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Theses

Random Structured Phylogenies

Abstract : This thesis consists of four self-contained chapters whose motivations stem from population genetics and evolutionary biology, and related to the theory of fragmentation or coalescent processes. Chapter 2 introduces an infinite random binary tree equipped with Poissonian mutations along its branches and with a finite measure on its boundary. The allelic partition – partition of the boundary into groups carrying the same combination of mutations – is defined and its intensity measure is described. Chapters 3 and 4 are devoted to the study of nested – i.e. taking values in the space of nested pairs of partitions – coalescent and fragmentation processes, respectively. These Markov processes are analogs of Λ-coalescents and homogeneous fragmentations. Nested coalescents are characterized in terms of Kingman coefficients and coagulation measures; nested fragmentations are characterized in terms of erosion coefficients and dislocation measures. Chapter 5 gives a construction of fragmentation processes with speed marks, which are fragmentation processes where each fragment is given a mark that speeds up or slows down its rate of fragmentation, and where the marks evolve as positive self-similar Markov processes. A Lévy-Khinchin representation of these generalized fragmentation processes is given, as well as sufficient conditions for their absorption in finite time to a frozen state.
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Submitted on : Wednesday, February 19, 2020 - 6:45:51 PM
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Jean-Jil Duchamps. Random Structured Phylogenies. Probability [math.PR]. Sorbonne Université, 2019. English. ⟨tel-02485010⟩

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