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Chemins confinés dans un quadrant

Abstract : The PhD thesis "Paths confined to a quadrant" deals with two different aspects of the walks with small steps (i.e. to the eight nearest neighbors) that are confined to a quarter plane. First, we study the combinatorics of counting the planar walks which, while moving according to a given step set, remain in a quadrant. We then concentrate on the following problems: - to make explicit the generating function of the numbers of such trajectories; - to analyse the fashion in which that series depends on the step set, and in particular its nature (rational, algebraic, (non-)holonomic). Second, we study the random walks with values in a quadrant, which are taken homogeneous inside and killed at the boundary. For these processes, we are interested in the following probabilistic problems: - to make explicit the absorption probabilities at a fixed time in a certain site of the boundary, and in particular the absorption probabilities in some point of the boundary; - to find the asymptotic behavior of these probabilities; - to make explicit the probabilities that the process, at a fixed time, is in a certain site belonging to the interior of the quadrant, and in particular the Green functions; - to compute the precise asymptotics of these Green functions along all trajectories; - to obtain all non-negative harmonic functions, as well as the Martin compactification; - to analyse the hitting time of the axes, and in particular evaluating its asymptotic tail distribution. The methods that we use in order to solve these problems are partially based on complex analysis.
Type de document :
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Contributeur : Kilian Raschel <>
Soumis le : jeudi 25 novembre 2010 - 16:19:17
Dernière modification le : vendredi 29 mai 2020 - 16:01:21
Document(s) archivé(s) le : vendredi 26 octobre 2012 - 16:50:55


  • HAL Id : tel-00539964, version 1


Kilian Raschel. Chemins confinés dans un quadrant. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2010. English. ⟨tel-00539964⟩



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