Comparison Theorems on Regular Points for Multidimensional Markov Processes of Transient Type

Abstract

The study of regular points for the Dirichlet problem has a long history. The probabilistic approach to regular points is originated by Doob [2] and [3] for Brownian motion and the heat process. The extension to general Markov processes is discussed in Dynkin [4] and [5]. They also clarified the relation between the fine topology and regular points.Regular points are by definition reflected in the behaviour of sample paths of Markov processes. Further the inclusion relation of collections of regular points for open sets determines the strength and the weakness of fine topologies between two processes. Hence it is meaningful to compare the collections of regular points for compact or open sets between two Markov processes apart from the Dirichlet problem.