On the equivalence of the static and dynamic points of view for diffusions in a random environment

We study the equivalence of the static and dynamic points of view for diffusions in a random environment in dimension one. First we prove that the static and dynamic distributions are equivalent 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 attached to the diffusion. We moreover show that the equivalence of the static and dynamic points of view is characterized by the existence of so-called "almost linear coordinates".