| keyword(s) : Processus de naissances et de morts – systèmes de particules en interaction – dynamique des populations – structure d'âge – asymptotique des grandes populations – théorème central limite – grandes déviations – dynamiques adaptatives – solutions statistiques – Navier-Stokes 2D – McKean-Vlasov – ondelettes – schéma de discrétisation numérique |
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| Stochastic particle models for problems of adaptive evolution and for the approximations of statistical solutions |
| This thesis is divided into two independent parts. In the first one, we are interested in a microscopic individual-based model for the description of a population structured by traits and ages. We study the ecology of the system (population dynamics problems) in a large population asymptotics. Under appropriate renormalizations, the microscopic process converges to the measure solution of a deterministic evolution equation. A Central Limit Theorem and the exponential deviations of this convergence are studied. These results are used to generalize some evolution models from the recent theory of adaptive dynamics to age-structured populations. These models describe the evolution of the trait structure of a population on large time scales and under the assumptions of rare (and possibly small) mutations and large populations. In the second part of this thesis, we consider McKean-Vlasov and 2D Navier-Stokes partial differential equations with random initial conditions. The law of the solutions, which are then random variables, is called statistical solution. Using a probabilistic approach for these equations, we propose original stochastic wavelet particle approximations for the moments of order 1 of the statistical solutions, and study the convergence rates of the proposed procedures. |
| english keyword(s) : Birth and death processes – interacting particle systems – population dynamics – age structure – large population asymptotics – central limit theorem – large deviations – adaptive dynamics – statistical solutions – 2D Navier-Stokes – McKean-Vlasov – wavelets – numerical discretization scheme |
