Multi-level reflecting Brownian motion on the half line and its stationary distribution

TL;DR

This paper studies a semi-martingale reflecting Brownian motion on the half-line with state-dependent drift and variance changes, establishing its unique stationary distribution and analyzing its stability and irreducibility.

Contribution

It introduces a new analysis of a state-dependent reflecting Brownian motion, providing existence, uniqueness, and explicit stationary distribution results.

Findings

Unique weak solution for the reflecting diffusion process.

Explicit analytic expression for the stationary distribution.

Conditions for Harris irreducibility and positive recurrence.

Abstract

A semi-martingale reflecting Brownian motion is a popular process for diffusion approximations of queueing models including their networks. In this paper, we are concerned with the case that it lives on the nonnegative half-line, but the drift and variance of its Brownian component discontinuously change at its finitely many states. This reflecting diffusion process naturally arises from a state-dependent single server queue, studied by the author (2024). Our main interest is in its stationary distribution, which is important for application. We define this reflecting diffusion process as the solution of a stochastic integral equation, and show that it uniquely exists in the weak sense. This result is also proved in a different way by Atar, Castiel and Reiman (2022,2023). In this paper, we consider its Harris irreducibility and stability, that is, positive recurrence, and derive its stationary distribution under this stability condition. The stationary distribution has a simple analytic expression, likely extendable to a more general state-dependent SRBM. Our proofs rely on the generalized Ito formula for a convex function and local time.