This paper studies questions of changes of variables in a class of
hyperbolic stochastic partial differential equations in two variables driven by
white noise. Two types of changes of variables are considered: naive changes of
variables which do not involve a change of filtration, which affect the
equation much as though it were deterministic, and changes of variables that do
involve a change of filtration, such as time-reversals. In particular, if the
process in reversed coordinates does satisfy an s.p.d.e., then we show how this
s.p.d.e. is related to the original one. Time-reversals for the Brownian sheet
and for equations with constant coefficients are considered in detail. A
necessary and sufficient condition is provided under which the reversal of the
solution to the simplest hyperbolic s.p.d.e. with certain random initial
conditions again satisfies such an s.p.d.e. This yields a negative conclusion
concerning the reversal in time of the solution to the stochastic wave equation
(in one spatial dimension) driven by white noise.
Publié le : 2002-01-14
Classification:
Hyperbolic stochastic partial differential
equations,
time reversal,
changes of variables,
Brownian sheet,
infinite dimensional diffusions,
60H15,
60G15,
35R60
@article{1020107766,
author = {Dalang, Robert C. and Walsh, John B.},
title = {Time-Reversal in Hyperbolic S.P.D.E.'s},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 213-252},
language = {en},
url = {http://dml.mathdoc.fr/item/1020107766}
}
Dalang, Robert C.; Walsh, John B. Time-Reversal in Hyperbolic S.P.D.E.'s. Ann. Probab., Tome 30 (2002) no. 1, pp. 213-252. http://gdmltest.u-ga.fr/item/1020107766/