Excursion theory revisited

Abstract

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 $\widehat X$ arises naturally and plays an important role. This allows the basic quantities of excursion theory such as the Laplace-L\'evy 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 $\widehat X$ are honest, then the resolvent of $X$ may be expressed entirely in terms of quantities that depend only on $X$ and $\widehat X$ killed when they first hit $b$.