Intersection local times, loop soups, and permanental wick powers / Yves Le Jan, Michael B. Marcus and Jay Rosen.

Includes bibliographical references.

Chapter 1. Introduction --
Chapter 2. Loop measures and renormalized intersection local times --
Chapter 3. Continuity of intersection local time processes --
Chapter 4. Loop soup and permanental chaos --
Chapter 5. Isomorphism Theorem I --
Chapter 6. Permanental Wick powers --
Chapter 7. Poisson chaos decomposition, I --
Chapter 8. Loop soup decomposition of permanental Wick powers --
Chapter 9. Poisson chaos decomposition, II --
Chapter 10. Convolutions of regularly varying functions.

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 $\mathcal{V}$ endowed with a metric $d$. Sufficient conditions are obtained for the continuity of these processes on $(\mathcal{V},d)$. The processes include $n$-fold self-intersection local times of transient 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 properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.