Choquet-type integral representation of polysupermedian measures

Abstract

Some aspects of multi-parameter potential theory are developed: we give a Choquet-type integral representation for measures which are supermedian for a countable family of submarkovian resolvents of commuting kernels on a Radon measurable space. For the subclass of polysupermedian measures we prove a Riesz-type decomposition, and we show that there is a ‘unique’ integral representation by minimal polysupermedian measures. The setting covers a variety of very different examples like random fields, measures on product spaces which are supermedian for resolvents on the factor spaces, and completely supermedian measures.