Excursion theory for Markov processes indexed by Levy trees

TL;DR

This paper develops an excursion theory for Markov processes indexed by Levy trees, extending classical results and encoding genealogies of excursions through Levy trees, with applications to Brownian motion indexed by Brownian trees.

Contribution

It introduces a new excursion theory for Levy tree-indexed Markov processes, generalizing classical excursion frameworks and connecting genealogical structures with Levy trees.

Findings

Established a local time-based excursion theory for Levy tree-indexed Markov processes.

Encoded the genealogy of excursions using Levy trees called the tree coded by local time.

Reproduced known excursion theory for Brownian motion on Brownian trees using new methods.

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 $\mathbb{R}_+$-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.