The Smoluchowski–Kramer approximation of a generalized Langevin equation with state-dependent damping

Abstract

The small-mass limit is derived for a generalized Langevin equation (GLE) with state-dependent damping and a kernel expressed as an infinite sum of exponentials. The state dependence includes both the current and the past. By some bounded estimates and tightness, as the mass tends to 0, the GLE is shown to converge in distribution to a limit equation with additional drift terms that come from the current and the past state dependence, respectively.