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

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. Institut de mathématiques, Ecole Polytechnique Fédérale, Station 8, 1015 Lausanne, Switzerland. [email protected] Partially supported by the Swiss National Foundation for Scientific Research. Department of Mathematics, University of Rochester, Rochester, NY 14627, USA. [email protected] Partially supported by an NSF grant. Department of Mathematics, University of Warwick, CV4 7AL, UK. [email protected] MSC 2000 Subject Classifications. Primary: 60H15; Secondary: 60H20.