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Intersection Local Times, Loop Soups and Permanental Wick Powers

About this Title

Yves Le Jan, Equipe Probabilités et Statistiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France, Michael B. Marcus, Department of Mathematics, City College, CUNY, New York, New York 10031 and Jay Rosen, Department of Mathematics, College of Staten Island, CUNY, Staten Island, New York 10314

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)
Published electronically: January 3, 2017
Keywords: Loop soups, Markov processes, intersection local times
MSC: Primary 60K99, 60J55; Secondary 60G17

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Table of Contents


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


Several stochastic processes related to transient Lévy processes with potential densities $u(x,y)=u(y-x)$, that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures $\mathcal {V}$ endowed with a metric $d$. Sufficient conditions are obtained for the continuity of these processes on $(\mathcal {V},d)$. The processes include $n$-fold self-intersection local times of transient Lévy processes and permanental chaoses, which are ‘loop soup $n$-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 $n$-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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