From ballistic to diffusive behavior in periodic potentials
Probability
2009-11-13 v1 Functional Analysis
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.
Cite
@article{arxiv.0707.2352,
title = {From ballistic to diffusive behavior in periodic potentials},
author = {Martin Hairer and Grigorios Pavliotis},
journal= {arXiv preprint arXiv:0707.2352},
year = {2009}
}