Branching random walks with selection

Abstract : We consider branching Brownian motion which is a mathematical object modeling the evolution of a population. In this system, particles diffuse on the real line according to Brownian motions and branch independently into two particles at rate $1$. We are interested in the rightmost (resp. leftmost) position at time $t$, which is defined as the maximum (resp. minimum) among the positions occupied by the particles alive at this time. According to \citet{Lalley-Sellke1987}, every particle born in this system will have a descendant reaching the rightmost position at some future time. We study this phenomenon quantitatively, by estimating the first time when every particle alive at time $s$ has had such a descendant. We then study an analogous model the branching random walk in discrete time, in which random walks are indexed by a Galton-Watson tree. Similarly, we define the rightmost and the leftmost positions at the $n$-th generation. We consider the walk starting from the root which ends at the leftmost position. We show that this work, after being properly rescaled, converges in law to a normalized Brownian excursion. The last part of the thesis concerns the evolution of a population with selection. Given a regular tree in which each individual has $N$ children, we attach to each individual a random variable. All these variables are i.i.d., uniformly distributed in $[0,1]$. Selection applies as follows. An individual is kept if along the shortest path from the root to the individual, the attached random variables are increasing. All other individuals are killed. We study the asymptotic behaviors of the evolution of the population when $N$ goes to infinity.
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Xinxin Chen. Branching random walks with selection. Probability [math.PR]. Université Pierre et Marie Curie - Paris VI, 2013. English. ⟨tel-00920308⟩

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