Journal of Statistical Physics

, Volume 131, Issue 1, pp 175–202 | Cite as

From Ballistic to Diffusive Behavior in Periodic Potentials

  • M. HairerEmail author
  • G. A. Pavliotis


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.


Homogenization Hypoelliptic diffusion Hypocoercivity 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Mathematics InstituteThe University of WarwickCoventryUK
  2. 2.Department of MathematicsImperial CollegeLondonUK

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