novel semimartingale criteria for studying explosions and establishing strong laws, which are of independent interest.