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February 2015 Permanental fields, loop soups and continuous additive functionals
Yves Le Jan, Michael B. Marcus, Jay Rosen
Ann. Probab. 43(1): 44-84 (February 2015). DOI: 10.1214/13-AOP893

Abstract

A permanental field, $\psi=\{\psi(\nu),\nu\in{ \mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, 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 $t$), $L=\{L_{t}^{\nu},(\nu,t)\in{ \mathcal{V}}\times R_{+}\}$. Sufficient conditions are obtained for the continuity of $L$ on ${ \mathcal{V}}\times R_{+}$. The metric on ${ \mathcal{V}}$ is given by a proper norm.

Citation

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Yves Le Jan. Michael B. Marcus. Jay Rosen. "Permanental fields, loop soups and continuous additive functionals." Ann. Probab. 43 (1) 44 - 84, February 2015. https://doi.org/10.1214/13-AOP893

Information

Published: February 2015
First available in Project Euclid: 12 November 2014

zbMATH: 1316.60075
MathSciNet: MR3298468
Digital Object Identifier: 10.1214/13-AOP893

Subjects:
Primary: 60G17, 60J55, 60K99

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 1 • February 2015
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