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A fragmentation process connected to Brownian motion


Bertoin, Jean (2000). A fragmentation process connected to Brownian motion. Probability Theory and Related Fields, 117(2):289-301.

Abstract

Let (B s , s≥ 0) be a standard Brownian motion and T 1 its first passage time at level 1. For every t≥ 0, we consider ladder time set ℒ (t) of the Brownian motion with drift t, B (t) s = B s + ts, and the decreasing sequence F(t) = (F 1(t), F 2(t), …) of lengths of the intervals of the random partition of [0, T 1] induced by ℒ (t) . The main result of this work is that (F(t), t≥ 0) is a fragmentation process, in the sense that for 0 ≤t < t′, F(t′) is obtained from F(t) by breaking randomly into pieces each component of F(t) according to a law that only depends on the length of this component, and independently of the others. We identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by Aldous and Pitman [3].

Abstract

Let (B s , s≥ 0) be a standard Brownian motion and T 1 its first passage time at level 1. For every t≥ 0, we consider ladder time set ℒ (t) of the Brownian motion with drift t, B (t) s = B s + ts, and the decreasing sequence F(t) = (F 1(t), F 2(t), …) of lengths of the intervals of the random partition of [0, T 1] induced by ℒ (t) . The main result of this work is that (F(t), t≥ 0) is a fragmentation process, in the sense that for 0 ≤t < t′, F(t′) is obtained from F(t) by breaking randomly into pieces each component of F(t) according to a law that only depends on the length of this component, and independently of the others. We identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by Aldous and Pitman [3].

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1 June 2000
Deposited On:24 Jul 2013 08:34
Last Modified:07 Dec 2017 21:43
Publisher:Springer
ISSN:0178-8051
Publisher DOI:https://doi.org/10.1007/s004400050008
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1771665
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0965.60072

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