Stationary Distribution of Brownian Motion in the Half-Plane with Two-sided Reflections

TL;DR

This paper characterizes the stationary distribution of a reflecting Brownian motion in the upper half-plane with constant reflection directions, deriving explicit formulas and analyzing boundary behaviors.

Contribution

It provides an explicit expression for the Laplace transform of the stationary distribution and analyzes its boundary behavior using advanced complex analysis techniques.

Findings

Explicit Laplace transform of the stationary distribution derived.

Boundary behavior at the origin and axes characterized.

Asymptotic properties of the stationary density established.

Abstract

We investigate the unique stationary measure of a positive recurrent reflecting Brownian motion in the upper half-plane, where the direction of reflection is constant on each half-axis. The Laplace transform of the stationary distribution is characterized by a functional equation, whose resolution is reduced to solving a discontinuous Riemann boundary value problem. By applying the Sokhotski-Plemelj formulas, we derive an explicit expression for the Laplace transform. Finally, we establish the local behavior of the stationary density at the origin and its asymptotics along the boundary axes using Tauberian theorems and asymptotic analysis.