Fractal Geometry and Stochastics IV
Over the last fifteen years fractal geometry has established itself as a substantial mathematical theory in its own right. The interplay between fractal geometry, analysis and stochastics has highly influenced recent developments in mathematical modeling
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Series Editors Charles Newman Sidney I. Resnick
Fractal Geometry and Stochastics IV
Christoph Bandt Peter Mörters Martina Zähle Editors
Birkhäuser Basel · Boston · Berlin
Editors: Christoph Bandt Institut für Mathematik und Informatik Ernst-Moritz-Arndt-Universität 17487 Greifswald Germany e-mail: [email protected]
Peter Mörters Department of Mathematical Sciences University of Bath Bath BA2 7AY UK e-mail: [email protected]
Martina Zähle Mathematisches Institut Friedrich-Schiller-Universität 07740 Jena Germany e-mail: [email protected]
2000 Mathematics Subject Classification: 20E08, 28A80, 35K05, 37B10, 37D20, 37E35, 60B15, 60D05, 60G15, 60G17, 60G18, 60G50, 60G51, 60G52, 60H15, 60J45, 60J50, 60J80, 60J85, 60K35, 60K37, 82B27
Library of Congress Control Number: 2009931383 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISBN 978-3-0346-0029-3 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.
© 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-0346-0029-3
e-ISBN 978-3-0346-0030-9
987654321
www.birkhauser.ch
Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Part 1: Analysis on Fractals A. Grigor’yan, J. Hu and K.-S. Lau Heat Kernels on Metric Spaces with Doubling Measure . . . . . . . . . . . . . .
3
V.A. Kaimanovich Self-similarity and Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Part 2: Conformal Dynamics and Schramm-Loewner Evolution G.F. Lawler Multifractal Analysis of the Reverse Flow for the Schramm-Loewner Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Part 3: Random Fractal Processes D. Khoshnevisan From Fractals and Probability to L´evy Processes and Stochastic PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Part 4: Emergence of Fractals in Complex Systems J.E. Steif A Survey of Dynamical Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 J. Blath Measure-valued Processes, Self-similarity and Flickering Random Measures . . . . . . . . . . . . . . . . . . . . . . . .
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