Scaling Limits for the Generalized Langevin Equation

In this paper, we study the diffusive limit of solutions to generalized Langevin equation (GLE) in a periodic potential. Under assumption quasi-Markovianity, obtain sharp longtime equilibration estimates for GLE using techniques from theory hypocoercivity. We then show asymptotic results effective diffusion coefficient small correlation time regime, as well overdamped and underdamped limits. Finally, employ recently developed numerical method (Roussel Stoltz ESAIM Math Model Numer Anal 52(3):1051–1083, 2018) calculate wide range (effective) friction coefficients, confirming our results.