Optimal diffusive transport in a tilted periodic potential

We study the diffusive motion of an overdamped Brownian particle in a tilted periodic potential. Mapping the continuous dynamics onto. a discrete cumulative process we find exact expressions for the diffusion coefficient and the Peclet number which characterize the transport. At a sufficiently strong but subcritical bias an optimized transport with respect to the noise strength is observed. These results are confirmed by numerical solution of the Fokker-Planck equation.