Sigma-convergence of semilinear stochastic wave equations

TL;DR

This paper develops a sigma-convergence method to analyze the homogenization of semilinear stochastic wave equations with oscillating coefficients, integrating both random and deterministic aspects.

Contribution

It introduces a novel sigma-convergence approach for hyperbolic SPDEs in ergodic algebras, including schemes for approximating effective coefficients.

Findings

Successfully homogenized various classes of oscillating coefficients

Extended sigma-convergence to stochastic hyperbolic equations

Provided concrete examples like periodic and almost periodic homogenization

Abstract

We address the homogenization of a semilinear hyperbolic stochastic partial differential equation with highly oscillating coefficients, in the context of ergodic algebras with mean value. To achieve our goal, we use a suitable variant of the sigma-convergence concept that takes into account both the random and deterministic behaviours of the phenomenon modelled by the underlying problem. We also provide an appropriate scheme for the approximation of the effective coefficients. To illustrate our approach, we work out some concrete problems such as the periodic homogenization problem, the almost periodic and the asymptotically almost periodic ones, and many more besides.