Annals of Probability
- Ann. Probab.
- Volume 30, Number 1 (2002), 213-252.
Time-Reversal in Hyperbolic S.P.D.E.'s
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.
Ann. Probab., Volume 30, Number 1 (2002), 213-252.
First available in Project Euclid: 29 April 2002
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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