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Grandes déviations pour les temps locaux d'auto-intersections de marches aléatoires

Clément Laurent 1
1 Probabilités
LATP - Laboratoire d'Analyse, Topologie, Probabilités
Abstract : In this thesis we are interested in the self-intersection local times of random walks. This quantity is defined as the $p$-norm to the power of $p$ of the local times of the random walk. It measures how much the trajectory of the random walk intersects itself. The self-intersection local times is connected with various physical models as polymer models or problems of anomalous dispersion in layered random flows, but it is also linked with the mathematical model of random walks in random sceneries. More precisely, we are interested in the large deviations of the self-intersection local times, i.e. we work on the probability for the intersections to be larger than expected. This question that has been studied a lot during the 2000's is divided in three cases, the subcritical one, the critical one and the super critical one. We improve the knowledge about this question by two complete results and a partial one. First, we have proved a large deviation principle in the critical and super critical cases of $\alpha$-stable random walks, then we have improved the deviations' scales to the entire subcritical case of simple random walk, finally we are extending this last result to the $\alpha$-stable random walks. The three proofs are based on a version due to Eisenbaum of a Dynkin isomorphism theorem. This method which has been first introduced by Castell in the critical case, is extended here to the others cases. Thus, we have succeeded to unify the methods of proof by this isomorphism theorem.
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Submitted on : Monday, November 28, 2011 - 4:03:24 PM
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Clément Laurent. Grandes déviations pour les temps locaux d'auto-intersections de marches aléatoires. Probabilités [math.PR]. Université de Provence - Aix-Marseille I, 2011. Français. ⟨tel-00645783⟩

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