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Homogenization principle and numerical analysis for fractional stochastic differential equations with different scales

Zhaoyang Wang, Ping Lin

TL;DR

The paper addresses Caputo fractional stochastic differential equations with multiscale time structure and non-Lipschitz drift, proving existence and uniqueness and establishing a mean-square homogenization principle via temporal averaging of fast-time coefficients. It introduces averaged coefficients $\overline f$ and $\overline g$, proves convergence of the original solution to a homogenized solution as $\epsilon\to0$, and provides an Euler–Maruyama scheme with detailed error bounds for both the original and homogenized systems. Numerical experiments verify the theory and showcase computational advantages of solving the homogenized autonomous system, especially when scale separation is strong. The results enhance tractability of long-time, multiscale fractional systems and offer a framework for efficient numerical methods, with planned extensions to stochastic partial differential equations.

Abstract

This work is concerned with fractional stochastic differential equations with different scales. We establish the existence and uniqueness of solutions for Caputo fractional stochastic differential systems under the non-Lipschitz condition. Based on the idea of temporal homogenization, we prove that the homogenization principle (averaging principle) holds in the sense of mean square ($L^2$ norm) convergence under a novel homogenization assumption. Furthermore, an Euler-Maruyama scheme for the non-autonomous system is constructed and its numerical error is analyzed. Finally, two numerical examples are presented to verify the theoretical results. Different from the existing literature, we demonstrate the computational advantages of the homogenized autonomous system from a numerical perspective.

Homogenization principle and numerical analysis for fractional stochastic differential equations with different scales

TL;DR

The paper addresses Caputo fractional stochastic differential equations with multiscale time structure and non-Lipschitz drift, proving existence and uniqueness and establishing a mean-square homogenization principle via temporal averaging of fast-time coefficients. It introduces averaged coefficients and , proves convergence of the original solution to a homogenized solution as , and provides an Euler–Maruyama scheme with detailed error bounds for both the original and homogenized systems. Numerical experiments verify the theory and showcase computational advantages of solving the homogenized autonomous system, especially when scale separation is strong. The results enhance tractability of long-time, multiscale fractional systems and offer a framework for efficient numerical methods, with planned extensions to stochastic partial differential equations.

Abstract

This work is concerned with fractional stochastic differential equations with different scales. We establish the existence and uniqueness of solutions for Caputo fractional stochastic differential systems under the non-Lipschitz condition. Based on the idea of temporal homogenization, we prove that the homogenization principle (averaging principle) holds in the sense of mean square ( norm) convergence under a novel homogenization assumption. Furthermore, an Euler-Maruyama scheme for the non-autonomous system is constructed and its numerical error is analyzed. Finally, two numerical examples are presented to verify the theoretical results. Different from the existing literature, we demonstrate the computational advantages of the homogenized autonomous system from a numerical perspective.
Paper Structure (13 sections, 8 theorems, 83 equations, 1 figure, 2 tables)

This paper contains 13 sections, 8 theorems, 83 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Well-posedness
  4. Homogenization principle
  5. Main result
  6. An illustrative example
  7. Numerical analysis
  8. Euler-Maruyama scheme
  9. Error analysis
  10. Numerical experiments
  11. Convergence test
  12. Stochastic fractional diffusion equation
  13. Conclusions and remarks

Key Result

Lemma 2.6

\newlabellemma2-60 (The Bihari inequality ouaddah2021fractional) Let $u:[0,T]\rightarrow [0,\infty]$ be a continuous function, $\psi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a nondecreasing continuous function, and $v$ be a nonnegative integrable function on $[0,T]$. If there exist a constant $ then Here, $W$ is defined by $W^{-1}$ is the inverse function of $W$, and for every $t\in[0,T]$,

Figures (1)

  • Figure 1: The error of the original variable $x_\epsilon$ and the homogenized variable $y_\epsilon$ with respect to the reference solution at different scales.

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.5
  • Lemma 2.6
  • Theorem 3.1
  • Proof 1
  • Remark 4.2
  • Lemma 4.3
  • Proof 2
  • ...and 12 more