Density functions for the overdamped generalized Langevin equation and its Euler--Maruyama method: smoothness and convergence

TL;DR

This paper investigates the convergence rate of the density function for the Euler--Maruyama method applied to the overdamped generalized Langevin equation with fractional noise, establishing smoothness and providing quantitative convergence estimates.

Contribution

It improves the upper bound for total variation distance, proves the existence and smoothness of densities, and derives convergence rates for the numerical density function.

Findings

Established smoothness of density functions for exact and numerical solutions

Derived convergence rates depending on noise and kernel regularity

Provided bounds for total variation distance between solutions

Abstract

This paper focuses on studying the convergence rate of the density function of the Euler--Maruyama (EM) method, when applied to the overdamped generalized Langevin equation with fractional noise which serves as an important model in many fields. Firstly, we give an improved upper bound estimate for the total variation distance between random variables by their Malliavin--Sobolev norms. Secondly, we establish the existence and smoothness of the density function for both the exact solution and the numerical one. Based on the above results, the convergence rate of the density function of the numerical solution is obtained, which relies on the regularity of the noise and kernel. This convergence result provides a powerful support for numerically capturing the statistical information of the exact solution through the EM method.