Annals of Probability

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

Robert C. Dalang and John B. Walsh

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 29 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1020107766

Digital Object Identifier
doi:10.1214/aop/1020107766

Mathematical Reviews number (MathSciNet)
MR1894106

Zentralblatt MATH identifier
1019.60063

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60G15: Gaussian processes 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Keywords
Hyperbolic stochastic partial differential equations time reversal changes of variables Brownian sheet infinite dimensional diffusions

Citation

Dalang, Robert C.; Walsh, John B. Time-Reversal in Hyperbolic S.P.D.E.'s. Ann. Probab. 30 (2002), no. 1, 213--252. doi:10.1214/aop/1020107766. https://projecteuclid.org/euclid.aop/1020107766


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