Spatial Branching In Random Environments and With Interaction PDF
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SPATIAL BRANCHING IN RANDOM ENVIRONMENTS AND WITH INTERACTION 8991hc_9789814569835_tp.indd 1 12/5/14 9:30 am ADVANCED SERIES ON STATISTICAL SCIENCE & APPLIED PROBABILITY Editor: Ole E. Barndorff-Nielsen Published Vol. 9 Non-Gaussian Merton–Black–Scholes Theory by S. I. Boyarchenko and S. Z. Levendorskii Vol. 10 Limit Theorems for Associated Random Fields and Related Systems by A. Bulinski and A. Shashkin Vol. 11 Stochastic Modeling of Electricity and Related Markets . by F. E. Benth, J. Šaltyte Benth and S. Koekebakker Vol. 12 An Elementary Introduction to Stochastic Interest Rate Modeling by N. Privault Vol. 13 Change of Time and Change of Measure by O. E. Barndorff-Nielsen and A. Shiryaev Vol. 14 Ruin Probabilities (2nd Edition) by S. Asmussen and H. Albrecher Vol. 15 Hedging Derivatives by T. Rheinländer and J. Sexton Vol. 16 An Elementary Introduction to Stochastic Interest Rate Modeling (2nd Edition) by N. 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Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Engländer, Janos. Spatial branching in random environments and with interaction / by Janos Engländer, University of Colorado Boulder, USA. pages cm. -- (Advanced series on statistical science and applied probability ; vol. 20) Includes bibliographical references. ISBN 978-981-4569-83-5 (hardcover : alk. paper) 1. Mathematical statistics. 2. Branching processes. 3. Law of large numbers. I. Title. QA276.E54 2014 519.2'34--dc23 2014014879 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore EH - Spatial Branching in Random Env.indd 2 15/10/2014 10:52:11 AM October13,2014 15:59 BC:8991–SpatialBranchinginRandomEnvironments JancsiKonyv pagev This book is dedicated to the memory of my parents, Katalin and Tibor Engl¨ander, Z”L Istandattheseashore,alone,andstarttothink. Therearethe rushingwaves... mountainsofmolecules,eachstupidlyminding its own business ... trillions apart ... yet forming white surf in unison. Richard Feynman It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better. Henri Poincar´e v May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk October13,2014 15:59 BC:8991–SpatialBranchinginRandomEnvironments JancsiKonyv pagevii Preface I felt honored and happy to receive an invitation from World Scientific to write lecture notes on the talk that I gave at the University of Illinois at Urbana-Champaign. The talk was based on certain particle models with a particulartypeofinteraction. Iwasevenmoreexcitedtoreadthefollowing suggestion: Although your talk is specialized, I hope that you can write something related to yourarea of research... Such a proposal gives an author the opportunity to write about his/her favoriteobsession! Inthecaseofthisauthor,thatobsessionconcernsspatial branching models with interactions and in random environments. My conversations with biologists convinced me that even though many such models constitute serious challenges to mathematicians, they are still ridiculously simplified compared to models showing up in population biol- ogy. Now,thebiologist,ofcourse,shrugs: afterall,she‘knows’theanswer, by using simulations. She feels being justified by the power of modern computer clusters andthe almighty Law(of the largenumbers); we, math- ematicians, however would still like to see proofs, in no small part because they giveaninsightintothe reasonsofthephenomenaobserved. Secondly, thehighertheorderoftheasymptoticsoneinvestigates,thelessconvincing the simulation result. In this volume I will present a number of such models, in the hope that it will inspire others to pursue research in this field of contemporary probability theory. (My other hope is that the reader will be kind enough to find my Hunglish amusing rather than annoying.) An outline of the contents follows. In Chapter 1, we review the preliminaries on Brownian motion and diffusion, branching processes, branching diffusion and superdiffusion, and vii October13,2014 15:59 BC:8991–SpatialBranchinginRandomEnvironments JancsiKonyv pageviii viii Spatial Branching in Random Environments and with Interaction some analytical tools. This chapter became quite lengthy, even though several results are presented without proofs. Nevertheless, the expert in probability can easily skip many well-known topics. Chapter2presentsaStrongLawofLargeNumbersforbranchingdiffu- sions and, as a main tool, the ‘spine decomposition.’ Chapter 3 illustrates the result through a number of examples. Chapter 4 investigates the behavior of the center of mass for spatial branchingprocessesandtreatsaspatialbranchingmodel with interactions between the particles. In Chapters 5, 6 and 7, spatial branching models are considered in random media. This topic can be considered a generalization of the well- studied model of a Brownian particle moving among random obstacles. Finally, Appendix A discusses path continuity for Brownian motion, whileAppendix Bpresentssomeuseful maximumprinciplesforsemi-linear operators. Each chapter is accompanied by a number of exercises. The best way to digest the material is to try to solve them. Some of them, especially in the first chapter, are well known facts; others are likely to be found only here. How to read this book (and its first chapter)? I had three types of audience in mind: (1) Graduate students in mathematics or statistics, with the background of, say, the typical North American student in those programs. (2) Researchersinprobability,especiallythoseinterestedinspatialstochas- tic models. (3) Population biologists with some background in mathematics (but not necessarily in probability). Ifyouareinthesecondcategory,thenyouwillprobablyskipmanysections when reading Chapter 1, which is really just a smorgasbord of various tools in probability and analysis that are needed for the rest of the book. However, if you are in the first or third category, then I would advise you to try to go throughmost ofit. (And if youare a student, I recommend to read Appendix A too.) If you do not immerse yourself in the intricacies of theconstructionofBrownianmotion,youcanstillenjoythelaterchapters, but if you are not familiar with, say, martingales or some basic concepts for secondorderelliptic operators,thenthere is no wayyoucanappreciate the content of this book. October13,2014 15:59 BC:8991–SpatialBranchinginRandomEnvironments JancsiKonyv pageix Preface ix Asforbeinga‘smorgasbord’: hoppingfromonetopictoanother(seem- inglyunrelated)one,mightbeabitannoying. Theauthorherebyapologizes for that! However,adding more connecting argumentswouldhave resulted in inflating the already pretty lengthy introductory chapter. What should you do if you find typos or errors? Please keep calm and sendyourcommentstomyemailaddressbelow. Also,recallGeorgePo´lya’s famous saying: The traditional professor writes a,says b, means c; but it should bed. Several discussions on these models and collaborations in various projects are gratefully acknowledged. I am thus indebted to the following colleagues: JulienBerestycki,MineC¸agˇlar,Zhen-QingChen,ChrisCosner, Bill Fagan,SimonHarris,FrankdenHollander, SergeiKuznetsov,Andreas Kyprianou,1 MehmetO¨z,RossPinsky,2 YanxiaRen,N´andorSieben,3 Ren- ming Song, Dima Turaev and Anita Winter. My student Liang Zhang has been great in finding typos and gaps, for which I am very grateful to him. I am very much obliged to Ms. E. Chionh at World Scientific for her professionalism and patience in handling the manuscript. Finally, I owe thanks to my wife, Kati, for her patience and support during the creation of this book, and to our three children for being a continuing source of happiness in our life. Boulder, USA, 2014 Ja´nos Engl¨ander [email protected] 1WhoevencorrectedmyEnglishinthispreface. 2Theauthor’sPh.D.advisorinthe1990s. 3Hishelpwithcomputer simulationsandpictureswasinvaluable.
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