The area of a self-similar fragmentation

We consider the area $A=\int_0^{\infty}\left(\sum_{i=1}^{\infty} X_i(t)\right) \d t$ofaself-similarfragmentationprocess$\X=(\X(t), t\geq 0)$withnegativeindex\mathrm{.}Wecharacterizethelawof$A$ 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.