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