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

Krzysztof Burdzy; Zhen Qing Chen; John Sylvester

doi:10.1214/aop/1079021464

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

Krzysztof Burdzy^*, Zhen Qing Chen, John Sylvester

^*Corresponding author for this work

University of Washington

Research output: Contribution to journal › Article › peer-review

48 Citations (Scopus)

Abstract

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 C³-smooth time-dependent domains in the n-dimensional Euclidean space Rⁿ. 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.

Original language	English
Pages (from-to)	775-804
Number of pages	30
Journal	Annals of Probability
Volume	32
Issue number	1 B
DOIs	https://doi.org/10.1214/aop/1079021464
Publication status	Published - Jan 2004
Externally published	Yes

Keywords

Feynman-kac formula
Girsanov transform
Heat equation with boundary conditions
Local time
Probabilistic representation
Reflecting brownian motion
Sko-rohod decomposition
Time-dependent domain
Time-inhomogeneous strong markov process
Time-reversal

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author = "Krzysztof Burdzy and Chen, {Zhen Qing} and John Sylvester",

year = "2004",

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T1 - The heat equation and reflected Brownian motion in time-dependent domains

AU - Burdzy, Krzysztof

AU - Chen, Zhen Qing

AU - Sylvester, John

PY - 2004/1

Y1 - 2004/1

N2 - 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 C3-smooth time-dependent domains in the n-dimensional Euclidean space Rn. 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.

AB - 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 C3-smooth time-dependent domains in the n-dimensional Euclidean space Rn. 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.

KW - Feynman-kac formula

KW - Girsanov transform

KW - Heat equation with boundary conditions

KW - Local time

KW - Probabilistic representation

KW - Reflecting brownian motion

KW - Sko-rohod decomposition

KW - Time-dependent domain

KW - Time-inhomogeneous strong markov process

KW - Time-reversal

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DO - 10.1214/aop/1079021464

M3 - Article

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SN - 0091-1798

VL - 32

SP - 775

EP - 804

JO - Annals of Probability

JF - Annals of Probability

IS - 1 B

ER -

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

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Keywords

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