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

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.