The area of a self-similar fragmentation

Jean Bertoin (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider the area $A=\int_0^{\infty}\left(\sum_{i=1}^{\infty} X_i(t)\right) \d t$ of a self-similar fragmentation process $\X=(\X(t), t\geq 0)$ with negative index. We characterize the law of $A$ by an integro-differential equation. The latter may be viewed as the infinitesimal version of a recursive distribution equation that arises naturally in this setting. In the case of binary splitting, this yields a recursive formula for the entire moments of $A$ which generalizes known results for the area of the Brownian excursion.

Mots clés: Self-similar fragmentation; area; recursive distributional equation.; recursive distributional equation

Domaines: Probabilités [math.PR]

Jean Bertoin. The area of a self-similar fragmentation. 2011. ⟨hal-00557785⟩