Excursion theory for Markov processes indexed by Levy trees

2024/11/19 by Riera, Armand, Rosales-Ortiz, Alejandro
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2411.12717

Abstract

We develop an excursion theory that describes the evolution of a Markov process indexed by a Levy tree away from a regular and instantaneous point x of the state space. The theory builds upon a notion of local time at x that was recently introduced in [37]. Despite the radically different setting, our results exhibit striking similarities to the classical excursion theory for ℝ₊-indexed Markov processes. We then show that the genealogy of the excursions can be encoded in a Levy tree called the tree coded by the local time. In particular, we recover by different methods the excursion theory of Abraham and Le Gall [2], which was developed for Brownian motion indexed by the Brownian tree.

Related

• The structure of the local time of Markov processes indexed by Levy trees
• Excursion theory for Brownian motion indexed by the Brownian tree
• The Markov property of local times of Brownian motion indexed by the Brownian tree
• Ray-Knight Theorems for Spectrally Negative Lévy Processes
• A tree-valued Markov processes associated with an admissible family of branching mechanisms