We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani’s lemma and Lamperti’s representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571–609] who studied the corresponding problem for Sinai’s random walk in random environment.
"Moderate deviations for diffusions with Brownian potentials." Ann. Probab. 32 (4) 3191 - 3220, October 2004. https://doi.org/10.1214/009117904000000829