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

Abstract

ABSTRACT In this paper, we characterize the topological support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable. This note is an extension of Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597]. The result presented here characterize a more general type of stochastic wave equations in 3-d space variable than those considered in Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597]. Here we extend these two previous results in the folowing sense. 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 of the mean Hölder continuity of such covariance function. As in Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597] 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.