Diffusion in a periodic potential with a local perturbation

We consider the diffusion of a particle at X/sub t/ in a drift field derived from a smooth potential of the form V + B, where V is periodic and B is a bump of compact support. With no bump, B = 0, the mean squared displacment E(t) triple bond E absolute value X/sub t/-X02 = D(V)t + C + O(e/sup -λt/), λ >0, in any dimension. When B 0, we establish in one dimension the asymptotic expansion E(t) = D(V)t + α root t + C + (1/root t) Σ/sub n=0//sup ∞/ α/sub n//t/sup n/, α 0, as t ??? ∞. Our analysis relies on the Nash estimates developed in previous work for the transition density of the process and their consequences for the analytic structure of the Laplace transform E(s) of E(t)