| keyword(s) : modèles d'évolution – dynamiques adaptatives – système de particules en interaction – processus à valeurs mesure – séparation d'échelles de temps – processus de sauts – diffusion dégénérée – grandes déviations – problème de sortie de domaine – problème d'atteinte de points isolés – existence faible – unicité en loi – propriété de Markov forte |
| Alternative location : http://www-sop.inria.fr/tosca/personnel/Nicolas.Champagnat/Thesis_Champagnat.pdf |
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| Mathematical study of stochastic models of evolution belonging to the ecological theory of adaptive dynamics |
| This thesis is interested in the probabilistic study ofecological models belonging to the recent theory of "adaptive dynamics". After having presented and generalized the scope and biological heuristics of these models, we obtain a microscopic justification of a jump process modelizing evolution from a measure-valued interacting particle system describing the population dynamics at the individual level. This is a time scale separation result based on two asymptotics: rare mutations and large population. Then, we obtain an ordinary differential equation known as the "canonical equation of adaptive dynamics" by applying an asymptotic of small jumps to the preceding process. This limit leads us to introduce a diffusion model of evolution as a diffusion approximation of the jump process, which coefficients present bad regularity properties: discontinuous drift and degenerate diffusion parameter at the same points. We then study the weak existence, uniqueness in law and strong Markov property for these processes, which are linked to the question whether these diffusions can reach particular isolated points of the space in finite time or not. Finally, we prove a large deviation principle for these degenerate diffusions, allowing to study the problem of exit from an attracting domain, which is a fundamental biologial question. |
| english keyword(s) : models of evolution – adaptive dynamics – interacting particle systems – measure-valued processes – time scale separation – jump processes – degenerate diffusions – large deviations – problem of diffusion exit from a domain – polarity of isolated points – weak existence – uniqueness in law – strong Markov property |
