The Art of Random Walks
Einstein proved that the mean square displacement of Brownian motion is proportional to time. He also proved that the diffusion constant depends on the mass and on the conductivity (sometimes referred to Einstein’s relation). The main aim of this book is
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The Art of Random Walks
ABC
Author András Telcs Department of Computer Science and Information Theory Budapest University of Technology Electrical Engineering and Informatics Magyar tudósok körútja 2, 1117 Budapest Hungary e-mail: [email protected]
Library of Congress Control Number: 2006922866 Mathematics Subject Classification (2000): 60 10 60 45 35K 05 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-33027-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33027-1 Springer Berlin Heidelberg New York DOI 10.1007/b134090 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A EX package Typesetting: by the authors and SPI Publisher Services using a Springer LT Cover design: design & production GmbH, Heidelberg
Printed on acid-free paper
SPIN: 11688020
VA41/3100/SPI
543210
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The beginnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1
2
Basic definitions and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mean exit time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Model fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 10 13 14 16 18
Part I Potential theory and isoperimetric inequalities 3
Some elements of potential theory . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Electric network model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Harnack inequality and the Green kernel . . . . . . . . . . . . . . . . . . . 3.4 Resistance regularity . .
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