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

TL;DR

This paper studies the stochastic wave equation driven by fractional noise in time and Riesz kernel spatial covariance, establishing existence, regularity, and Malliavin differentiability of solutions using a Malliavin calculus approach.

Contribution

It extends the analysis of stochastic wave equations to fractional-in-time noise with Riesz kernel covariance, providing existence conditions and regularity results.

Findings

Existence of solutions under the condition lpha > d-2.

Solutions exhibit Hf6lder continuity.

Solutions are infinitely Malliavin differentiable.

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 H\"older continuity, and we show that the solution is Malliavin differentiable of any order. When $d \leq 2$, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.