Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s
Robert C. Dalang, Ecole Polytechnique Fédérale
Abstract
We extend the definition of Walsh's martingale measure
stochastic integral so as to be able to solve stochastic partial differential
equations whose Green's function is not a function but a Schwartz distribution.
This is the case for the wave equation in dimensions greater than two.
Even when the integrand is a distribution, the value of our stochastic
integral process is a real-valued martingale. We use this extended integral
to recover necessary and sufficient conditions under which the linear wave
equation driven by spatially homogeneous Gaussian noise has a process solution,
and this in any spatial dimension. Under this condition, the non-linear
three dimensional wave equation has a global solution. The same methods
apply to the damped wave equation, to the heat equation and to various
parabolic equations.
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