Growth of the Brownian forest

Jim Pitman; Matthias Winkel

doi:10.1214/009117905000000422

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Home > Journals > Ann. Probab. > Volume 33 > Issue 6 > Article

November 2005 Growth of the Brownian forest

Jim Pitman, Matthias Winkel

Ann. Probab. 33(6): 2188-2211 (November 2005). DOI: 10.1214/009117905000000422

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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.

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Jim Pitman. Matthias Winkel. "Growth of the Brownian forest." Ann. Probab. 33 (6) 2188 - 2211, November 2005. https://doi.org/10.1214/009117905000000422