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Sdes for Embedded Successful Genealogies SDEs for embedded successful genealogies submitted by Dorottya Fekete for the degree of Doctor of Philosophy of the University of Bath Department of Mathematical Sciences July 2019 COPYRIGHT Attention is drawn to the fact that copyright of this thesis rests with the author. A copy of this thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that they must not copy it or use material from it except as permitted by law or with the consent of the author. This thesis may be made available for consultation within the University Library and may be photocopied or lent to other libraries for the purposes of consultation. Signature of Author . Dorottya Fekete Summary This thesis concerns different skeletal decompositions of various branching processes. First we develop a stochastic differential equation (SDE) approach to describe the fitness of certain sub-populations in asexual high-density stochastic population models. Initially we only consider continuous-state branching processes, then we extend the SDE approach to the spatial setting of superprocesses. In both cases the SDE can be used to simultaneously describe the total mass, and those embedded genealogies that propagate prolific traits in surviving populations, where `survival' can be interpreted in different ways. For example, it can mean survival beyond a certain time-horizon, but it can also mean survival according to some spatial criteria. In the second part of the thesis we construct the prolific backbone decomposition of multitype superprocesses by extending the semigroup approach already available for one-type branching processes. 2 Acknowledgements I'm grateful to so many people who one way or another helped me over the course of my years in Bath. First of all I would like to thank my supervisor, Andreas Kyprianou for his friendship, continuous guidance and support; your enthusiasm towards research, and the friendly atmosphere of our meetings always got me excited about academia, no matter how I'd felt hours before. I'm especially grateful for taking me to all those conferences around the world, where apart from learning a lot I also got to know a lot of inspiring probabilists, many of whom I'm lucky to call my friend now. These trips played a massive role in the completion of this thesis, in fact, the most productive work during my PhD was probably done on those 10 hour long flights to Mexico. A massive thanks goes to my second supervisor, Juan Carlos Pardo, whose support, just like Andreas's, went way beyond the boundaries of academia. Thank you for all the fruitful academic discussions, and for finding the time to go for a coffee, whenever I needed it the most. I feel incredibly lucky to have worked with people who supported not only my academic development, but also cared for me on a personal level. I would also like to thank Peter M¨orters,who while being my supervisor, always made time for me despite his extremely busy schedule. A big thank you to my other co-authors, Joaqu´ınFontbona, Jos´eLuis P´erezand Sandra Palau from whom I have learnt a lot. I would also like to thank Sandra for being such a good friend; I enjoyed every minute we have spent together. I would also like to acknowledge all the members of Prob-L@b for providing an inspiring and friendly environment. Many thanks goes to my examiners Matt Roberts and Jean-Fran¸cois Delmas for taking the time to read the thesis and making very valuable suggestions for the final version. I would like to give a huge thank you to Paul, Anna, Jess and especially Susie for having your door open all time, putting up with my personal drama, and truly going out of your way to make my SAMBa years more enjoyable. I will never forget the birthday cake you gave me in my first year being abroad, it meant a lot. I am also grateful to EPSRC for the four years of financial support. A very special thank you to Emma, Tsogii and Alice for their friendship, encouragement and making my time in Bath unforgettable, and to Ramses for always being there for me during my time in Mexico, muchas gracias! I would also like to thank The Climbing Academy and all my friends there for providing a pleasant escape from the PhD struggles. My warmest thanks I owe to my family, especially my parents Katalin and S´andor,for their unconditional love, enduring effort and endless support. Words cannot express how grateful I am for believing in me and working so hard to make this journey possible. This thesis is dedicated to you. 4 Contents 1 Introduction8 1.1 Examples of backbones in literature....................9 1.1.1 Galton-Watson process........................9 1.1.2 Branching Brownian motion in a strip............... 11 1.1.3 Continuous-state branching processes............... 14 1.1.4 Superprocesses............................ 18 1.2 Summary of chapters............................. 21 1.2.1 Skeletal stochastic differential equations for continuous-state branch- ing processes (Chapter2)....................... 21 1.2.2 Skeletal stochastic differential equations for superprocesses (Chap- ter3).................................. 23 1.2.3 Backbone decomposition of multitype superprocesses (Chapter4). 23 2 Skeletal stochastic differential equations for continuous-state branch- ing process 30 2.1 Introduction.................................. 31 2.2 Main results.................................. 35 2.3 Thinning of the CSBP SDE......................... 40 2.4 λ-Skeleton: Proof of Theorem 2.2.1..................... 41 2.5 Exploration of subcritical CSBPs...................... 47 2.6 Finite-time horizon Skeleton: Proof of Theorem 2.2.2........... 52 2.7 Thinning the skeleton to a spine: Proof of Theorem 2.2.3........ 53 3 Skeletal stochastic differential equations for superprocesses 66 3.1 Introduction.................................. 66 3.2 Skeletal decomposition............................ 70 3.3 SDE representation of the dressed tree................... 74 3.4 Sketch proof.................................. 79 4 Backbone decomposition of multitype superprocesses 98 4.1 Introduction and main results........................ 98 4.1.1 Multitype superprocesses....................... 101 4.1.2 The multitype supercritical superdiffusion conditioned on extinc- tion................................... 106 4.1.3 Dynkin-Kuznetsov measure...................... 107 4.1.4 Prolific individuals........................... 108 4.1.5 The backbone decomposition..................... 110 4.2 Proofs..................................... 112 6 Chapter 1 Introduction Historically branching processes have been used as stochastic models for the evolution of a population. They were originally introduced by Watson and Galton [32] in the 19th century to investigate the extinction of British aristocratic family names. Since then the original model has led to many generalisations, and branching processes have been a subject of extensive research. Their popularity is due to the fact that they can be widely applied in the physical and biological sciences; in modelling nuclear chain reactions, cosmic radiation, or bacterial reproduction, to mention just a few. Even though many aspects of branching processes are now well understood, survival remains a central topic of interest. As we will see \survival" can be interpreted in many different ways; it might simply mean the existence of individuals that have an infinite genealogical line of descent, but in a different setup \survival" can also mean the existence of genealogies that stay in a strip, or visit a compact domain infinitely often. A common thread is that on the event of \survival", certain genealogies do something infinitely often while others do not, and thus we can split the individuals of the population into two categories. The so called prolific individuals are part of the surviving genealogies, and as such they form the backbone of the process. The rest of the population can be thought of as a \dressing" of this backbone, since they can be seen as genealogies grafted onto the trajectories of the prolific skeleton. Individuals of the dressing have no long term contribution to \survival", and therefore many results are true for the original process if and only if they are true for the backbone. But since the latter is usually a stochastically \thinner" object with almost sure survival, results are generally easier to prove for the backbone. Decomposing different branching processes along their infinite genealogies has been an intensely researched topic in the recent years, which unavoidably led to not entirely consistent terminology. The expressions \backbone" and \skeleton" both appeared in research papers, signifying the same object. As in this thesis we study several different branching processes, while drawing inspiration from the vast literature already available on the topic, we will use the terms \skeleton" and \backbone" interchangeably. To motivate our results, we start by giving some examples for stochastic processes and their backbones that already exist in the literature. We try to cover a wide range of processes by discussing branching processes in both discrete and continuous setup, as well as the spatial extension of these. But as our main aim here is to give an intuitive picture, we won't be presenting the results in full rigour, for more detail the reader is referred to the referenced literature. 8 1.1 Examples of backbones in literature 1.1.1 Galton-Watson process Let us start with considering a Galton-Watson (GW) process Z = (Zt; t ≥ 0) with offspring distribution A, and initial population Z0 = 1. Here by survival we mean the event f8t ≥ 0 : Zt > 0g, that is when we have at least one individual in the population at any time t. It is well known that the probability of this event is positive if and only if the mean offspring number m = E[A] is greater than one, in which case we call the process supercritical. When m ≤ 1, then the process goes extinct almost surely, and Z is said to be (sub)critical (critical when m = 1, and subcritical when m < 1).
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