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

Cristina Costantini

TL;DR

This work addresses the long-standing question of uniqueness in law for semimartingale obliquely reflecting Brownian motion (ORBM) in nonpolyhedral, piecewise smooth cones, with direct relevance to diffusion limits for bandwidth-sharing networks. It develops sufficient conditions (G) and (auxfunc) and employs a novel reverse ergodic theorem for inhomogeneous, killed Markov chains to establish uniqueness, avoiding the Krein-Rutman framework. The authors verify these conditions in a concrete bandwidth-sharing diffusion example, demonstrating unique characterizations of the conjectured limit under certain parameter regimes and detailing the origin's behavior (immediate departure or never hitting) as part of the argument. This advances diffusion-approximation theory for networks with curved cone geometries and cusps, providing a rigorous foundation for the conjectured ORBM limits and guiding future invariance-principle results for nonpolyhedral domains.

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).

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

TL;DR

This work addresses the long-standing question of uniqueness in law for semimartingale obliquely reflecting Brownian motion (ORBM) in nonpolyhedral, piecewise smooth cones, with direct relevance to diffusion limits for bandwidth-sharing networks. It develops sufficient conditions (G) and (auxfunc) and employs a novel reverse ergodic theorem for inhomogeneous, killed Markov chains to establish uniqueness, avoiding the Krein-Rutman framework. The authors verify these conditions in a concrete bandwidth-sharing diffusion example, demonstrating unique characterizations of the conjectured limit under certain parameter regimes and detailing the origin's behavior (immediate departure or never hitting) as part of the argument. This advances diffusion-approximation theory for networks with curved cone geometries and cusps, providing a rigorous foundation for the conjectured ORBM limits and guiding future invariance-principle results for nonpolyhedral domains.

Abstract

This work gives sufficient conditions for uniqueness in law of semimartingale, obliquely reflecting Brownian motion in a nonpolyhedral, piecewise 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).
Paper Structure (22 sections, 15 theorems, 152 equations)

This paper contains 22 sections, 15 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. An example from diffusion approximation of bandwidth sharing queues
  4. Diffusion approximation of bandwidth sharing queues
  5. The example
  6. Outline of the proof of Theorem \ref{['th:exuniq']}
  7. Uniqueness in a domain ${\cal W}\cap(\overline {B_{\delta}})^ c\cap B_{\delta'}$
  8. Starting off the origin, the origin is never reached
  9. Starting at the origin, the origin is immediately left
  10. If any two solutions starting at the origin have the same hitting distributions on $\partial B_{\delta}$, uniqueness holds for solutions starting at the origin
  11. The ergodic argument and conclusion of the proof of Theorem \ref{['th:exuniq']}
  12. Uniqueness for semimartingale ORBM in a piecewise ${\cal C}^2$ cone
  13. Formulation of the problem and main result
  14. Constrained martingale problems
  15. Localization
  16. ...and 7 more sections

Key Result

Theorem 2.3

If $\mu >\sqrt {\frac{3}{\sqrt {2}-1}}$, uniqueness in distribution holds for $(eq:bandSDER)$. If $X$ starts at $x\in\overline {{\cal W}}-\{0\}$, If $X$ starts at $0$,

Theorems & Definitions (27)

  • Conjecture 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1
  • Remark 3.2
  • Remark 3.4
  • Theorem 3.6
  • ...and 17 more