Energy image density property and the lent particle method for Poisson measures

Abstract

We introduce a new approach to absolute continuity of laws of Poisson functionals. It is based on the energy image density property for Dirichlet forms. The associated gradient is a local operator and gives rise to a nice formula called the lent particle method which consists in adding a particle and taking it back after some calculation.

Application of the lent particle method to Poisson-driven SDEs

“…We have obtained results which simplify highly the approach with local operators (cf [7]). Based on Dirichlet forms on the general Poisson space, they possess the advantages of Dirichlet form methods and simplicity of use.…”

“…In fact, no case is known where (EID) fails for a local Dirichlet structure with carré du champ. Following [7], we suppose that the bottom structure satisfies the following hypotheses:…”

“…-Hypothesis (H0) implies what we call the bottom core hypothesis in [7] and denote (BC). It is a technical condition due to the fact that the carré du champ takes its values in L 1 , and we need a set of test functions on which it has its values in L 2 .…”

“…V.2]) -The need for hypotheses (H1) to (H3) lies in the argument followed in [7] to prove (EID) on the Poisson space. -In the classical applications, hypotheses (H0) to (H4) are fulfilled.…”

“…2.2, 2.3). The only difference with our treatment in [7] is that we work here on a product probability space in order to be able to put also a Brownian motion or other semi-martingales in the studied SDE. With respect to the note [4] we have introduced a clearer new notation, as in [7] the operator ε − is shared from the integration by N .…”