Numerical approximation and fast evaluation of the overdamped generalized Langevin equation with fractional noise

The generalized Langevin equation (GLE) is a stochastic integro-differential equation that has been used to describe the movement of microparticles with sub-diffusion phenomenon. It has been proved that with fractional Gaussian noise (fGn) mostly considered by biologists, the overdamped Generalized Langevin equation satisfying fluctuation dissipation theorem can be written as a fractional stochastic differential equation (FSDE). In this work, we present both a direct and a fast algorithm respectively for this FSDE model in order to numerically study ergodicity. The strong orders of convergence are proven for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then verify the convergence to Gibbs measure algebraically for the double well potentials in both 1D and 2D setups. Our work is new in numerical analysis of FSDEs and provides a useful tool for studying ergodicity. The idea can also be used for other stochastic models involving memory.