The Annals of Probability

Permanental fields, loop soups and continuous additive functionals

Yves Le Jan, Michael B. Marcus, and Jay Rosen

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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.

Article information

Ann. Probab., Volume 43, Number 1 (2015), 44-84.

First available in Project Euclid: 12 November 2014

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Primary: 60K99: None of the above, but in this section 60J55: Local time and additive functionals 60G17: Sample path properties

Permanental fields Markov processes loop soups continuous additive functionals


Le Jan, Yves; Marcus, Michael B.; Rosen, Jay. Permanental fields, loop soups and continuous additive functionals. Ann. Probab. 43 (2015), no. 1, 44--84. doi:10.1214/13-AOP893.

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