Martin boundary of a degenerate Reflected Brownian Motion in a wedge

TL;DR

This paper explicitly computes the Laplace transforms and asymptotics of Green's functions for a degenerate reflected Brownian motion in a wedge, and characterizes its Martin boundary and harmonic functions.

Contribution

It provides explicit formulas for Green's functions, their asymptotics, and the Martin boundary for a specific degenerate reflected Brownian motion.

Findings

Laplace transforms of Green's functions expressed as an infinite sum

Asymptotics of Green's functions along all paths derived

Explicit harmonic functions associated with the process provided

Abstract

We consider an outward degenerate drifted Brownian motion in the quarter plane with oblique reflections on the boundaries. In this article, we explicitly compute the Laplace transforms of the Green's functions associated with the process. These Laplace transforms are expressed as an infinite sum of products by iterating a functional equation, which is deeply linked to the compensation method. We also derive the asymptotics of the Green's functions along all possible paths and determine the (minimal) Martin boundary. Finally, we provide explicit formulae for all the corresponding harmonic functions.