Drichlet forms for Poisson measures and L\'evy processes : the lent particle method

TL;DR

This paper introduces a novel method using local Dirichlet forms to analyze the absolute continuity of Poisson functionals, involving a particle addition and removal technique, with applications to Poisson-driven SDEs.

Contribution

It develops a new explicit calculus based on Dirichlet forms for Poisson measures, providing a fresh approach to studying their laws and properties.

Findings

New explicit calculus for Poisson functionals

Application to SDEs driven by Poisson measures

Enhanced understanding of absolute continuity in Poisson settings

Abstract

We present a new approach to absolute continuity of laws of Poisson functionals. The theoretical framework is that of local Dirichlet forms as a tool to study probability spaces. The method gives rise to a new explicit calculus that we show first on some simple examples : it consists in adding a particle and taking it back after computing the gradient. Then we apply it to SDE's driven by Poisson measure.