The Stochastic Wave Equation with Multiplicative Fractional Noise: a Malliavin calculus approach

Abstract

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index $H>1/2$, and has a homogeneous spatial covariance structure given by the Riesz kernel of order $\alpha$. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is $\alpha >d-2$, which coincides with the condition obtained in Dalang (1999), when the noise is white in time. Under this condition, we obtain estimates for the $p$-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When $d\le 2$, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.