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

Yoann Offret (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 weighted 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 (1986) and those investigated by Gradinaru and Offret (2011). We point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations.

Mots clés: Fluctuating stationary distribution; Foster-Lyapunov drift condition; Time-dependent random environment; Time-inhomogeneous Brox's diffusion; Random dynamical system

Domaines: Probabilités [math.PR]

Yoann Offret. Invariant distributions and scaling limits for some diffusions in time-varying random environments. Probability Theory and Related Fields, 2014, 158 (1-2), pp.1-38. ⟨10.1007/s00440-012-0475-7⟩. ⟨hal-00690386v2⟩