Table Of ContentSPATIAL BRANCHING IN
RANDOM ENVIRONMENTS
AND WITH INTERACTION
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Advanced Series on
Statistical Science &
Vol. 20
Applied Probability
SPATIAL BRANCHING IN
RANDOM ENVIRONMENTS
AND WITH INTERACTION
János Engländer
University of Colorado Boulder, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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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,
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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
janos.englander@colorado.edu
1WhoevencorrectedmyEnglishinthispreface.
2Theauthor’sPh.D.advisorinthe1990s.
3Hishelpwithcomputer simulationsandpictureswasinvaluable.