Growth of the Brownian forest

Abstract

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton--Watson forests with i.i.d. exponential branch lengths. We give elementary proofs of this composition rule and explain how it is intimately linked with Williams' decomposition for Brownian motion with drift.

Published at http://dx.doi.org/10.1214/009117905000000422 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Path decompositions for Markov chains
• Rayleigh processes, real trees, and root growth with re-grafting

Cited by in corpus (5)

• Growth of Levy trees
• Binary trees, exploration processes, and an extented Ray--Knight Theorem
• Williams' decomposition of the Lévy continuous random tree and simultaneous extinction probability for populations with neutral mutations
• Patterns in random walks and Brownian motion
• On the genealogy on conditioned stable Lévy forest