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Theses

Temps local et diffusion en environnement aléatoire

Abstract : A diffusion in random environment is the solution of the following stochastic differential equation: dX(t) = dB(t) − 1/2 W’(X(t))dt where B is a standard Brownian motion and W a càd-làg process which is not necessarily differentiable (the previous SDE has then only a formal sense). Schumacher [69] and Brox [17] have shown that the diffusion X has a sub-diffusive behavior when W is also a standard Brownian motion. Moreover they point out a localization phenomena for X. This thesis is principally devoted to the description of the asymptotic behavior of the local time process of X. The local time LX(t, x) represents the time spent by X before t at point x. This is thereby a useful tool to study the localization of the diffusion. Here is described the limit law of the local time when the environment is a Brownian motion or more generally a stable Lévy process. We are also interested in the time spent by X in the neighborhood of the most visited points and in the almost sure asymptotic behavior of the maximum of the local time. In the last chapter of the thesis the notion of local time is used in a discrete version of the model to obtain informations on the environment. The goal is to apply this model to DNA sequencing.
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Roland Diel. Temps local et diffusion en environnement aléatoire. Mathématiques générales [math.GM]. Université d'Orléans, 2010. Français. ⟨NNT : 2010ORLE2036⟩. ⟨tel-00590440⟩

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