Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case

Abstract

This paper is a continuation of (Bernoulli 20 (2014) 2169–2216) where we prove a characterization of the support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable and null initial conditions. Here, we allow for non-null initial conditions and, therefore, the solution does not possess a stationary property in space. As in (Bernoulli 20 (2014) 2169–2216), the support theorem is a consequence of an approximation result, in the convergence of probability, of a sequence of evolution equations driven by a family of regularizations of the driving noise. However, the method of the proof differs from (Bernoulli 20 (2014) 2169–2216) since arguments based on the stationarity property of the solution cannot be used.