Malliavin differentiability of solutions of SPDEs with L\'evy white noise

TL;DR

This paper studies the Malliavin differentiability of solutions to SPDEs driven by Le9vy white noise, establishing conditions for existence, uniqueness, and differentiability, with applications to heat and wave equations.

Contribution

It introduces Malliavin calculus for Le9vy white noise SPDEs and proves differentiability results for solutions with affine c, extending stochastic analysis tools.

Findings

Unique solutions for SPDEs with Le9vy white noise under certain conditions

Malliavin differentiability established for solutions when c is affine

Solution derivatives satisfy stochastic integral equations

Abstract

In this article, we consider a stochastic partial differential equation (SPDE) driven by a L\'evy white noise, with Lipschitz multiplicative term $\sigma$. We prove that under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the L\'evy white noise. If $\sigma$ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.