Permanental fields, loop soups and continuous additive functionals

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