## 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 = {L^{ν}_{t}, (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 language | English (US) |
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Pages (from-to) | 44-84 |

Number of pages | 41 |

Journal | Annals of Probability |

Volume | 43 |

Issue number | 1 |

DOIs | |

State | Published - 2015 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty