Generalized sweeping-out and probability

AbstractIn this paper, in further confirmation of the close relation between potential theory and probability, generalized sweeping-out (balayage) will be investigated from a probabilistic point of view. The key concept is that of a supermartingale with values in a compact space K. A specified reference family S of functions from K to the reals determines a partial ordering of measures on K; a measure following a measure is a balayage of the latter. It is shown that under appropriate hypotheses on S an ordered (totally ordered) family of measures is the family of marginal distributions of a supermartingale with state space K. If the measure family is maximal in the order, if K is metrizable and if the supermartingale is chosen properly, the supermartingale sample paths are right-continuous paths to the Choquet boundary of K relative to S. The functions in S are generalized superharmonic functions and the supermartingale paths to the Choquet boundary of K are analogs of Brownian paths to the Martin boundary of a Green space in the classical potential theoretic context