Long time asymptotics for constrained diffusions in polyhedral domains

• Published in: Stochastic processes and their applications Vol. 117; no. 8; pp. 1014 - 1036
• Main Authors: Budhiraja, Amarjit, Lee, Chihoon
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.08.2007; Elsevier Science; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: [formula omitted]-Irreducibility; [formula omitted]-Uniform ergodicity; Constrained diffusions; Distribution theory; Exact sciences and technology; Functional central limit theorems; Geometric ergodicity; Heavy traffic; Limit theorems; Mathematics; Moment stability; Poisson equation; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Semimartingale reflecting Brownian motion; Semimartingale reflecting Brownian motion [phi]-Irreducibility V-Uniform ergodicity Geometric ergodicity Constrained diffusions Heavy traffic Poisson equation Moment stability Functional central limit theorems; Statistics; Stochastic analysis; Stochastic processes; secondary; V -Uniform ergodicity; Functional central limit theorems; Heavy traffic; Moment stability; Geometric ergodicity; Poisson equation; φ -Irreducibility; Semimartingale reflecting Brownian motion; Constrained diffusions; primary; primary 60J65; secondary 60H35; 60K25; 93E15; Probability distribution; Dynamical system; Stochastic process; Queueing network; Invariant measure; Diffusion coefficient; Semimartingale; Vector field; Numerical stability; Generating function; Statistical moment; Asymptotic behavior; Probability; Ergodicity; Polynomial time; Functional; Brownian motion; Semimartingale reflecting Brownian motion; φ-Irreducibility; V-Uniform ergodicity; Geometric ergodicity; Constrained diffusions; Heavy traffic; Poisson equation; Moment stability; Functional central limit theorems; Central limit theorem; Deterministic system; Application; Lyapunov function
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2006.11.007

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.