The area of a self-similar fragmentation

TL;DR

This paper studies the total area under a self-similar fragmentation process with negative index, characterizing its distribution through an integro-differential equation and deriving recursive formulas for its moments.

Contribution

It introduces a novel integro-differential equation to characterize the law of the area and generalizes known results for Brownian excursions to self-similar fragmentations.

Findings

Derived an integro-differential equation for the area distribution.

Established recursive formulas for moments in binary splitting cases.

Extended results to a broader class of self-similar fragmentation processes.

Abstract

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.