Intersection local times, loop soups and permanental wick powers

Yves Le Jan, Michael B. Marcus, Jay Rosen

Research output: Contribution to journalArticlepeer-review

Abstract

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 V endowed with a metric d. Sufficient conditions are obtained for the continuity of these processes on (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.

Original languageEnglish (US)
Pages (from-to)1-92
Number of pages92
JournalMemoirs of the American Mathematical Society
Volume247
Issue number1171
DOIs
StatePublished - May 2017

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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