Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm

Abstract

A characterization of the support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable is proved. 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.

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

“…the regularity of the probability measure induced by the solution [26,24,31], large deviation principles [25,19], Varadhan estimates [20,30], support theorems [21,16], path properties such as Hölder continuity [29,12] and much more. See also the references in these works for a more detailed account.…”

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

“…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.…”

“…The main objective in [9] is to prove that, in the particular case v 0 =ṽ 0 = 0, the topological support of the law of the solution to (7) in the space E ρ ([t 0 , T ] × K) with ρ ∈ 0, 2−β 2 is the closure in the Hölder norm of the set {Φ h , h ∈ H T }, for any t 0 > 0. (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.…”

“…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.…”