Numerical study of ergodicity for the overdamped Generalized Langevin Equation with fractional noise

TL;DR

This paper investigates the ergodic properties of the overdamped Generalized Langevin Equation driven by fractional Gaussian noise, providing numerical algorithms and convergence proofs, with applications to double well potentials in 1D and 2D.

Contribution

It introduces both a direct and a fast numerical scheme for the fractional stochastic differential equation and analyzes ergodicity for nonlinear forces.

Findings

Convergence orders are established for the proposed algorithms.

Numerical verification confirms theoretical convergence results.

Ergodicity is demonstrated for double well potentials in 1D and 2D.

Abstract

The Generalized Langevin Equation, in history, arises as a natural fix for the rather traditional Langevin equation when the random force is no longer memoryless. 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). While the ergodicity is clear for linear forces, it remains less transparent for nonlinear forces. In this work, we present both a direct and a fast algorithm respectively to this FSDE model. The strong orders of convergence are proved for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then present the ergodicity study of the double well potentials in both 1D and 2D setups.