Asymptotic analysis for the generalized Langevin equation

M. Ottobre, G. A. Pavliotis

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36 Citations (Scopus)

Abstract

Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in Eckmann J-P et al 1999 Commun. Math. Phys. 201 657-97. Ergodicity, exponentially fast convergence to equilibrium, short time asymptotics, a homogenization theorem (invariance principle) and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity (Villani C 2009 Mem. Am. Math. Soc. 202 iv, 141) is made.

Original languageEnglish
Pages (from-to)1629-1653
Number of pages25
JournalNonlinearity
Volume24
Issue number5
DOIs
Publication statusPublished - May 2011

Keywords

  • NONEQUILIBRIUM STATISTICAL-MECHANICS
  • SMOLUCHOWSKI-KRAMERS APPROXIMATION
  • KAC-ZWANZIG MODEL
  • HEAT BATH MODELS
  • ERGODIC PROPERTIES
  • ANHARMONIC CHAINS
  • LIMIT-THEOREM
  • SYSTEMS
  • HOMOGENIZATION
  • OSCILLATORS

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