Excessive Functions

Abstract

In this chapter we present the basic concepts concerning the potential theory associated with a sub-Markovian resolvent of kernels on a Radon measurable space (E, B). We begin in Section 1.1 with the sub-Markovian resolvent of kernels and we complete the section with a characterization of those proper kernels which are generating sub-Markovian resolvents. Section 1.2 is reserved to the study of the excessive functions with respect to a given sub-Markovian resolvent U. We conclude that the natural context will be whenever the set ε(U) ∩ pB of all U-excessive functions which are B-measurable is min-stable, contains the positive constant functions and generates B. In Section 1.3 we introduce the fine topology and start to establish the reduction operation on the measurable sets. Section 1.4 is concerned with the set of excessive measures with respect to U. Important for the further use are the energy functional and the decomposition of every U-excessive measure as a sum of its potential and harmonic components. In Section 1.5 we introduce the Ray topologies, the associated Ray compactifications and we realize some extensions of the resolvent U. Section 1.6 points out supplementary properties of the reduction operation, related to the induced Choquet capacities. In Section 1.7 we treat different types of negligible sets with respect to a positive finite measure λ: the λ-polar, λ-semipolar and λ-mince sets. The main result of the last section asserts that a proper sub-Markovian resolvent of kernels on a Radon measurable (E, B) space may be considered, after a possible extension of E as the resolvent associated with a transient right process. Based on this result we shall state in the next chapters a fruitful dialog between analytic and probabilistic potential theoretical aspects.