Asymptotic analysis for the generalized Langevin equation
M Ottobre, G A Pavliotis · Nonlinearity · 2011
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