Intersection Local Times, Loop Soups and Permanental Wick Powers
- ISBN-10
- 1470436957
- ISBN-13
- 9781470436957
- Category
- Gaussian processes
- Pages
- 78
- Language
- English
- Published
- 2017-04-25
- Publisher
- American Mathematical Soc.
- Authors
- Michael B. Marcus, Yves Le Jan, Jay Rosen
Description
Several stochastic processes related to transient Lévy processes with potential densities , that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures endowed with a metric . Sufficient conditions are obtained for the continuity of these processes on . The processes include -fold self-intersection local times of transient Lévy processes and permanental chaoses, which are `loop soup -fold self-intersection local times' constructed from the loop soup of the Lévy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of -th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.
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