Centro de Investigacion´ en Matematicas´ Generalisations of Continuous State Branching Processes THESIS In partial fulfilment of the requirements for the degree of: Doctor of Sciences with orientation in Probability and Statistic PRESENTED BY: Sandra Palau Calderon´ ADVISORS: Dr. Juan Carlos Pardo Millan´ Dr. Andreas Kyprianou Guanajuato, M´exico. August, 2016 2 Integrantes del jurado Dr. Jos´eAlfredo L´opez Mimbela (CIMAT), Presidente Dr. V´ıctorManuel Rivero Mercado (CIMAT), Secretario Dr. Ger´onimoFrancisco Uribe Bravo (UNAM), Vocal Dr. Goetz Dietrich Kersting (Univ. de Frankfurt), Vocal Dr. Juan Carlos Pardo Mill´an(CIMAT), Vocal y Co-Director de la Tesis Dr. Andreas Kyprianou (Univ. de Bath), Lector Especial y Co-Director de la Tesis Dr. Vincent Roger Gilbert Bansaye (Ecole Polytechnique), Lector Especial. i ii Summary This thesis concerns on new developments of continuous-state branching processes. In particular, we focus on two different topics on this subject. The first topic concerns continuous-state branch- ing processes in a L´evyrandom environment. In order to define this class of processes, we study the existence and uniqueness of strong solutions of a particular class of non-negative stochas- tic differential equations driven by Brownian motions and Poisson random measures which are mutually independent. The long-term behaviours of absorption and explosion are also studied. The second topic is related to multi-type continuous-state branching processes with a count- able infinite number of types. We define this kind of processes as super Markov chains with both local and non-local branching mechanisms. Special attention is given to extinction events; in particular local and global extinction are studied. iii iv Acknowledgements Quiero agradecer a mis pap´asporque siempre me han impulsado para que sea feliz, en particular me han apoyado econ´omicay mentalmente para que siga este camino. A mis hermanos por todo su cari~noy sus palabras de amor. A Ro porque ha hecho que este andar sea mucho m´as agradable. He tenido la suerte de encontrar alguien que me entiende y comparte la misma pasi´on que yo. Tambi´enquiero agradecer a mis tios Fili y Heidi porque cuando empec´econ la idea de estudiar matem´aticas,siempre me apoyaron y durante mi estancia en el DF me hicieron sentir como en casa. Y a todas las pericas por nuestro chat lleno de amor. Quiero agradecer a Juan Carlos Pardo. Es un asesor impecable, la verdad estuve m´asque consentida. Me ayud´oa crear matem´aticas,me consigui´ovi´aticospara todos los viajes que tuve y me impulso a hacer cosas que jam´ascre´ırealizar. Adem´assiempre ha sido un gran consejero y una persona que ayuda y se preocupa por los dem´as. Agradezco a Victor Rivero por todos los consejos que me dio y porque sin duda, sin ´ely Maika mi vida en Bath no hubiera sido tan amena como lo fue. Siempre estar´eagradecida con Ger´onimo Uribe por todos los consejos y pl´aticasque tuvimos, tanto acad´emicascomo no acad´emicas.De hecho, puedo decir que mi decisi´onde hacer un doctorado en M´exicoy en particular en CIMAT fue influenciada por algunas de las pl´aticas.Y, fue una gran decisi´on. A special recognition to the Mexican people who made possible the existence of postgraduate fellowships. Through the Mexican Council of Science, CONACyT, I received the grant number 351643 to do my PhD. I also want to thanks the Department of Mathematical Sciences of the University of Bath where part of this thesis was done. Andreas Kyprianou was a really good advisor that helped me a lot before and after my time at Bath. I am sure that without him and Juan Carlos, my Ph.D. would have been more difficult. I also want to thanks the reviewers of my thesis. Thank you for took your time and checked the details. Es una suerte haber conocido a Javier Fern´andez,de ´elaprend´ıel amor por las matem´aticas. La formalidad de ´estasla aprend´ıcon Guillermo Grabinsky. Agradezco a Airam por estar juntas en el doctorado, fue una suerte compartir cub´ıculo,es una gran compa~nerade risas y pl´aticas profundas. Finalmente, quiero agradecer a todos mis amigos, a aquellos que a´unfrecuento y aquellos que el camino distancio. Gracias por las pl´aticas,las risas, las salidas y bailes, por formar parte de mi vida y ense~narmea disfrutar los distintos matices de ´esta. En particular gracias a Adriana, Alfredo, Amauri, Arturo, Dami´an,Daniel, Ehyter, Elia, George, Ger´onimo, v Gris, Harold, Jan, Jos´eLuis, Lu, Mario, Octavio, Osvaldo, To~no,...I cannot forget my friends at Bath; Matt D., Sam, Dorka and Marcus, thanks for making my time there more enjoyable. vi Contents Integrantes del jurado i Summary iii Acknowledgements v Introduction ix Outline . xvi 1 Branching processes in a L´evyrandom environment 1 1.1 Stochastic differential equations . .1 1.2 CBI-processes with competition in a L´evyrandom environment . .5 2 Long term behavior 13 2.1 CB-processes in a L´evyrandom environment . 13 2.2 Competition model in a L´evyrandom environment . 22 3 Asymptotic behaviour of exponential functionals of L´evyprocesses 29 3.1 Introduction and main results . 29 3.2 Applications . 33 3.2.1 Competition model in a L´evyrandom environment . 33 3.2.2 Diffusion processes in a L´evyrandom environment . 34 3.3 Proofs of Theorems 2, 3 and 4. 37 4 Stable CBLRE 47 4.1 Introduction . 47 4.2 Speed of explosion of SCBLRE . 49 4.3 Speed of absorption of SCBLRE . 52 4.4 Conditioned processes . 56 4.4.1 The process conditioned to be never absorbed . 56 4.4.2 The process conditioned on eventual absorption . 60 5 Multi-type continuous-state branching processes 65 5.1 Introduction and main results . 65 5.2 MCBPs as a superprocess . 69 5.3 Spectral properties of the moment semigroup . 72 vii 5.4 Spine decomposition . 76 5.5 Martingale convergence . 80 5.6 Local and global extinction . 82 5.7 Examples . 84 A 87 A.1 Lemmas of Chapter 1 . 87 A.2 Lemmas of Chapter 3 . 97 A.3 Lemmas of Chapter 4 . 98 A.4 Lemmas of Chapter 5 . 99 viii Introduction In many biological systems, when the population size is large enough, many birth and death events occur. Therefore, the dynamics of the population become difficult to describe. Under this scenario, continuous-state models are good approximations of these systems and sometimes they can be simpler and computationally more tractable. Moreover, the qualitative behaviour of the approximate models may be easier to understand. The simplest branching model in continuous time and space is perhaps the so called continuous- state branching process (or CB-process for short). They have been the subject of intensive study since their introduction by Jiˇrina[58]. This model arises as the limit of Galton-Watson processes; where individuals behave independently one from each other and each individual gives birth to a random number of offspring, with the same offspring distribution (see for instance Grimvall [49], for a general background see Chapter 12 of [64] or Chapter 3 of [70]). More precisely, a [0; 1]-valued strong Markov process Y = (Yt; t ≥ 0) with probabilities (Px; x ≥ 0) is called a continuous-state branching process if it has paths that are right-continuous with left limits and its law observes the branching property: for all θ ≥ 0 and x; y ≥ 0, h i h i h i −θYt −θYt −θYt Ex+y e = Ex e Ey e ; t ≥ 0: Moreover, its law is completely characterized by the latter identity, i.e. h i −λYt −xut(λ) Ex e = e ; t; λ ≥ 0; (1) where u is a differentiable function in t satisfying @u (λ) t = − (u (λ)); u (λ) = λ, (2) @t t 0 and satisfies the celebrated L´evy-Khintchine formula, i.e. Z 2 2 −λx (λ) = −q − aλ + γ λ + e − 1 + λx1fx<1g µ(dx); λ ≥ 0; (0;1) R 2 where a 2 R, q; γ ≥ 0 and µ is a measure concentrated on (0; 1) such that (0;1) 1 ^ x µ(dx) is finite. The function is convex and is known as the branching mechanism of Y . Let T0 = infft ≥ 0 : Yt = 0g and T1 = infft ≥ 0 : Yt = 1g denote the absorption and explosion times, respectively. Then Yt = 0 for every t ≥ T0 and Yt = 1 for every t ≥ T1. Plainly, equation (2) can be solved in terms of , and this readily yields the law of the absorption and extinction times ( see Grey [48]). More precisely, let η be the largest root of the branching mechanism , i.e. η = supfθ ≥ 0 : (θ) = 0g, (with 1 = supf;g). Then for every x > 0: ix R i) if η = 0 or if 0+ dθ=j (θ)j = 1, we have Px(T1 < 1) = 0, R ii) if η > 0 and 0+ dθ=j (θ)j < 1, we define Z t dθ g(t) = − ; t 2 (0; η): 0 (θ) The mapping g : (0; η) ! (0; 1) is bijective, and we write γ : (0; 1) ! (0; η) for its right- continuous inverse. Thus Px(T1 > t) = exp{−xγ(t)g; x; t > 0: R 1 iii) if (1) < 0 or if dθ= (θ) = 1, we have Px(T0 < 1) = 0, iv) if (1) = 1 and R 1 dθ= (θ) < 1, we define Z 1 dθ φ(t) = ; t 2 (η; 1): t (θ) The mapping φ :(η; 1) ! (0; 1) is bijective, we write ' : (0; 1) ! (η; 1) for its right- continuous inverse. Thus Px(T0 < t) = exp{−x'(t)g; x; t > 0: From (ii), we get that Px(T1 < 1) = 1 − exp{−xηg.