The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity

Abstract

We consider the class of non-linear stochastic partial differential equations studied in [Conus-Dalang, 2008]. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point $(t,x)\in[0,T]\times \mathbb{R}^d$ is differentiable in the Malliavin sense. For this, an extension of the integration theory in [Conus-Dalang, 2008] to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise integrals are proved. In the particular case of equations with additive noise, we establish the existence of density for the law of the solution at $(t,x)\in]0,T]\times\mathbb{R}^d$. The results apply to the stochastic wave equation in spatial dimension $d\ge 4$.