Some Non-Linear S.P.D.E's That Are Second Order In Time
Robert C. Dalang, Ecole Polytechnique Fédérale
Carl Mueller, University of Rochester
Abstract
We extend J.B. Walsh's theory of martingale measures
in order to deal with stochastic partial differential equations that
are second order in time, such as the wave equation and the beam
equation, and driven by spatially homogeneous Gaussian noise. For such
equations, the fundamental solution can be a distribution in the sense
of Schwartz, which appears as an integrand in the reformulation of the
s.p.d.e. as a stochastic integral equation. Our approach provides an
alternative to the Hilbert space integrals of Hilbert-Schmidt
operators. We give several examples, including the beam equation and
the wave equation, with nonlinear multiplicative noise terms.
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