Asymptotics for the Green's functions of a transient reflected Brownian motion in a wedge

TL;DR

This paper derives precise asymptotic expansions for Green's functions of a transient obliquely reflected Brownian motion in a wedge, using complex analysis and saddle point methods on the Laplace transform's Riemann surface.

Contribution

It provides a novel asymptotic analysis of Green's functions for reflected Brownian motion in a wedge, including a kernel functional equation and singularity analysis.

Findings

Asymptotic expansions valid in all directions

Explicit characterization of the Laplace transform singularities

Application of saddle point method on Riemann surface

Abstract

We consider a transient Brownian motion reflected obliquely in a two-dimensional wedge. A precise asymptotic expansion of Green's functions is found in all directions. To this end, we first determine a kernel functional equation connecting the Laplace transforms of the Green's functions. We then extend the Laplace transforms analytically and study its singularities. We obtain the asymptotics applying the saddle point method to the inverse Laplace transform on the Riemann surface generated by the kernel.