  We study a model of diffusion in a brownian potential. This model was firstly
introduced by T. Brox (1986) as a continuous time analogue of random walk in
random environment. We estimate the deviations of this process above or under
its typical behavior. Our results rely on different tools such as a
representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani's lemma,
introduced at first by K. Kawazu and H. Tanaka (1997), and a decomposition of
hitting times developed in a recent article by A. Fribergh, N. Gantert and S.
Popov (2008). Our results are in agreement with their results in the discrete
case.
