# The area of a self-similar fragmentation

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.
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2011
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Cited literature [19 references]

https://hal.archives-ouvertes.fr/hal-00557785
Contributor : Jean Bertoin <>
Submitted on : Thursday, January 20, 2011 - 7:03:15 AM
Last modification on : Wednesday, October 12, 2016 - 1:04:01 AM
Document(s) archivé(s) le : Thursday, April 21, 2011 - 2:42:00 AM

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### Identifiers

• HAL Id : hal-00557785, version 1
• ARXIV : 1101.3965

### Citation

Jean Bertoin. The area of a self-similar fragmentation. 2011. 〈hal-00557785〉

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