The Annals of Probability

Discrete approximations to reflected Brownian motion

Krzysztof Burdzy and Zhen-Qing Chen

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Abstract

In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in ℝn that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on D∩2kn moving at the rate 2−2k with stationary initial distribution converge weakly in the space D([0, 1], ℝn), equipped with the Skorokhod topology, to the law of the stationary reflected Brownian motion on D. We further show that the following “myopic conditioning” algorithm generates, in the limit, a reflected Brownian motion on any bounded domain D. For every integer k≥1, let {Xkj2k, j=0, 1, 2, …} be a discrete time Markov chain with one-step transition probabilities being the same as those for the Brownian motion in D conditioned not to exit D before time 2k. We prove that the laws of Xk converge to that of the reflected Brownian motion on D. These approximation schemes give not only new ways of constructing reflected Brownian motion but also implementable algorithms to simulate reflected Brownian motion.

Article information

Source
Ann. Probab., Volume 36, Number 2 (2008), 698-727.

Dates
First available in Project Euclid: 29 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1204306964

Digital Object Identifier
doi:10.1214/009117907000000240

Mathematical Reviews number (MathSciNet)
MR2393994

Zentralblatt MATH identifier
1141.60014

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60J60: Diffusion processes [See also 58J65] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 31C25: Dirichlet spaces

Keywords
Reflected Brownian motion random walk killed Brownian motion conditioning martingale tightness Skorokhod space Dirichlet form

Citation

Burdzy, Krzysztof; Chen, Zhen-Qing. Discrete approximations to reflected Brownian motion. Ann. Probab. 36 (2008), no. 2, 698--727. doi:10.1214/009117907000000240. https://projecteuclid.org/euclid.aop/1204306964


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References

  • Bogdan, K., Burdzy, K. and Chen, Z.-Q. (2003). Censored stable processes. Probab. Theory Related Fields 127 89–152.
  • Burdzy, K. and Chen, Z.-Q. (1998). Weak convergence of reflecting Brownian motions. Electron. Comm. Probab. 3 29–33.
  • Burdzy, K., Hołyst, R. and March, P. (2000). A Fleming–Viot particle representation of Dirichlet Laplacian. Comm. Math. Phys. 214 679–703.
  • Burdzy, K. and Quastel, J. (2006). An annihilating–branching particle model for the heat equation with average temperature zero. Ann. Probab. 34 2382–2405.
  • Chen, Z.-Q. (1993). On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Related Fields 94 281–351.
  • Chen, Z.-Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T.-S. (2008). Stochastic calculus for symetric Markov processes. Ann. Probab. To appear.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Fukushima, M. (1967). A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math. 4 183–215.
  • Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • Jerison, D. and Kenig, C. (1982). Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. in Math. 46 80–147.
  • Jones, P. (1981). Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 71–88.
  • Silverstein, M. L. (1974). Symmetric Markov Processes. Springer, Berlin.
  • Stroock, D. W. (1988). Diffusion semigroups corresponding to uniformly elliptic divergence form operator. Lect. Notes Math. 1321 316–347. Springer, Berlin.
  • Stroock, D. W. and Varadhan, S. R. S. (1971). Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 147–225.
  • Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, Berlin.
  • Takeda, M. (1996). Two classes of extensions for generalized Schrödinger operators. Potential Anal. 5 1–13.
  • Varopoulos, N. Th. (2003). Marches aléatoires et théorie du potentiel dans les domaines lipschitziens. C. R. Math. Acad. Sci. Paris 337 615–618.