Analytic approach for reflected Brownian motion in the quadrant

TL;DR

This paper extends an analytic approach from discrete quarter-plane random walks to reflected Brownian motion in the quadrant, providing explicit formulas and comparative analysis of the two contexts.

Contribution

It develops an analytic framework for reflected Brownian motion in the quadrant, analogous to the discrete case, and compares discrete and continuous approaches.

Findings

Explicit expressions for generating functions of reflected Brownian motion

Asymptotic analysis of coefficients in the continuous setting

Comparison of discrete and continuous analytic methods

Abstract

Random walks in the quarter plane are an important object both of combinatorics and probability theory. Of particular interest for their study, there is an analytic approach initiated by Fayolle, Iasnogorodski and Malyshev, and further developed by the last two authors of this note. The outcomes of this method are explicit expressions for the generating functions of interest, asymptotic analysis of their coefficients, etc. Although there is an important literature on reflected Brownian motion in the quarter plane (the continuous counterpart of quadrant random walks), an analogue of the analytic approach has not been fully developed to that context. The aim of this note is twofold: it is first an extended abstract of two recent articles of the authors of this paper, which propose such an approach; we further compare various aspects of the discrete and continuous analytic approaches.