Study of some fractal and pathwise properties of continuous branching processes

Abstract : This thesis investigates some fractal and pathwise properties of branching processes with continuous time and state-space. Informally, this kind of process can be described by considering the evolution of a population where individuals reproduce and die over time, randomly. The first chapter deals with the class of continuous branching processes with immigration. We provide a semi-explicit formula for the hitting times and a necessary and sufficient condition for the process to be recurrent or transient. Those two results illustrate the competition between branching and immigration. The second chapter deals with the Brownian tree and its local time measures : the level-sets measures. We show that they can be obtained as the restriction, with an explicit multiplicative constant, of a Hausdorff measure on the tree. The result holds uniformly for all levels. The third chapter study the Super-Brownian motion associated with a general branching mechanism. Its total occupation measure is obtained as the restriction to the total range, of a given packing measure on the euclidean space. The result is valid for large dimensions. The condition on the dimension is discussed by computing the packing dimension of the total range. This is done under a weak assumption on the regularity of the branching mechanism.
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Submitted on : Tuesday, January 13, 2015 - 1:36:58 PM
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Jean-Pierre Duhalde. Study of some fractal and pathwise properties of continuous branching processes. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI, 2015. English. ⟨NNT : 2015PA066029⟩. ⟨tel-01102714⟩



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