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Abstract: Given a divergence operator δ on a probability space such that the law of is infinitely divisible with characteristic exponent we derive a family of Laplace transform identities for the derivative when u is a non-necessarily adapted process. These expressions are based on intrinsic geometric tools such as the Carleman–Fredholm determinant of a covariant derivative operator and the characteristic exponent (0.1), in a general framework that includes the Wiener space, the path space over a Lie group, and the Poisson space. We use these expressions for measure characterization and to prove the invariance of transformations having a quasi-nilpotent covariant derivative, for Gaussian and other infinitely divisible distributions. [Copyright &y& Elsevier]

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 identifies of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Ùstünel and Zakai (Probab. Theory Related Fields 103 (1995) 409-429) 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 [ABSTRACT FROM AUTHOR]