Permanental fields, loop soups and continuous additive functionals

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.