Invariance of Poisson measures under random transformations

Abstract

We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields 103 (1995) 409-429) on the Wiener space, although the corresponding algebra is more complex than in theWiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure. © 2012 Association des Publications de l'Institut Henri Poincaré.