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January 2004 The heat equation and reflected Brownian motion in time-dependent domains
Krzysztof Burdzy, Zhen-Qing Chen, John Sylvester
Ann. Probab. 32(1B): 775-804 (January 2004). DOI: 10.1214/aop/1079021464


The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as "noncylindrical domains,'' and its connections with partial differential equations. Construction is given for RBM in $C^3$-smooth time-dependent domains in the n-dimensional Euclidean space $\R^n$. We present various sample path properties of the process, two-sided estimates for its transition density function, and a probabilistic representation of solutions to some partial differential equations. Furthermore, the one-dimensional case is thoroughly studied, with the assumptions on the smoothness of the boundary drastically relaxed.


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Krzysztof Burdzy. Zhen-Qing Chen. John Sylvester. "The heat equation and reflected Brownian motion in time-dependent domains." Ann. Probab. 32 (1B) 775 - 804, January 2004.


Published: January 2004
First available in Project Euclid: 11 March 2004

zbMATH: 1046.60060
MathSciNet: MR2039943
Digital Object Identifier: 10.1214/aop/1079021464

Primary: 35K20 , 60H30 , 60J45
Secondary: 60J50 , 60J60

Keywords: Feynman--Kac formula , Girsanov transform , heat equation with boundary conditions , Local time , probabilistic representation , Reflecting Brownian motion , Skorohod decomposition , time-dependent domain , time-inhomogeneous strong Markov process , Time-reversal

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 1B • January 2004
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