Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann-Hilbert problem via Fourier transforms

TL;DR

This paper develops a method to analyze stationary reflected Brownian motion in a three-quarter plane by reducing the problem to a Riemann-Hilbert boundary value problem using Fourier transforms, extending techniques from quarter plane models.

Contribution

It introduces a novel approach using Fourier transforms to solve for the stationary distribution in a three-quarter plane, connecting it to known quarter plane boundary value problems.

Findings

Stationary distribution can be obtained via a boundary value problem similar to quarter plane models.

Fourier transforms facilitate deriving a functional equation in complex variables.

The approach reveals dualities and symmetries in the three-quarter plane model.

Abstract

The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the stationary distribution can indeed be found by solving a boundary value problem of the same kind as the one encountered in the quarter plane, up to various dualities and symmetries. The main idea is to start from Fourier (and not Laplace) transforms, allowing to get a functional equation for a single function of two complex variables.