  In this note we present some examples of diffusions in random environment
whose asymptotic behavior is rather surprising. We construct a family of
diffusions that are small perturbations of Brownian motion with non-vanishing
expected local drift under the static measure of the environment but where the
ballistic behavior is lost. As slight modifications of this collection of
diffusions we also provide examples with ballistic behavior where the
non-vanishing limiting velocity points to a direction opposite to the expected
local drift under the static measure.
