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Invariant distributions and scaling limits for some diffusions in time-varying random environments

Offret, Yoann

In: Probability Theory and Related Fields, 2014, vol. 158, no. 1-2, p. 1-38

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    Titre
    • Invariant distributions and scaling limits for some diffusions in time-varying random environments
    Auteur
    • Offret, Yoann. Institut de Mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • Probability Theory and Related Fields, 2014, vol. 158, no. 1-2, p. 1-38. Springer Berlin Heidelberg
    Autre version électronique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:325440
    Summary
    We consider a family of one-dimensional diffusions, in dynamical Wiener mediums, which are random perturbations of the Ornstein-Uhlenbeck diffusion process. We prove quenched and annealed convergences in distribution and under weigh-ted total variation norms. We find two kind of stationary probability measures, which are either the standard normal distribution or a quasi-invariant measure, depending on the environment, and which is naturally connected to a random dynamical system. We apply these results to the study of a model of time-inhomogeneous Brox's diffusions, which generalizes the diffusion studied by Brox (Ann Probab 14(4):1206-1218, 1986) and those investigated by Gradinaru and Offret (Ann Inst Henri Poincaré Probab Stat, 2011). We point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations.