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 we derive a family of Laplace transform identities for the derivative ∂E[eλδ(u)]/∂λ when u is a non-necessarily adapted process. These expressions are based on intrinsi...

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Bibliographic Details
Main Author: Privault, Nicolas
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2013
Subjects:
Online Access:https://hdl.handle.net/10356/97911
http://hdl.handle.net/10220/17123
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Institution: Nanyang Technological University
Language: English
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Summary:Given a divergence operator δ on a probability space such that the law of δ(h) is infinitely divisible with characteristic exponent we derive a family of Laplace transform identities for the derivative ∂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.