A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.’s

Dalang, Robert; Mueller, Carl; Tribe, Roger

doi:10.1090/S0002-9947-08-04351-1

A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.’s
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by Robert C. Dalang, Carl Mueller and Roger Tribe PDF

Trans. Amer. Math. Soc. 360 (2008), 4681-4703 Request permission

Abstract:

We establish a probabilistic representation for a wide class of linear deterministic p.d.e.’s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution.

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