On the equivalence of three potential principles for right Markov processes

Summary 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 processX with distinguished excessive measurem. We make no duality hypotheses andm need not be a reference measure. Our main tools are the stationary process (Y, Q _m) associated withX andm, and a correspondence between potentials U and certain random measures over (Y, Q _m).Research supported in part by NSF Grant 8419377