Obliquely reflecting Brownian motion in nonpolyhedral, piecewise smooth cones, with an example of application to diffusion approximation of bandwidth sharing queues

TL;DR

This paper establishes conditions for the uniqueness of obliquely reflecting Brownian motion in nonpolyhedral cones, crucial for diffusion approximations in bandwidth sharing networks, using advanced ergodic theorems.

Contribution

It provides new sufficient conditions for the uniqueness in law of obliquely reflecting Brownian motion in nonpolyhedral cones, extending previous methods with recent ergodic theorems.

Findings

Verified conditions for diffusion approximation in bandwidth sharing networks.

Ensured the conjectured limit's uniqueness in complex cone domains.

Replaced classical theorems with modern ergodic results for broader applicability.

Abstract

This work gives sufficient conditions for uniqueness in law of semimartingale, obliquely reflecting Brownian motion in a nonpolyhedral, piecewise ${\cal C}^2$ cone, with radially constant, Lipschitz continuous direction of reflection on each face. The conditions are shown to be verified by the conjectured diffusion approximation to the workload in a class of bandwidth sharing networks, thus ensuring that the conjectured limit is uniquely characterized. This is a key step in proving the diffusion approximation. This uniqueness result is made possible by replacing the Krein-Rutman theorem used by Kwon and Williams (1993) in a smooth cone with the recent reverse ergodic theorem for inhomogeneous, killed Markov chains of Costantini and Kurtz (2024).