Abstract
We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.
Highlights
A Markovian loop soup is a particular Poisson point process L on paths associated to a Markov process X
Loop measure for Brownian motion was introduced by Symanzik in his seminal paper [25] on Euclidean quantum field theory, where it is referred to as ‘blob measure’, and is a basic building block in his construction of quantum fields
Le Jan extended the notion of loop soups to other Markov processes [12], and this has been generalized further in [14, 15]
Summary
A Markovian loop soup is a particular Poisson point process L on paths associated to a Markov process X. Our motivation in studying Markovian loop soups is to better understand the wonderful and mysterious Isomorphism Theorem of Dynkin, [7, 8], which connects the family of total local times L = {Lx∞, x ∈ S} of a symmetric Markov process X in S with the Gaussian process G = {Gx, x ∈ S} of covariance u(x, y). Recent work, [14, 15], uses loop soups for Markov processes with potential densities u(x, y) which may be infinite on the diagonal In this case there are no local times and no permanental processes. Loop soups are used to prove the existence of permanental fields (indexed by measures rather than points in S) with which to establish Isomorphism Theorems: for continuous additive functionals in [14], and for intersection local times in [15].
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