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

TL;DR

This paper studies one-dimensional diffusions in time-varying random environments, establishing convergence results, identifying stationary measures, and analyzing diffusive behaviors in generalized Brox's models.

Contribution

It introduces new convergence results for diffusions in dynamical environments and characterizes their stationary measures, extending Brox's diffusion analysis.

Findings

Proved quenched and annealed convergence in distribution.

Identified two types of stationary measures: normal and quasi-invariant.

Described two diffusive behaviors and convergence speeds.

Abstract

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.