PERMANENTAL FIELDS, LOOP SOUPS AND CONTINUOUS ADDITIVE FUNCTIONALS

• 成果类型: Article
• 署名作者: Le Jan, Yves; Marcus, Michael B.; Rosen, Jay
• 署名单位: Universite Paris Saclay; City University of New York (CUNY) System; City College of New York (CUNY); City University of New York (CUNY) System; College of Staten Island (CUNY)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP893
• 发表日期: 2015
• 页码: 44-84
• 关键词: sample path properties; MARKOV-PROCESSES; local-times

摘要

A permanental field, psi = {psi(v), v is an element of 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 is an element of 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 psi 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 psi to continuous additive functionals of X (continuous in t), L = {L-t(v),(v,t) is an element of V x R+}. Sufficient conditions are obtained for the continuity of L on V x R+. The metric on V is given by a proper norm.