Invariance of Poisson measures under random transformations

Nicolas Privault

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arXiv · 1004.2588

Invariance of Poisson measures under random transformations

Nicolas Privault

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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.

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Nicolas Privault. 2011-02-23. Invariance of Poisson measures under random transformations. https://arxiv.org/abs/1004.2588

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