## The Annals of Probability

### Compensated fragmentation processes and limits of dilated fragmentations

Jean Bertoin

#### Abstract

A new class of fragmentation-type random processes is introduced, in which, roughly speaking, the accumulation of small dislocations which would instantaneously shatter the mass into dust, is compensated by an adequate dilation of the components. An important feature of these compensated fragmentations is that the dislocation measure $\nu$ which governs their evolutions has only to fulfill the integral condition $\int_{\mathcal{P}}(1-p_{1})^{2}\nu({\mathrm{d}}{\mathbf{p}})<\infty$, where ${\mathbf{p}}=(p_{1},\ldots)$ denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $\int_{\mathcal{P}}(1-p_{1})\nu({\mathrm{d}}{\mathbf{p}})<\infty$ for $\nu$ to be the dislocation measure of a homogeneous fragmentation. Our main results show that such compensated fragmentations naturally arise as limits of homogeneous dilated fragmentations, and bear close connections to spectrally negative Lévy processes.

#### Article information

Source
Ann. Probab., Volume 44, Number 2 (2016), 1254-1284.

Dates
Revised: December 2014
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.aop/1457960395

Digital Object Identifier
doi:10.1214/14-AOP1000

Mathematical Reviews number (MathSciNet)
MR3474471

Zentralblatt MATH identifier
1344.60033

#### Citation

Bertoin, Jean. Compensated fragmentation processes and limits of dilated fragmentations. Ann. Probab. 44 (2016), no. 2, 1254--1284. doi:10.1214/14-AOP1000. https://projecteuclid.org/euclid.aop/1457960395

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