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Quelques interactions entre analyse, probabilités et fractals


Bertoin, J; Berestycki, J; Haas, B; Miermont, C (2010). Quelques interactions entre analyse, probabilités et fractals. Panoramas et Syntheses, 32:191-243.

Abstract

Following the seminal contributions of Benoît Mandelbrot in the 70s, concepts derived from fractal geometry gave a new impulse to several areas of mathematics. The goal of this volume is to present syntheses on two subjects where important advances occurred in the last 15 years: multiplicative processes and fragmentation. One arose from harmonic analysis (Riesz products) and the other from a probabilistic model proposed by N. Kolmogorov in order to explain experimental observations on rock fragmentation; however they share analogies and use common mathematical tools issued from the study of random fractals. The first text introduces basic concepts in fractal analysis. It starts with the description of the historical developments that led to their introduction and interactions. The definitions of fractional dimensions are introduced, and pertinent tools in geometric measure theory are recalled. Examples of multifractal functions and measures are studied. Finally, ubiquity systems, which play an increasing role in multifractal analysis, are introduced. The second text deals with fine geometric properties of measures obtained as limits of multiplicative processes. One starts by showing in which contexts they appear, and what are their key properties. The notions of dimension of a measure and of multifractal analysis are introduced in a general setting, and illustrated on the aforementioned examples. Finally, one shows the efficiency of these measures for the description of percolation on trees, and for dynamical or random coverings. The third text describes the time evolution of objects that disaggregate in a random way, and the fragments of which evolve independently. A statistical self-similarity assumption endows them with a structure of random fractal. The foundations of fragmentation theory are given, and the laws of these processes are shown to be characterized by a self-similarity index, a dislocation measure and an erosion coefficient. Then, one considers a random tree endowed with a distance, that allows to describe the genealogy of the process. Finally, one studies the speed with which the fragment containing a given point decays. This leads to the introduction of a multifractal spectrum of speeds of fragmentation.

Abstract

Following the seminal contributions of Benoît Mandelbrot in the 70s, concepts derived from fractal geometry gave a new impulse to several areas of mathematics. The goal of this volume is to present syntheses on two subjects where important advances occurred in the last 15 years: multiplicative processes and fragmentation. One arose from harmonic analysis (Riesz products) and the other from a probabilistic model proposed by N. Kolmogorov in order to explain experimental observations on rock fragmentation; however they share analogies and use common mathematical tools issued from the study of random fractals. The first text introduces basic concepts in fractal analysis. It starts with the description of the historical developments that led to their introduction and interactions. The definitions of fractional dimensions are introduced, and pertinent tools in geometric measure theory are recalled. Examples of multifractal functions and measures are studied. Finally, ubiquity systems, which play an increasing role in multifractal analysis, are introduced. The second text deals with fine geometric properties of measures obtained as limits of multiplicative processes. One starts by showing in which contexts they appear, and what are their key properties. The notions of dimension of a measure and of multifractal analysis are introduced in a general setting, and illustrated on the aforementioned examples. Finally, one shows the efficiency of these measures for the description of percolation on trees, and for dynamical or random coverings. The third text describes the time evolution of objects that disaggregate in a random way, and the fragments of which evolve independently. A statistical self-similarity assumption endows them with a structure of random fractal. The foundations of fragmentation theory are given, and the laws of these processes are shown to be characterized by a self-similarity index, a dislocation measure and an erosion coefficient. Then, one considers a random tree endowed with a distance, that allows to describe the genealogy of the process. Finally, one studies the speed with which the fragment containing a given point decays. This leads to the introduction of a multifractal spectrum of speeds of fragmentation.

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Additional indexing

Other titles:Some interactions between analysis, probabilities and fractals
Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2010
Deposited On:24 Apr 2013 11:30
Last Modified:05 Apr 2016 16:43
Publisher:Societe Mathematique de France
ISSN:1272-3835
Official URL:http://smf4.emath.fr/en/Publications/PanoramasSyntheses/2010/32/html/smf_pano-synth_32.php

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