Obliquely reflected Brownian motion in non-smooth planar domains

TL;DR

This paper develops a method to construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth ones, using conformal mappings and excursion theory, and characterizes these processes via harmonic functions.

Contribution

It introduces a new construction technique for obliquely reflected Brownian motions in non-smooth domains and provides an alternative characterization in smooth domains.

Findings

Constructed reflected Brownian motions in all bounded simply connected planar domains.

Provided an alternative characterization using harmonic functions and rotation rates.

Showed convergence of reflected Brownian motions in smooth domains to those in Jordan domains.

Abstract

We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary distribution of the process, and a real number that represents, in a suitable sense, the asymptotic rate of rotation of the process around a reference point in the domain. Furthermore, we also show that any obliquely reflected Brownian motion in a simply connected Jordan domain can be obtained as a suitable limit of obliquely reflected Brownian motions in smooth domains.