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January 2002 Time-Reversal in Hyperbolic S.P.D.E.'s
Robert C. Dalang, John B. Walsh
Ann. Probab. 30(1): 213-252 (January 2002). DOI: 10.1214/aop/1020107766


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.


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Robert C. Dalang. John B. Walsh. "Time-Reversal in Hyperbolic S.P.D.E.'s." Ann. Probab. 30 (1) 213 - 252, January 2002.


Published: January 2002
First available in Project Euclid: 29 April 2002

zbMATH: 1019.60063
Digital Object Identifier: 10.1214/aop/1020107766

Primary: 60H15
Secondary: 35R60 , 60G15

Keywords: Brownian sheet , changes of variables , hyperbolic stochastic partial differential equations , infinite dimensional diffusions , Time reversal

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.30 • No. 1 • January 2002
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