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)δ(h) is infinitely divisible with characteristic exponentequation(0.1)h↦−12∫0∞ht2dt,or∫0∞(eih(t)−ih(t)−1)dt,h∈L2(R+), we derive a family of Laplace transform identities for the derivative ∂E[eλδ(u)]/∂λ∂E[eλδ(u)]/∂λ 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.