One-dimensional diffusions in random environment and invariant measures for the environment process

We study one-dimensional diffusions in a stationary and ergodic random environment. We introduce the environment process, an auxiliary process that describes the environment as it is seen by the random walker. A key issue is the existence of an invariant measure for this process that is absolutely continuous w.r.t. the "static" distribution of the environment, as it provides for instance a law of large numbers. We characterize the existence of such an invariant measure. It exists if and only if either the speed in the law of large numbers does not vanish, or b/a is a.s. the gradient of a stationary function, where $a$ and $b$ are the covariance coefficient resp. the local drift attaches to the diffusion.