Capacity theory without duality

Summary The paper develops a theory of capacity for a Borel right process without duality assumptions. The basic tool in this approach is a stationary process ralative to an excessive measure.IfP_t)t0 denotes the semigroup of the process on the state spaceE and ifm is an excessive measure onE, then there exists a processY = (Y_t)_t onE with random birth and death and a -finite measureQ_m such thatY is stationary underQ_m and Markov with respect to (P_t).For a setB inE the hitting (resp. last exit) time ofY is denoted by _B (resp._B), andB is called transient (resp. cotransient) ifQ_m(_B=)= 0 (resp.Q_m(_B= – )=0. The main theorem then states that for a both transient and contransient setB the distributions of_B and _B underQ_m are the same. For suchB the capacity is denfined byC(B):=Q_m(_B[0, 1] and the cocapacity by(B):=Q_m(_B[0, 1], and it is shown that these definitions in fact generalize previous definitions under duality assumptions.Without duality assumption there is no representation of the capacitary potential in terms of a capacitary measure, but there exists a cocapacitary entrance law _t^Bwhich generalizes the notion of a cocapacitary measure such that(B)= _t^B(1).The paper contains investigations of transience and cotransience, a decomposition of the cocapacitrary entrance law, some remarks on left versions, and furthermore a generalization of Spitzer's asymptotic formula.Research supported in part by NSF Grant DMS 8419377Research carried out while visiting University of California, San Diego, during Spring 1985