Intersection local times, loop soups and permanental Wick powers

TL;DR

This paper studies stochastic processes related to transient Levy processes, including local times, loop soups, and permanental Wick powers, establishing continuity conditions, isomorphism theorems, and chaos decompositions.

Contribution

It introduces new processes based on loop soups and permanental Wick powers, extending Gaussian chaoses, and provides foundational theorems connecting these processes.

Findings

Established continuity conditions for the processes

Derived Dynkin type isomorphism theorems

Proved Poisson chaos decompositions for permanental Wick powers

Abstract

Several stochastic processes related to transient L\'evy processes with potential densities $u(x,y)=u(y-x)$, that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures $V$ endowed with a metric $d$. Sufficient conditions are obtained for the continuity of these processes on $(V,d)$. The processes include $n$-fold self-intersection local times of transient L\'evy processes and permanental chaoses, which are `loop soup $n$-fold self-intersection local times' constructed from the loop soup of the L\'evy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of $n$-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.