From Ballistic to Diffusive Behavior in Periodic Potentials |
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Authors: | M Hairer G A Pavliotis |
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Institution: | (1) Mathematics Institute, The University of Warwick, Coventry, CV4 7AL, UK;(2) Department of Mathematics, Imperial College, London, SW7 2AZ, UK |
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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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Keywords: | Homogenization Hypoelliptic diffusion Hypocoercivity |
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