Excursion theory for Brownian motion indexed by the Brownian tree

TL;DR

This paper develops an excursion theory for Brownian motion indexed by the Brownian tree, establishing connections with continuous-state branching processes and providing explicit distributions for boundary lengths and excursions.

Contribution

It introduces a novel excursion theory for tree-indexed Brownian motion, linking boundary lengths to branching processes and characterizing the distribution of excursions.

Findings

Boundary lengths match jumps of a specific continuous-state branching process.

Conditional independence of excursions given boundary lengths.

Explicit distributions for boundary length and excursion mass.

Abstract

We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical It\^o theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connectedcomponent is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism $\psi(u)=\sqrt{8/3}\,u^{3/2}$. Furthermore, conditionally on the boundary lengths, the different excursions are independent, and we determine their conditional distribution in terms of an excursion measure $\mathbb{M}_0$ which is the analog of the It\^o measure of Brownian excursions. We provide various descriptions of the excursion measure $\mathbb{M}_0$, and we also determine several explicit distributions, such as the joint distribution of the boundary length and the mass of an excursion under $\mathbb{M}_0$. We use the Brownian snake as a convenient tool for defining and analysing the excursions of our tree-indexed Brownian motion.