Permanental fields, loop soups and continuous additive functionals

A permanental field, .={.(.),..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.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 . 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 . to continuous additive functionals of X (continuous in t), L={L.t,(.,t).V.R+}. Sufficient conditions are obtained for the continuity of L on V.R+. The metric on V is given by a proper norm.