From Ballistic to Diffusive Behavior in Periodic Potentials

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• 作者：M. Hairer and G. A. Pavliotis
• 关键词：Homogenization ; Hypoelliptic diffusion ; Hypocoercivity
• 刊名：Journal of Statistical Physics
• 出版年：2008
• 出版时间：April, 2008
• 年：2008
• 卷：131
• 期：1
• 页码：175-202
• 全文大小：718.8 KB

文摘

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.