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

Abstract

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.

A Discretization of Caputo Derivatives with Application to Time Fractional SDEs and Gradient Flows

“…Since this is not our focus, we assume the Lipschitz condition for simplicity. As proved in [3] and [23], with assumption (4.9), the fractional SDE (4.5) has a unique strong solution and for any T > 0 there exists C(T ) > 0 such that the following hold.…”

“…For general cases, whether it converges to the Gibbs measure is unknown. Recently, in the case of overdamped GLE, some numerical experiments indicate that the law of X still converges algebraically to the corresponding Gibbs measure for general potential ( [23]).…”

“…In fact, this kind of distribution stays for long time (t ∼ 10 2 ). We expect that when t is very large, it can be close to the Gibbs distribution as in [23]. We choose not to do the simulation for H = 0.8 and t ∼ 10 3 since the complexity for obtaining a sample path is O(N 2 ) and the simulation is expensive (we need 10 4 samples).…”

“…We choose not to do the simulation for H = 0.8 and t ∼ 10 3 since the complexity for obtaining a sample path is O(N 2 ) and the simulation is expensive (we need 10 4 samples). For long time simulation when H is close to 1 (like H = 0.8 for T 100), it is good to adopt the fast scheme in [23]. However, our scheme here is appropriate for dissipative problems due to its good stability properties, and can be used to analyze the time continuous problems compared with the one in [23].…”

“…The general potential V cases seem hard to justify. In [23], numerical methods have been designed for the overdamped GLE and the numerical results there give positive evidence. One of our goals in this paper is to use the numerical schemes to prove that the limiting measure is unique if the potential V is strongly convex.…”