Error calculus and regularity of Poisson functionals : the lent particle method

We propose a new method to apply the Lipschitz functional calculus of local Dirichlet forms to Poisson random measures. Résumé Calcul d’erreur et régularité des fonctionnelles de Poisson : la méthode de la particule prêtée. Nous proposons une nouvelle méthode pour appliquer le calcul fonctionnel lipschitzien des formes de Dirichlet locales aux mesures aléatoires de Poisson. 1 Notation and basic formulae. Let us consider a local Dirichlet structure with carré du champ (X,X , ν,d, γ) where (X,X , ν) is a σ-finite measured space called bottom-space. Singletons are in X and ν is diffuse, d is the domain of the Dirichlet form ǫ[u] = 1/2 ∫ γ[u]dν. We denote (a,D(a)) the generator in L(ν) (cf. [3]). A random Poisson measure associated to (X,X , ν) is denoted N . Ω is the configuration space of countable sums of Dirac masses on X and A is the σ-field generated by N , of law P on Ω. The space (Ω,A,P) is called the up-space. We write N(f) for ∫ fdN . If p ∈ [1,∞[ the set {e : f real, f ∈ L ∩ L(ν)} is total in L C (Ω,A,P). We put Ñ = N − ν. The relation E(Ñf) = ∫ f dν extends and gives sense to Ñ(f), f ∈ L(ν). The Laplace functional and the differential calculus with γ yield ∀f ∈ d, ∀h ∈ D(a) E[e(Ñ(a[h]) + i 2 N(γ[f, h])] = 0. (1) 2 Product, particle by particle, of a Poisson random measure by a probability measure. Given a probability space (R,R, ρ), let us consider a Poisson random measure N ⊙ ρ on (X × R,X × R) with intensity ν × ρ such that for f ∈ L(ν) and g ∈ L(ρ) if ∗Ecole des Ponts, Paris-Est, ParisTech. email: [email protected]