A Feynman-Kac-type formula for the deterministic and stochastic wave equations

TL;DR

This paper develops a probabilistic Feynman-Kac-type formula applicable to various linear PDEs, including wave, heat, telegraph, and beam equations, and extends to stochastic cases with Gaussian noise.

Contribution

It introduces a unified probabilistic representation for linear PDEs with potential terms, encompassing both deterministic and stochastic equations, including the wave equation in multiple dimensions.

Findings

Provides a probabilistic formula for wave equations in 1-3 dimensions.

Extends the Feynman-Kac formula to the heat, telegraph, and beam equations.

Derives expressions for moments of solutions under Gaussian noise.

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.