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Compensated fragmentation processes and limits of dilated fragmentations


Bertoin, Jean (2016). Compensated fragmentation processes and limits of dilated fragmentations. The Annals of Probability, 44(2):1254-1284.

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 $\int_{\mathit{p}}$ (1-$\mathit{p}_{1}$)$^{2}\nu$(d$\mathbf{p}$ < $\infty$, where $\mathbf{p}$ = ($\mathit{p}_{1}$,…) denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $\int_{\mathit{p}}$ (1-$\mathit{p}_{1}$)$^{2}\nu$(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.

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 $\int_{\mathit{p}}$ (1-$\mathit{p}_{1}$)$^{2}\nu$(d$\mathbf{p}$ < $\infty$, where $\mathbf{p}$ = ($\mathit{p}_{1}$,…) denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $\int_{\mathit{p}}$ (1-$\mathit{p}_{1}$)$^{2}\nu$(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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:March 2016
Deposited On:01 Feb 2017 07:42
Last Modified:21 Nov 2017 18:57
Publisher:Institute of Mathematical Statistics
ISSN:0091-1798
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/14-AOP1000

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