Excursion theory for Markov processes indexed by Lévy trees

Abstract We develop an excursion theory that describes the evolution of a Markov process indexed by a Lévy tree away from a regular and instantaneous point of the state space. The theory builds upon a notion of local time at that was recently introduced in the companion paper [Probab. Theory Related Fields. 189 (2024), 1–99]. Despite the radically different setting, our results exhibit striking similarities to the classical excursion theory for ‐indexed Markov processes. We then show that the genealogy of the excursions can be encoded in a Lévy tree called the tree coded by the local time. In particular, we recover by different methods the excursion theory of Abraham and Le Gall, which was developed for Brownian motion indexed by the Brownian tree.