A fragmentation process connected to Brownian motion

Let (B-s,s greater than or equal to 0) be a standard Brownian motion and T-1 its first passage time at level 1. For every t greater than or equal to 0, we consider ladder time set L-(t) of the Brownian motion with drift t, B-s((t)), = 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 greater than or equal to 0) is a fragmentation process, in the sense that for 0 less than or equal to 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].