Behaviour of Excessive Functions of Certain Diffusions under the Action of the Transition Semi-Group

Abstract

In earlier papers [4],[5], it has been shown that under certain analytic conditions concerning its potential kernel, a strong Markov process, which is transient and with continuous sample paths, has all of its excessive harmonic functions, which are not identically infinite, continuous. Also, it has been shown that under the same conditions the excessiveness of harmonic functions of the process is automatic. In this paper we are studying the behaviour of excessive functions of the process under the action of the transition semi-group of the process. For example, all excessive functions for the Brownian motion semi-group are transformed into continuous functions by the semi-group. It seems that even this classical case does not appear in the literature. This will be shown below under a more general setting.