Laplace transform identities and measure-preserving transformations on the Lie–Wiener–Poisson spaces

Given a divergence operator δ on a probability space such that the law of δ(h) is infinitely divisible with characteristic exponent h −→−1 2 ∞ 0 h2t dt, or ∞ 0 eih(t) −ih(t) −1 dt, h ∈ L2(R+), (0.1) we derive a family of Laplace transform identities for the derivative ∂E[eλδ(u)]/∂λ when u is a nonnecessarily 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. © 2012 Elsevier Inc. All rights reserved.