Intersection Local Times, Loop Soups and Permanental Wick Powers

Abstract

Several stochastic processes related to transient Levy 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. Su- cient conditions are obtained for the continuity of these processes on (V,d). The processes include n-fold self-intersection local times of tran- sient Levy processes and permanental chaoses, which are 'loop soup n- fold self-intersection local times' constructed from the loop soup of the Levy 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 proper- ties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.