Stratonovich solution for the wave equation

TL;DR

This paper constructs a Stratonovich solution for the stochastic wave equation in low dimensions with specific noise conditions, providing a probabilistic representation and comparing chaos expansions with existing solutions.

Contribution

It introduces a novel Stratonovich solution for the stochastic wave equation with spatially homogeneous noise satisfying a strong integrability condition, extending previous frameworks.

Findings

Probabilistic representation similar to Feynman-Kac formula.

Chaos expansion of the Stratonovich solution provided.

Comparison with Skorohod solution chaos expansion.

Abstract

In this article, we construct a Stratonovich solution for the stochastic wave equation in spatial dimension $d \leq 2$, with time-independent noise and linear term $\sigma(u)=u$ multiplying the noise. The noise is spatially homogeneous and its spectral measure satisfies an integrability condition which is stronger than Dalang's condition. We give a probabilistic representation for this solution, similar to the Feynman-Kac-type formula given in Dalang, Mueller and Tribe (2008) for the solution of the stochastic wave equation with spatially homogeneous Gaussian noise, that is white in time. We also give the chaos expansion of the Stratonovich solution and we compare it with the chaos expansion of the Skorohod solution from Balan, Chen and Chen (2020).