Splitting AVF method for generalized Langevin equations: probability density function and geometric ergodicity

TL;DR

This paper introduces a structure-preserving splitting AVF method for generalized Langevin equations, ensuring accurate probability density functions and geometric ergodicity, with theoretical proofs and numerical validation.

Contribution

It develops a novel splitting AVF numerical scheme that preserves key properties of GLEs and proves its convergence and ergodicity.

Findings

Probability density function converges with first-order accuracy.

Numerical solution maintains exponential integrability and Malliavin differentiability.

The method is proven to be geometrically ergodic.

Abstract

The generalized Langevin equation (GLE) constitutes a fundamental model for describing nonequilibrium dynamics with memory effects. To overcome the numerical challenges arising from superquadratically growing potentials and degenerate noise, we propose and analyze a structure-preserving splitting averaged vector field (AVF) method for a quasi-Markovian GLE. The core advantage of this method lies in its ability to simultaneously preserve the exponential integrability, Malliavin differentiability, and ergodicity of the underlying continuous system. Notably, by leveraging exponential integrability, Malliavin differentiability, and uniform non-degeneracy of the numerical solution, we obtain the existence and smoothness of its probability density function, which converges to that of the exact solution with first-order accuracy. Furthermore, by validating the Lyapunov condition and the minorization condition using a localized technique, we establish the geometric ergodicity of the numerical solution. Finally, numerical experiments are conducted to confirm the theoretical results.