Dirichlet Forms for Poisson Measures and Lévy Processes: The Lent Particle Method

With respect to the analysis on Wiener space in the spirit of Malliavin calculus, what brings the Dirichlet forms approach is threefold: a) The arguments hold under only Lipschitz hypotheses, b) A general criterion exists, the energy image density property, (proved on the Wiener space for the Ornstein-Uhlenbeck form but still a conjecture in general since 1986 cf [3]) c) Dirichlet forms are easy to construct in the infinite dimensional frameworks encountered in probability theory and this yields a theory of errors propagation through the stochastic calculus, especially for finance and physics cf [5], but also for numerical analysis of pde and spde cf [6]. Extensions of Malliavin calculus to the case of stochastic differential equations with jumps have been soon proposed and gave rise to an extensive literature. The approach is either dealing with local operators acting on the size of the jumps (cf [7-9] etc.) or based on the Fock space representation of the Poisson space and finite difference operators (cf [10-12] etc.). Jointly with Laurent Denis we have obtained results which simplify highly the approach with local op-erators. They are based on Dirichlet forms on the general Poisson space. Roughly speaking, in order to calculate the Malliavin matrix, we add a particle to the system, compute the gradient of the functional on this particle, and take back the particle before integrating by the Poisson measure. We call this method the lent particle method. Let (X,X, ν,d, γ) be a local symmetric Dirichlet structure which admits a carre ́ du champ operator i.e. (X,X, ν) is a measured space called the bottom space, ν is σ-finite and the bilinear form e[f, g]