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Rates of Convergence to Stationarity for Reflected Brownian Motion

Author

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  • Jose Blanchet

    (Stanford University, Stanford, California 94305;)

  • Xinyun Chen

    (The Chinese University of Hong Kong, Shenzhen, Shenzhen, China)

Abstract

We provide the first rate of convergence to stationarity analysis for reflected Brownian motion (RBM) as the dimension grows under some uniformity conditions. In particular, if the underlying routing matrix is uniformly contractive, uniform stability of the drift vector holds, and the variances of the underlying Brownian motion (BM) are bounded, then we show that the RBM converges exponentially fast to stationarity with a relaxation time of order O ( d 4 ( l o g ( d ) ) 3 ) as the dimension d → ∞. Our bound for the relaxation time follows as a corollary of the nonasymptotic bound we obtain for the initial transient effect, which is explicit in terms of the RBM parameters.

Suggested Citation

  • Jose Blanchet & Xinyun Chen, 2020. "Rates of Convergence to Stationarity for Reflected Brownian Motion," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 660-681, May.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:2:p:660-681
    DOI: 10.1287/moor.2019.1006
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    References listed on IDEAS

    as
    1. Budhiraja, Amarjit & Lee, Chihoon, 2007. "Long time asymptotics for constrained diffusions in polyhedral domains," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 1014-1036, August.
    2. Jose Blanchet & Xinyun Chen, 2019. "Perfect Sampling of Generalized Jackson Networks," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 693-714, May.
    3. S. Creemers & M. Lambrecht, 2010. "Modeling a hospital queueing network," Post-Print hal-00814195, HAL.
    4. Amarjit Budhiraja & Chihoon Lee, 2009. "Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 45-56, February.
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    Cited by:

    1. Yanlin Qu & Jose Blanchet & Peter Glynn, 2026. "Deep learning for Markov chains: Lyapunov functions, Poisson’s equation, and stationary distributions," Queueing Systems: Theory and Applications, Springer, vol. 110(1), pages 1-25, March.

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