On the equivalence of three potential principles for right Markov processes
We examine three of the principles of probabilistic potential theory in a nonclassical setting. These are: (i) the bounded maximum principle, (ii) the positive definiteness of the energy (of measures of bounded potential), and (iii) the condition that each semipolar set is polar. These principles are known to be equivalent in the context of two Markov processes in strong duality, when excessive functions are lower semicontinuous. We show that when the principles are appropriately formulated their equivalence persists in the wider context of a Borel right Markov process X with distinguished excessive measure m. We make no duality hypotheses and m need not be a reference measure. Our main tools are the stationary process \((Y,Q_ m)\) associated with X and m, and a correspondence between potentials \(\mu\) U and certain random measures over \((Y,Q_ m)\).
principles of probabilistic potential theory
scientific article; zbMATH DE number 4094528
scientific article