Excursion theory for Brownian motion indexed by the Brownian tree

  • Céline Abraham

    Université Paris-Sud, Orsay, France
  • Jean-François Le Gall

    Université Paris-Sud, Orsay, France


We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Itô 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 connected component 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 . Furthermore, conditionally on the boundary lengths, the different excursions are independent, and we determine their conditional distribution in terms of an excursion measure which is the analog of the Itô measure of Brownian excursions. We provide various descriptions of M0, and we also determine several explicit distributions, such as the joint distribution of the boundary length and the mass of an excursion under . We use the Brownian snake as a convenient tool for defining and analysing the excursions of our tree-indexed Brownian motion.

Cite this article

Céline Abraham, Jean-François Le Gall, Excursion theory for Brownian motion indexed by the Brownian tree. J. Eur. Math. Soc. 20 (2018), no. 12, pp. 2951–3016

DOI 10.4171/JEMS/827