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.
Publié le : 2004-01-14
Classification:
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,
60H30,
60J45,
35K20,
60J50,
60J60
@article{1079021464,
author = {Burdzy, Krzysztof and Chen, Zhen-Qing and Sylvester, John},
title = {The heat equation and reflected Brownian motion in time-dependent domains},
journal = {Ann. Probab.},
volume = {32},
number = {1A},
year = {2004},
pages = { 775-804},
language = {en},
url = {http://dml.mathdoc.fr/item/1079021464}
}
Burdzy, Krzysztof; Chen, Zhen-Qing; Sylvester, John. The heat equation and reflected Brownian motion in time-dependent domains. Ann. Probab., Tome 32 (2004) no. 1A, pp. 775-804. http://gdmltest.u-ga.fr/item/1079021464/