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

Xinjie Dai, Diancong Jin

TL;DR

This work analyzes the density behavior of the overdamped generalized Langevin equation driven by fractional noise and its Euler–Maruyama discretization. By leveraging Malliavin calculus for fractional Brownian motion, it proves existence and smoothness of densities for both the exact solution and the EM scheme, and derives a quantitative density-convergence rate of $O(h^{\alpha+H-1})$ under $f\in C_b^2$. An improved upper bound for the total variation distance via Malliavin–Sobolev norms is established to support the convergence analysis. The results enable reliable numerical capture of the exact solution’s statistical properties and provide a rigorous link between kernel regularity, noise roughness, and density approximation fidelity.

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.

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

TL;DR

This work analyzes the density behavior of the overdamped generalized Langevin equation driven by fractional noise and its Euler–Maruyama discretization. By leveraging Malliavin calculus for fractional Brownian motion, it proves existence and smoothness of densities for both the exact solution and the EM scheme, and derives a quantitative density-convergence rate of under . An improved upper bound for the total variation distance via Malliavin–Sobolev norms is established to support the convergence analysis. The results enable reliable numerical capture of the exact solution’s statistical properties and provide a rigorous link between kernel regularity, noise roughness, and density approximation fidelity.

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.
Paper Structure (10 sections, 9 theorems, 59 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Preliminaries
  4. Main results
  5. Auxiliary lemmas
  6. Proofs of Theorems \ref{['thm.ExactDensityExis']}--\ref{['thm.DensityConvergence']}
  7. Proof of Theorem \ref{['thm.ExactDensityExis']}
  8. Proof of Theorem \ref{['thm.ExactDensitySmooth']}
  9. Proofs of Theorems \ref{['thm.EulerDensityExis']} and \ref{['thm.EulerDensitySmooth']}
  10. Proof of Theorem \ref{['thm.DensityConvergence']}

Key Result

Proposition 2.1

Let $\{F_n\}_{n\ge1}$ be a sequence in $\mathbb{D}^{1,2}$ and $F_\infty\in\mathbb{D}^{2,4}$ with $\|DF_\infty\|_{\mathcal{H}} \geq c$ a.s. for some constant $c > 0$. If the laws of $\{F_n\}_{n\ge1}$ and $F_\infty$ are absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$, then t where the constant $C>0$ is independent of $n$.

Figures (1)

  • Figure 1: The relationship between the main results of this paper. Here, the convergence of $x_n$ and $q_{x_n}$ as $h \rightarrow 0$ is uniform for all $n \in \{1,2,\cdots,N\}$.

Theorems & Definitions (12)

  • Proposition 2.1
  • proof
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.1
  • ...and 2 more