University of Bath PHD Branching diffusion processes Harris, John William Award date: 2006 Awarding institution: University of Bath Link to publication Alternative formats If you require this document in an alternative format, please contact: [email protected] General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? 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Box 1346 Ann Arbor, Ml 48106-1346 Sum m ary In this thesis we study aspects of three branching diffusion models: a branching Brownian motion with an absorbing barrier; a typed branching Brownian motion, in which each particle has a ‘type’ that controls its spatial variance and breeding rate; and branching Brownian motion in a quadratic breeding potential. Within the context of these processes, we present results on a range of themes from the literature on branching diffusions. The major topics of interest are: links with differential equations, particularly travelling wave solutions; questions on exponential growth or eventual extinction in different regions of the domain of a process; and the rate of spatial spread of a process. Additive martingales are fundamental to the study of branching diffusions, both for their convergence properties and their use in changes of measure, and consequently such martingales are used heavily throughout this thesis. Particularly important for us is the use of additive martingales in changes of measure known as spine constructions; in recent years this technique has been used to great effect in providing intuitive proofs of many important classical results, as well new ones, in the theory of branching processes. Acknowledgements I would like to thank my supervisor, Simon Harris, whose enthusiasm for the subject first sparked my interest in probability as an undergraduate, and whose insight and encouragement were invaluable during the course of my postgraduate study. I am also grateful to Andreas Kyprianou for many useful discussions on my research, and to all members (past and present) of the probability group at Bath, from whom I have learnt a great deal and who made Bath such an enjoyable place to work. Thanks also to the department secretarial and support staff, and to the EPSRC for funding my studentship. I would finally like to thank my parents for their support throughout my education. Contents Chapter 1 Introduction 1 1.1. Branching Brownian motion .......................................................................... 1 1.2. Branching diffusions and differential equations ........................................ 2 1.2.1. KPP equations and travelling w aves .............................................. 4 1.2.2. Branching Brownian motion and the FKPP equation .................. 5 1.2.3. The one-sided FKPP s y s te m ........................................................... 7 1.3. Introduction to the spine approach ............................................................. 9 1.3.1. The spine construction for the Galton-Watson process ............... 9 1.3.2. A first spine construction for branching Brownian motion .... 11 1.3.3. More applications of s p in e s .............................................................. 14 1.4. Some other branching diffusion m odels ....................................................... 16 1.4.1. Typed branching diffusions .............................................................. 16 1.4.2. BBM with quadratic breeding ra te ................................................. 18 Chapter 2 The one-sided FKPP equation 19 2.1. Introduction and summary of results .......................................................... 19 2.2. Spine constructions for B B M ....................................................................... 23 2.3. Killed branching Brownian m o tio n .............................................................. 23 2.4. Non-existence for p > y/2/3 27 2.5. Existence and uniqueness for —oo < p < y/2j3 ........................................... 28 2.6. Asymptotic when —oo < p < \/2j3 .............................................................. 29 2.7. Right-most particle asymptotic —oo < p <y/2 0 ......................................... 34 Chapter 3 Further analysis of the BBM with absorption 36 3.1. W\ and statement of results .......................................................................... 36 3.2. The Hardy and Harris spine construction ................................................. 41 3.2.1. Spine n o ta tio n .................................................................................... 41 iii 3.2.2. New measures for B B M ..................................................................... 45 3.3. Proofs of the martingale resu lts .................................................................... 49 3.4. Proof of Theorem 3.1.6 52 3.5. A Yaglom-type limit th e o re m ....................................................................... 59 3.6. An alternative interpretation of Qa ............................................................. Chapter 4 Survival probabilities for sub-critical BBM with absorption 65 4.1. Introduction and summary of results .......................................................... 65 4.2. Survival probabilities ....................................................................................... 70 4.3. Proof of Proposition 4 .2 .1 ............................................................................. 72 4.4. Martingale results and spine decompositions .............................................. 79 Chapter 5 Exponential growth rates for a typed branching diffusion 87 5.1. Introduction ...................................................................................................... 88 5.2. Review of previous work on this m odel ....................................................... 90 5.3. Proof of Theorem 5.1.1: upper bound ......................................................... 98 5.4. Proof of Theorem 5.1.1: lower b o u n d ............................................................ 99 5.5. The spine construction ...................................................................................... 104 5.6. Proof of Theorem 5.4.1: the short climb probability ...................................... 109 Chapter 6 BBM in a quadratic breeding potential 120 6.1. Introduction .........................................................................................................120 6.2. Additive martingales for the (/?t/2;R )-B B M .................................................121 6.3. A lower bound for the right-most p a rtic le ....................................................126 6.4. The asymptotic exponential speed of the right-most particle ...................... 127 References 127 Chapter 1 Introduction In this introductory chapter we discuss the themes in the branching-diffusion literature that are the focus of this thesis, and describe some of the most relevant work that has been done in these areas. Our intention here is to introduce some important concepts in the theory of branching processes, and also to provide motivation for the problems considered later on; the results that represent the original work of the thesis will be given chapter by chapter. 1.1 Branching Brownian motion We begin with a description of the simplest branching diffusion — branching Brownian motion (BBM) on M. It is a compound of three stochastic elements. (i) The spatial motion. As