We consider a one-dimensional diffusion process in a random medium, which is itself generated by a Wiener process. The asymptotic behaviour is investigated, using the self-similarity of the medium process. The results are analogous to the lattice case.
"A One-Dimensional Diffusion Process in a Wiener Medium." Ann. Probab. 14 (4) 1206 - 1218, October, 1986. https://doi.org/10.1214/aop/1176992363