Poisson Malliavin calculus in Hilbert space with an application to SPDE

TL;DR

This paper develops a Hilbert space-valued Malliavin calculus for Poisson random measures, providing a simple framework applicable to SPDEs and analyzing convergence rates for equations with stable noise.

Contribution

It introduces a novel, elementary approach to Malliavin calculus for Poisson measures in Hilbert spaces, with applications to stochastic PDEs and convergence analysis.

Findings

Applicable to space-time SPDEs with minimal conditions

Weak convergence order is α times the strong order for stable noise

Simplifies the treatment of Malliavin operators in Poisson settings

Abstract

In this paper we introduce a Hilbert space-valued Malliavin calculus for Poisson random measures. It is solely based on elementary principles from the theory of point processes and basic moment estimates, and thus allows for a simple treatment of the Malliavin operators. The main part of the theory is developed for general Poisson random measures, defined on a $\sigma$-finite measure space, with minimal conditions. The theory is shown to apply to a space-time setting, suitable for studying stochastic partial differential equations. As an application, we analyze the weak order of convergence of space-time approximations for a class of linear equations with $\alpha$-stable noise, $\alpha\in(1,2)$. For a suitable class of test functions, the weak order of convergence is found to be $\alpha$ times the strong order.