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A Selection of Stochastic Processes Emanating from Natural Sciences

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A Selection of Stochastic Processes Emanating from Natural Sciences A selection of stochastic processes emanating from natural sciences vorgelegt von Diplom-Mathematikerin Maite Isabel Wilke Berenguer Berlin Von der Fakult¨atII - Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation. Promotionsausschuss: Vorsitzender: Prof. Dr. math. J¨orgLiesen Berichter/Gutachter: Prof. Dr. rer. nat. Michael Scheutzow Berichter/Gutachter: Prof. Dr. rer. nat. Frank Aurzada Tag der wissenschaftlichen Aussprache: 07. Oktober 2016 Berlin 2016 Jack of all trades, master of none,... ... though oftentimes better than master of one. " 4 Contents I Percolation 13 1 Lipschitz Percolation 15 1.1 (Classic) Lipschitz Percolation . 16 1.1.1 Applications and related fields . 19 1.2 Lipschitz Percolation above tilted planes . 19 1.3 Asymptotic bounds on the critical probability . 23 1.3.1 A dual notion: λ-paths . 25 1.3.2 Proofs of lower bounds . 27 1.3.3 Proofs of upper bounds . 40 II Population Genetics 49 2 A novel seed-bank model 51 2.1 A famous model by Fisher and Wright and Kingman's dual . 54 2.2 Modelling a seed-bank . 64 2.3 The Wright-Fisher model with geometric seed-bank . 68 2.3.1 A forward scaling limit . 72 2.3.2 The dual of the seed-bank frequency process . 77 2.3.3 Long-term behavior and fixation probabilities . 78 2.4 The seed-bank coalescent . 81 2.4.1 Related coalescent models . 87 2.5 Properties of the seed-bank coalescent . 88 2.5.1 Coming down from infinity . 88 2.5.2 Bounds on the time to the most recent common ancestor 95 2.5.3 Recursions for important values . 105 2.6 Technical results . 112 2.6.1 Convergence of Generators . 112 2.6.2 Proofs of recursions . 116 5 Contents III Random Dynamical Systems 121 3 Volterra Stochastic Operators 123 3.1 Quadratic Stochastic Operators . 124 3.1.1 Biological origins, related developments and enhance- ments of the model . 125 3.2 Polynomial Stochastic Operators . 127 3.3 Randomization of the model . 134 3.4 A Martingale Lemma . 144 4 A Random Dynamical System 149 4.1 Introduction to Random Dynamical Systems and Attractors . 150 4.2 Evolution of the RDS forward in time . 156 4.2.1 Pullback attractors of our RDS . 156 4.2.2 Considering forward convergence . 158 4.3 Evolution backward in time . 167 4.4 Delta attractors - refining an established concept . .171 4.5 Auxiliary observations . 188 4.5.1 Observations related to measurability . 188 4.5.2 A helpful Markov chain . 190 4.5.3 A suitable metric . 191 4.5.4 Derivatives of Volterra PSOs . 196 6 Introduction The history of probability theory is no doubt closely intertwined with its application in diverse areas, which has been its boon and bane. One could say that mathematical probability started with the interest of Pascal and Fermat in the description of gambling games common at their time. This same motivation in mind, first Bernoulli and De Moivre and later Laplace continued the developement of the mathematical theory of games of chance and began applying it to statistics of human population and insurance ques- tions, without regard for the comparability of these to the original problem. As a result of these examples, the field of applications of probability theory expanded rapidly throughout the 19th century (creating new areas like sta- tistical mechanics or actuarial mathematics), while mathematical probability itself experienced a period of stagnation as the weakness of the conceptual foundation was neglected. As unfounded applications in social and moral questions began to emerge and Bertrand presented a series of Paradoxa the need for an axiomatization became evident. This was fully accomplished by Kolmogorov in the early 1930s. It still took some time for probability theory to shake its image as a non-rigorous science (outside of the Soviet Union), but nowadays it is a flourishing legitimate branch of mathematics, in addition to being applied in other fields as diverse as genetics, economics, psychology and engineering.1 This description suggests probability theory has a one-way profitable connection to other sciences. However, credit should also go tothe `applications' since the flow of inspiration often goes the other way round as new probabilistic, mathematically fascinating objects arise from an applied model. Both profit flows are present in this thesis. This work is constituted of three independent parts that can be read in any order. They all draw a link between the theory of stochastic processes indexed by discrete sets with a structure of independence and models in science. In this context, Part I: Percolation differs from the remainder of the text as here the index in question represents not time, but space. Parts II: Popu- 1This information can be found in [12] and [74] 7 Contents lation Genetics and III: Random Dynamical Systems are closer connected, as the independent structure results in Markovian processes (mostly) with dis- crete time-index and their joint origin lies in biological population genetics. However, they differ strongly in their present relation to biology. The results of Part II do indeed parallel current developement in the field of biological population genetics and have actual applications in that area (cf. [6]). Part III on the other hand, has the other type of relation to `applications'. That is, it focuses on the mathematical relevance of problems that arose from bio- logical population genetics to the extent of introducing a new mathematical object in the end, which serves as vindication for the different title of the parts. Since the three parts do differ in spite their commonalities, detailed in- troductions are left to the individual chapters. Part of this thesis has already been published in [6], [9], [22], [56]. This is detailed in the following, together with the structure of this work: Part I: Percolation. The first part of this thesis consists of the chapter on Lipschitz Percolation. It begins with a thorough introduction to (clas- sic) Lipschitz percolation outlining previous results on this model from [19] and [47] and discusses related models. Section 1.2 then introduces our novel contribution of Lipschitz percolation above tilted planes summarizing prelim- inary results. Section 1.3 is then devoted to the main result of this chapter: Theorem 1.14. Here, first, the exact bounds obtained are detailed in aseries of propositions, separated into upper and lower asymptotical bounds. The important notion of λ-paths in a sense dual to Lipschitz percolation is given in Section 1.3.1. For better readability, all proofs are grouped into Sections 1.3.2 and 1.3.3. The complete content of Sections 1.2 and 1.3 is joint work with Prof. M. Scheutzow and Prof. A. Drewitz and has been published in [22]. Part II: Population Genetics. The second part is made up of the chap- ter on a novel seed-bank model. This begins with a basic introduction to the two most important models in population genetics: the classic Wright- Fisher model and the Kingman coalescent. The purpose of Section 2.1 is to familiarize the reader new to this field with the type of questions asked in population genetics together with their corresponding answers that are classic results in this branch of mathematics. This also serves to highlight the parallels between our seed-bank model and the classic theory. Section 2.2 then provides an overview of the types of extended Wright-Fisher mod- els reformed to include the seed-bank phenomenon and known results so far, 8 Contents explaining also their limitations. Our own addition to this family of models is presented in Section 2.3, which also collects first results on it. In Section 2.3.1 we obtain a frequency process as a scaling limit of our model, that is the solution to a system of two- dimensional SDEs. We identify its dual-process in Section 2.3.2 and apply it in Section 2.3.3 to determine the diffusion’s longtime behaviour. In Section 2.4, we then define the most important object of this chapter which is a new seed-bank coalescent corresponding to the previously derived dual block-counting process. We explain how it describes the ancestry of the Wright-Fisher geometric seed-bank model and conclude the section with a comparison to other similar coalescents. Section 2.5 summarizes the most important properties of the seed-bank coalescent. We are able to prove that the seed-bank coalescent does not come down from infinity (cf. Section 2.5.1). In the following, we obtain that the expected time to the most recent common ancestor of a sample of k individuals is of asymptotic order log log k as k gets large. The section is concluded with recursions for selected quantities of interest of our model, needed in the scope of applications in order to execute simulations. Here, we also give a brief discussion of an extension of the seed- bank model that we studied in [6] by adding mutation. The last section of this chapter serves the purpose of an appendix, containing the technical details of calculations in Sections 2.3 and 2.5. We shall remark here that the results obtained in Sections 2.3 - 2.6 have previously been published in [9], with the exception of 2.5.3 and in part also 2.6.2 which are part of [6]. These two publications were joint work with Profes. N. Kurt and J. Blath, Dr. A. Gonz´alezCasanova and in the case of [6] also with Dr. B. Eldon and the results also feature in the dissertation of Dr.
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