Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise

TL;DR

This paper establishes sharp weak convergence rates for spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise, extending previous results limited to additive noise.

Contribution

It provides the first known weak convergence rates for semilinear stochastic wave equations with multiplicative noise, using advanced probabilistic and analytical techniques.

Findings

Sharp weak convergence rates are proven for multiplicative noise cases.

Results include essentially sharp rates for the hyperbolic Anderson model.

Methodology involves Kolmogorov equations and mild Itô formulas.

Abstract

Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the case of approximation results for strong convergence rates, semilinear stochastic wave equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic wave equation is constant, that is, it is assumed that the considered wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish sharp weak convergence rates for semilinear stochastic wave equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation, the H\"older-inequality for Schatten norms, and the mild It\^o formula.