A fragmentation process connected to Brownian motion

Abstract

Let (Bs, s ≥ 0) be a standard Brownian motion and T1 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 = Bs + ts, and the decreasing sequence F (t) = (F1(t), F2(t), . . .) of lengths of the intervals of the random partition of [0, T1] 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].