Moment identities for Poisson-Skorohod integrals and application to measure invariance

The classical invariance theorem for Poisson measures states that given a deterministic transformation τ : X → Y between measure spaces (X, σ) and (Y, μ) sending σ to μ, the corresponding transformation on point processes maps the Poisson distribution πσ with intensity σ(dx) on X to the Poisson distribution πμ with intensity μ(dy) on Y . In this note we present sufficient conditions for the invariance of random transformations τ : Ω ×X → Y of Poisson random measures on metric spaces. Our results are inspired by the treatment of the Wiener case in [8], see [6] for a recent simplified proof. However, the use of finite difference operators instead of derivation operators as in the continuous case makes the proofs and arguments more complex from an algebraic point of view. Here the almost sure isometry condition on R assumed in the Gaussian case will be replaced by an almost sure condition on the preservation of intensity measures and, as in the Wiener case, we will characterize probability measures via their moments.