Asymptotic analysis for the generalized Langevin equation

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.