A proof of a support theorem for stochastic wave equations in Holder norm with some general noises

TL;DR

This paper proves a support theorem for solutions to stochastic wave equations in Hölder norm, extending previous results to include multiplicative noise, non-zero initial conditions, and covariance functions with mean Hölder continuity.

Contribution

It extends the support theorem for stochastic wave equations to cover multiplicative noise, non-zero initial conditions, and more general covariance functions.

Findings

Characterized the topological support in Hölder norm of the solution law.

Proved an approximation theorem for evolution equations driven by regularized noise.

Extended previous results to more general noise and initial conditions.

Abstract

In this paper, we characterize the topological support in Holder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable is proved. This note is a continuation of [9] and [10]. The result is a consequence of an approximation theorem, in the convergence of probability, for a sequence of evolution equations driven by a family of regularizations of the driving noise. We extend two previous results on this subject. The first extension is that we cover the case of multiplicative noise and non-zero initial conditions. The second extension is related to the covariance function associated with the noise, here we follow the approach of Hu, Huang and Nualart and ask conditions in terms the of the mean Holder continuity of such covariance function.