Abstract
The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator.
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Hairer, M., Pavliotis, G.A. From Ballistic to Diffusive Behavior in Periodic Potentials. J Stat Phys 131, 175–202 (2008). https://doi.org/10.1007/s10955-008-9493-3
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DOI: https://doi.org/10.1007/s10955-008-9493-3