Excursion Theory Revisited

Excursions from a fixed point b are studied in the framework of a general Borel right process X, with a fixed excessive measure m serving as background measure; such a measure always exists if b is accessible from every point of the state space of X. In this context the left-continuous moderate Markov dual process b X arises naturally and plays an important role. This allows the basic quantities of excursion theory such as the Laplace-Lévy exponent of the inverse local time at b and the Laplace transform of the entrance law for the excursion process to be expressed as inner products involving simple hitting probabilities and expectations. In particular if X and b X are honest, then the resolvent of X may be expressed entirely in terms of quantities that depend only on X and b X killed when they first hit b.