On Choquet's Dichotomy of Capacity for Markov Processes

Abstract

Following Choquet, the capacity associated with a Markov process is said to be dichotomous if each compact set $K$ contains two disjoint sets with the same capacity as $K$. In the context of right processes, we prove that the dichotomy of capacity is equivalent to Hunt's hypothesis that semipolar sets are polar. We also show that a weaker form of the dichotomy is valid for any Levy process with absolutely continuous potential kernel.