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.

A support theorem for 3d-stochastic wave equations in Hölder norm with some general noises

“…(see [9,Theorem 3.1]). In [10] they obtained a similar result for the case of non zero initial conditions but restricted to the case of the function ς being a linear function, this is called the affine case. Their approach is based on the fractional Sobolev imbedding theorem and the Fourier transform technique.…”

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In this paper we characterize the topological support in Hölder norm of the law of the solution to a stochastic wave equation with threedimensional 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.

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“…This paper is an extension and (in some sense) also continuation of [9] and [10] where we prove a characterization of the topological support in Hölder norm for the law of the solution of a stochastic wave equation. The main difference in the method used here, with respect to the one used in [9] and [10], is very technical and it is described in the Remark 2.1 below.…”

“…As in [9] (see also [10]) we apply the approximation method of [17] to obtain a characterization of the topological support of the law of u (the solution to (7)) in the Hölder norm · ρ,t0,K . The core of the work consists of an approximation result for a family of equations more general than equation (7) by a sequence of pathwise evolution equations obtained by a smooth approximation of the driving process M .…”

Approximation of solutions of the stochastic wave equation by using the Fourier series

“…Convergence of solutions of stochastic wave equation by using the approximation of stochastic integrator was studied in [3,4]. Mild solutions of equations driven by the Gaussian random field in dimension three were considered in these papers.…”