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The purpose of these notes is to explore some simple relations between Markovian path and loop measures, the Poissonian ensembles of loops they determine, their occupation fields, uniform spanning trees, determinants, and Gaussian Markov fields such as the free field. These relations are first studied in complete generality for the finite discrete setting, then partly generalized to specific examples in infinite and continuous spaces.

Markov processes --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Markov processes. --- Stochastic processes. --- Random processes --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Mathematics. --- Potential theory (Mathematics). --- Probabilities. --- Probability Theory and Stochastic Processes. --- Potential Theory. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Math --- Science --- Probabilities --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions

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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.

Gaussian processes. --- Local times (Stochastic processes) --- Loop spaces.

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