Annealed large deviations for diffusions in a random Gaussian shear flow drift

We prove a full large deviations principle for the one-dimensional laws of the diffusion process with random drift Xtε=ε2/3Wt+ε1/3∫0tV(Xsε/ε) ds, where V is a centered Gaussian shear flow random field independent of the Brownian W. The large deviations principle is an annealed one, that is integrated over the randomnesses of V and W.