Several stochastic processes related to transient Lévy 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 Lévy processes and permanental chaoses, which are `loop soup \(n\)-fold self-intersection local times' constructed from the loop soup of the Lévy 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.