Integration by Parts Formula and Applications for SPDEs with Jumps

TL;DR

This paper develops an integration by parts formula for SPDEs with jumps using Malliavin calculus, leading to new inequalities and estimates, especially for SDEs driven by stable-like processes.

Contribution

It introduces a Driver-type integration by parts formula for SPDEs with jumps, expanding the analytical tools available for such equations.

Findings

Established a shift Harnack inequality for the semigroup.

Derived heat kernel estimates for SPDEs with jumps.

Applied results to SDEs driven by stable-like processes.

Abstract

By using the Malliavin calculus and finite jump approximations, the Driver-type integration by parts formula is established for the semigroup associated to stochastic (partial) differential equations with noises containing a subordinate Brownian motion. As applications, the shift Harnack inequality and heat kernel estimates are derived. The main results are illustrated by SDEs driven by $\aa$-stable like processes.