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Fragmentations, growth-fragmentations, and random structures


Shi, Quan. Fragmentations, growth-fragmentations, and random structures. 2016, University of Zurich, Faculty of Science.

Abstract

In this thesis we study the stochastic model of fragmentation phenomena. We focus on two themes: applications to random laminations of the disk, and growth-fragmentation processes. In the first part we use fragmentatio n theory as the principal tool to study Aldous' Brownian triangulation of the disk, that is a random set of non-crossing chords that divide the disk into triangles. We investigate the number of large triangles and the law of the length of the longest chord, and generalize these results to stable laminations. As part of the proof apparatus, we obtain new results on the number of large splitting events of self-similar fragmentations. The second part concerns growth-fragmentation processes, which describe particle systems in which each particle grows and splits randomly and independently of the others. We prove that the law of a self-similar growth-fragmentation is determined by a cumulant function and its index of self-similarity. We also introduce a new class of growth-fragmentations that are related to Lévy driven Ornstein-Uhlenbeck type processes and prove a law of large numbers for these growth-fragmentations.

Abstract

In this thesis we study the stochastic model of fragmentation phenomena. We focus on two themes: applications to random laminations of the disk, and growth-fragmentation processes. In the first part we use fragmentatio n theory as the principal tool to study Aldous' Brownian triangulation of the disk, that is a random set of non-crossing chords that divide the disk into triangles. We investigate the number of large triangles and the law of the length of the longest chord, and generalize these results to stable laminations. As part of the proof apparatus, we obtain new results on the number of large splitting events of self-similar fragmentations. The second part concerns growth-fragmentation processes, which describe particle systems in which each particle grows and splits randomly and independently of the others. We prove that the law of a self-similar growth-fragmentation is determined by a cumulant function and its index of self-similarity. We also introduce a new class of growth-fragmentations that are related to Lévy driven Ornstein-Uhlenbeck type processes and prove a law of large numbers for these growth-fragmentations.

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Additional indexing

Item Type:Dissertation
Referees:Bertoin Jean, Féray Valentin, Kortchemski Igor, Nikeghbali Ashkan, Winkel Matthias
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2016
Deposited On:27 Oct 2016 08:35
Last Modified:27 Oct 2016 08:35
Number of Pages:145
Related URLs:http://www.recherche-portal.ch/ZAD:default_scope:ebi01_prod010718627 (Library Catalogue)

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