Permanental fields, loop soups and continuous additive functionals

• Le Jan, Yves ^[1] ; Marcus, Michael B. ^[2] ; Rosen, Jay ^[2]
  1. [1] University of Paris-Sud

     University of Paris-Sud

     Arrondissement de Palaiseau, Francia

     
  2. [2] City University of New York

     City University of New York

     Estados Unidos

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 1, 2015, págs. 44-84
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • A permanental field, ψ={ψ(ν),ν∈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∈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)∈V×R+}. Sufficient conditions are obtained for the continuity of L on V×R+. The metric on V is given by a proper norm.