Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process

The potential kernel of a positive left additive functional (of a strong Markov process X) maps positive functions to strongly supermedian functions and satisfies a variant of the classical domination principle of potential theory. Such a kernel V is called a regular strongly supermedian kernel in recent work of L. Beznea and N. Boboc. We establish the converse: Every regular strongly supermedian kernel V is the potential kernel of a random measure homogeneous on [0,∞[. Under additional finiteness conditions such random measures give rise to left additive functionals. We investigate such random measures, their potential kernels, and their associated characteristic measures. Given a left additive functional A (not necessarily continuous), we give an explicit construction of a simple Markov process Z whose resolvent has initial kernel equal to the potential kernel U\!A. The theory we develop is the probabilistic counterpart of the work of Beznea and Boboc. Our main tool is the Kuznetsov process associated with X and a given excessive measure m.