Time-Reversal in Hyperbolic S.P.D.E.'s



The Annals of Probability

Time-Reversal in Hyperbolic S.P.D.E.'s

Robert C. Dalang and John B. Walsh

Source: Ann. Probab. Volume 30, Number 1 (2002), 213-252.

Abstract

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.

Primary Subjects: 60H15
Secondary Subjects: 60G15, 35R60
Keywords: Hyperbolic stochastic partial differential equations; time reversal; changes of variables; Brownian sheet; infinite dimensional diffusions

Full-text: Open access

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1020107766
Mathematical Reviews number (MathSciNet): MR
Digital Object Identifier: doi:10.1214/aop/1020107766
Zentralblatt MATH identifier: 1019.60063

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VANCOUVER, BC V6T1Z2 CANADA E-MAIL: walsh@math.ubc.ca

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