  We obtain a Liouville property for stationary diffusions in random
environment which are small, isotropic perturbations of Brownian motion in
spacial dimension greater than two. Precisely, we prove that, on a subset of
full probability, the constant functions are the only strictly sub-linear maps
which are invariant with respect to the evolution of the diffusion. And, we
prove that the constant functions are the only bounded, ancient maps which are
invariant under the evolution. These results depend upon the previous work of
Fehrman [3] and Sznitman and Zeitouni [7] and, in the first case, our methods
are motivated by the work, in the discrete setting, of Benjamini,
Duminil-Copin, Kozma and Yadin [1].
