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Dynamique de diffusions inhomogènes sous des conditions d'invariance d'échelle

Abstract : We study the asymptotic behaviour of some stochastic processes whose dynamics depends not only on position, but also time, and such that the diffusion term and the potential satisfy some scaling properties. We point out a general phase transition phenomenon, entirely determined by the self-similar parameters. The main idea is to consider an appropriate scaling transformation, taking full advantage of the scaling properties. In the first part, we investigate a family of one-dimensional diffusion processes, driven by a Brownian motion, whose drift is polynomial in time and space. These diffusions are continuous counterparts of the random walks studied by Menshikov and Volkov (2008) and related to theFriedman's urn model. We give, in terms of all scaling parameters, the iterated logarithm type laws, the scaling limits and the explosion times of these processes.The second part dealt with a family of diffusion processes in random environment, directed by a one dimensional Brownian motion, whose potential is Brownian in space and polynomialin time. This situation is a generalization of the time-homogeneous Brox's diffusion (86) studied in an extensive body of the literature. We obtain in the critical case a quasi-invariant and quasi stationary random measure for the time-inhomogeneous semi-group, deduced from the study of a underlying random dynamical system.
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Submitted on : Monday, September 10, 2012 - 4:19:08 PM
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  • HAL Id : tel-00730606, version 1

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Yoann Offret. Dynamique de diffusions inhomogènes sous des conditions d'invariance d'échelle. Probabilités [math.PR]. Université Rennes 1, 2012. Français. ⟨NNT : 2012REN1S022⟩. ⟨tel-00730606⟩

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