Permanental fields, loop soups and continuous additive functionals

Yves Le Jan, Michael B. Marcus, Jay Rosen

Research output: Contribution to journalArticlepeer-review

Abstract

A permanental field, ψ = {ψ(ν),ν ε V}, is a particular stochastic process indexed by a space of measures on a set S. It is determined by a kernel u(x, y), x,y ε S, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when u(x, y) is a potential density of a transient Markov process X in S. A permanental field ψ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of X, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relates ψ to continuous additive functionals of X (continuous in t ), L = {Lνt, (ν, t) ε V × R+}. Sufficient conditions are obtained for the continuity of L on V × R+. The metric on V is given by a proper norm.

Original languageEnglish (US)
Pages (from-to)44-84
Number of pages41
JournalAnnals of Probability
Volume43
Issue number1
DOIs
StatePublished - 2015

Keywords

  • Continuous additive functionals
  • Loop soups
  • Markov processes
  • Permanental fields

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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