Excursion theory for Brownian motion indexed by the Brownian tree

DOI10.4171/JEMS/827zbMATH Open1501.60046arXiv1509.06616MaRDI QIDQ1630636

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Publication date: 10 December 2018

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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 $psi(u)=sqrt8/3,u^{3/2}$. 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.

Full work available at URL: https://arxiv.org/abs/1509.06616

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