Long time asymptotics for constrained diffusions in polyhedral domains

Amarjit Budhiraja, Chihoon Lee

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39 Scopus citations

Abstract

We study long time asymptotic properties of constrained diffusions that arise in the heavy traffic analysis of multiclass queueing networks. We first consider the classical diffusion model with constant coefficients, namely a semimartingale reflecting Brownian motion (SRBM) in a d-dimensional positive orthant. Under a natural stability condition on a related deterministic dynamical system [P. Dupuis, R.J. Williams, Lyapunov functions for semimartingale reflecting brownian motions, Annals of Probability 22 (2) (1994) 680-702] showed that an SRBM is ergodic. We strengthen this result by establishing geometric ergodicity for the process. As consequences of geometric ergodicity we obtain finiteness of the moment generating function of the invariant measure in a neighborhood of zero, uniform time estimates for polynomial moments of all orders, and functional central limit results. Similar long time properties are obtained for a broad family of constrained diffusion models with state dependent coefficients under a natural condition on the drift vector field. Such models arise from heavy traffic analysis of queueing networks with state dependent arrival and service rates.

Original languageEnglish
Pages (from-to)1014-1036
Number of pages23
JournalStochastic Processes and their Applications
Volume117
Issue number8
DOIs
StatePublished - Aug 2007

Keywords

  • Constrained diffusions
  • Functional central limit theorems
  • Geometric ergodicity
  • Heavy traffic
  • Moment stability
  • Poisson equation
  • Semimartingale reflecting Brownian motion
  • V-Uniform ergodicity
  • φ-Irreducibility

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