Discontinuous additive functionals of dual processes

Abstract

Let X be a standard Markov process with state space E and let ~ be an excessive reference measure for X. In recent work ([6], E7]) Revuz associates with certain additive functionals of X measures on E which determine those additive functionals in case they are natural and X is in duality, relative to 3, with another standard process X. In this paper we use an analogous method to associate with every finite additive functional A ofX a measure v A on E x E which turns out to be a-finite and whose projection upon the second co-ordinate is Revuz's measure on E. Under the hypotheses of duality, but with no other assumptions as, for example, on the left-continuity of the fields or Feller properties of the resolvents, we give a formula for the bipotential of a finite additive functional A in terms ofv A and we construct a canonical measure v on E x E for the process X which reflects the behavior of the jumps of X. Using this canonical measure, we can prove that if A is a finite purely discontinuous quasi-left-continuous additive functional of X then A is of the form A t = ~ F(Xs_, Xs), a result due to Motoo (see Watanabe [8]) in the case S =0}, resolvent {U~; ~=>0} and lifetime ~. The a-fields ~, ~ and ~* are respectively the Borel sets, nearly Borel sets and universally measurable sets in E. The object of our attention here is an additive functional (AF) of X. We call A a finite AF of X if A t < ~ on [0, ~) a. s., and we denote by ~r the class of finite AF's of X. The restriction to finite AF's makes it possible to avoid a number of tricky points dealt with by Revuz [6]. We assume throughout that there is a a-finite measure ~ on E which is an excessive reference measure for X. For the main results of this paper, when X is assumed to be in duality with a standard process ){ relative to the a-finite measure 3, then ~ automatically possesses all the above named properties. All the regularity properties discussed in Chapter V of [-1] may be assumed. One particularly useful result is that if f and g are a-excessive and f< g a.e. (r then f< g everywhere.