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Intersection Local Times, Loop Soups and Permanental Wick Powers


About this Title

Yves Le Jan, Michael B. Marcus and Jay Rosen

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 247, Number 1171
ISBNs: 978-1-4704-3695-7 (print); 978-1-4704-3703-9 (online)
DOI: https://doi.org/10.1090/memo/1171
Published electronically: January 3, 2017
Keywords:Loop soups, Markov processes, intersection local times

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Loop measures and renormalized intersection local times
  • Chapter 3. Continuity of intersection local time processes
  • Chapter 4. Loop soup and permanental chaos
  • Chapter 5. Isomorphism Theorem I
  • Chapter 6. Permanental Wick powers
  • Chapter 7. Poisson chaos decomposition, I
  • Chapter 8. Loop soup decomposition of permanental Wick powers
  • Chapter 9. Poisson chaos decomposition, II
  • Chapter 10. Convolutions of regularly varying functions

Abstract


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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