The Annals of Probability

The heat equation and reflected Brownian motion in time-dependent domains

Krzysztof Burdzy, Zhen-Qing Chen, and John Sylvester

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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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Ann. Probab., Volume 32, Number 1B (2004), 775-804.

First available in Project Euclid: 11 March 2004

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Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 60J50: Boundary theory 60J60: Diffusion processes [See also 58J65]

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


Burdzy, Krzysztof; Chen, Zhen-Qing; Sylvester, John. The heat equation and reflected Brownian motion in time-dependent domains. Ann. Probab. 32 (2004), no. 1B, 775--804. doi:10.1214/aop/1079021464.

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