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We consider a family of one-dimensional diffusions, in dynamical Wienermediums, which are random perturbations of the Ornstein-Uhlenbeck diffusionprocess. We prove quenched and annealed convergences in distribution and underweighted total variation norms. We find two kind of stationary probabilitymeasures, which are either the standard normal distribution or aquasi-invariant measure, depending on the environment, and which is naturallyconnected to a random dynamical system. We apply these results to the study ofa model of time-inhomogeneous Brox's diffusions, which generalizes thediffusion studied by Brox (1986) and those investigated by Gradinaru and Offret(2011). We point out two distinct diffusive behaviours and we give the speed ofconvergences in the quenched situations.

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