The Generalized Langevin Equation in Harmonic Potentials: Anomalous Diffusion and Equipartition of Energy

We consider the generalized Langevin equation (GLE) in a harmonic potential with power law decay memory. study anomalous diffusion of particle's displacement and velocity. By comparison free particle situation which velocity was previously shown to be either diffusive or subdiffusive, we find that, when trapped potential, may superdiffusive. Under slightly stronger assumptions on memory kernel, namely, for kernels related broad class completely monotonic functions, show that both harmonically bounded GLE satisfy equipartition energy condition. This generalizes known results under particular kernel instances such as Rouse (exactly) function.