Invariance of Poisson measures under random transformations

TL;DR

This paper demonstrates that Poisson measures remain invariant under certain random transformations that preserve intensity, using moment identities and complex algebraic methods inspired by Wiener space techniques.

Contribution

It introduces a novel invariance result for Poisson measures under random transformations with a cyclic vanishing gradient condition, extending previous Wiener space methods.

Findings

Poisson measures are invariant under specific random transformations.

Moment identities for Poisson stochastic integrals are established.

Applications include transformations conditioned by random sets like convex hulls.

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 applied in [22] on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure.