Excursion theory for Markov processes indexed by Lévy trees

This talk concerns Markov processes indexed by Lévy trees. These processes play a
fundamental role in probability theory because of their relationships with superprocesses,
their appearance in various limit theorems, and their connections to growth–fragmentation
processes. Moreover, they can be used to construct a broad family of two-dimensional
random geometry models, which are referred to as Brownian surfaces. In particular,
Brownian motion indexed by the Brownian tree has served as a building block for the
celebrated Brownian map.

The purpose of the talk is to introduce the main elements of an excursion theory tailored
for this family of processes. This theory provides a unified framework for understanding
their evolution between visits to a reference point. It extends the classical excursion theory
for Markov processes indexed by the real half-line, and in the special case of Brownian
motion indexed by the Brownian tree, we recover previous results of Abraham and Le Gall
in a more precise form.

These results are part of joint work with Alejandro Rosales-Ortiz. No specialized background
is required.