Additive Functionals and Excessive Functions

Abstract

During the past five or six years there has been much work done on the problem of representing excessive functions as the potentials of additive functionals. This is a natural generalization to Markov processes of the classical theorem of Riesz dealing with the representation of non-negative superharmonic functions as the potentials of measures. These results (due mainly to Meyer [9], Sur [12], and Volkonski [13]) are now in a fairly definitive state, and the purpose of this paper is to give a cohesive account of them. In addition to their intrinsic importance these representation theorems have been very useful in several recent applications, for example, in the theory of time changes [2] and the theory of local times [4]. Although the general theory of Markov processes is extremely rich, it is necessary to set up a rather large amount of notation and machinery before coming to grips with the problems of interest. Consequently Section 1 contains a compact summary of the definitions and basic theorems of what is now called the theory of Hunt processes, that is, Markov processes satisfying Hypothesis (A) of Hunt's fundamental memoir [8]. The author feels that such a summary is worthwhile in itself, since it can serve as a source of definitions and notations for many current research papers. The reader familiar with this material should begin with Section 2 and refer back to Section 1 only as needed. The proofs of most of the results quoted in Section 1 can be found in [1], [6], [11], or, of course, [8]. Beginning with Section 2 the exposition becomes more leisurely. We have tried not only to state definitions and theorems, but also to give some insight into them and indicate some of the more important (in our opinion) open problems. The proofs of all results quoted in Sections 2-5 (with the exception of 2.6.1 (ii)) can be found in [7], as well as in the original papers cited in connection with each theorem.