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

TL;DR

This paper studies the Malliavin differentiability of solutions to high-dimensional nonlinear stochastic PDEs, establishing conditions for absolute continuity of the solution law, especially for the stochastic wave equation in dimensions four and higher.

Contribution

It extends the integration theory for Hilbert space-valued integrands and proves Malliavin differentiability and density existence for solutions of certain stochastic PDEs.

Findings

Solutions are Malliavin differentiable at fixed points.

Density of the solution law exists in additive noise cases.

Results apply to stochastic wave equations in dimensions ≥ 4.

Abstract

We consider the class of non-linear stochastic partial differential equations studied in \cite{conusdalang}. 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 $(t,x)\in[0,T]\times \Rd$ is differentiable in the Malliavin sense. For this, an extension of the integration theory in \cite{conusdalang} 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 $(t,x)\in]0,T]\times\Rd$. The results apply to the stochastic wave equation in spatial dimension $d\ge 4$.