Permanental fields, loop soups and continuous additive functionals

TL;DR

This paper introduces permanental fields linked to potential densities of transient Markov processes, constructs them via loop soups, and establishes a Dynkin-type isomorphism relating these fields to continuous additive functionals.

Contribution

It demonstrates the existence of permanental fields for non-symmetric kernels, constructs them through loop soups, and proves a new isomorphism theorem connecting these fields to additive functionals.

Findings

Existence of permanental fields for certain kernels.

Construction of permanental fields via loop soups.

A Dynkin-type isomorphism theorem relating fields and 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.