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Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity

Shan Huang, Xiaoyue Li

TL;DR

The paper tackles invariant-measure approximation for nonlinear SDEs with non-uniform (dissipativity-at-infinity) drift by introducing an explicit Truncated Euler-Maruyama (TEM) scheme. It combines a truncation-based discretization with a novel coupling strategy (synchronous plus Majka-like reflection) to prove TEM is exponentially ergodic in $\mathcal{W}_p$ and to obtain uniform-in-time $1/2$-order strong convergence of the numerical solution to the true solution, as well as a $1/2$-order convergence of the numerical invariant measure to the exact invariant measure in $\mathcal{W}_1$. The results provide explicit step-size bounds and robust error control for invariant-measure approximation in non-globally dissipative settings, and are validated by two numerical experiments. Overall, the work extends ergodic discretization theory to nonlinear SDEs with dissipativity at infinity, offering practical, provable tools for stochastic sampling and long-time statistics in high-dimensional systems.

Abstract

The approximation of invariant measures for nonlinear ergodic stochastic differential equations (SDEs) is a central problem in scientific computing, with important applications in stochastic sampling, physics, and ecology. We first propose an easily applicable explicit Truncated Euler-Maruyama (TEM) scheme and prove its numerical ergodicity in the $L^p$-Wasserstein distance ($p\geqslant 1$). Furthermore, by combining truncation techniques with the coupling method, we establish a uniform-in-time $1/2$-order convergence rate in moments for the TEM scheme. Additionally, leveraging the exponential ergodicity of both the numerical and exact solutions, we derive a $1/2$-order convergence rate for the invariant measures of the TEM scheme and the exact solution in the $L^1$-Wasserstein distance. Finally, two numerical experiments are conducted to validate our theoretical results.

Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity

TL;DR

The paper tackles invariant-measure approximation for nonlinear SDEs with non-uniform (dissipativity-at-infinity) drift by introducing an explicit Truncated Euler-Maruyama (TEM) scheme. It combines a truncation-based discretization with a novel coupling strategy (synchronous plus Majka-like reflection) to prove TEM is exponentially ergodic in and to obtain uniform-in-time -order strong convergence of the numerical solution to the true solution, as well as a -order convergence of the numerical invariant measure to the exact invariant measure in . The results provide explicit step-size bounds and robust error control for invariant-measure approximation in non-globally dissipative settings, and are validated by two numerical experiments. Overall, the work extends ergodic discretization theory to nonlinear SDEs with dissipativity at infinity, offering practical, provable tools for stochastic sampling and long-time statistics in high-dimensional systems.

Abstract

The approximation of invariant measures for nonlinear ergodic stochastic differential equations (SDEs) is a central problem in scientific computing, with important applications in stochastic sampling, physics, and ecology. We first propose an easily applicable explicit Truncated Euler-Maruyama (TEM) scheme and prove its numerical ergodicity in the -Wasserstein distance (). Furthermore, by combining truncation techniques with the coupling method, we establish a uniform-in-time -order convergence rate in moments for the TEM scheme. Additionally, leveraging the exponential ergodicity of both the numerical and exact solutions, we derive a -order convergence rate for the invariant measures of the TEM scheme and the exact solution in the -Wasserstein distance. Finally, two numerical experiments are conducted to validate our theoretical results.

Paper Structure

This paper contains 13 sections, 24 theorems, 293 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and assumptions
  4. Useful lemmas
  5. Truncated EM scheme and numerical ergodicity
  6. Construction of numerical scheme
  7. Numerical ergodicity
  8. Uniform-in-time error analysis
  9. Error bounds for $\mathbb{E}|X_{k}-\hat{X}_{k}|$ and $\mathbb{E} |\hat{S}_{k}-x_{kh}|$
  10. Construction of coupling
  11. Contractivity of coupling
  12. Uniform-in-time error analysis
  13. Numerical examples

Key Result

Lemma 2.4

(IMA) Under Assumption A, the SDE (E) with initial value $x_0\in\mathbb{R}^d$ has a unique global solution $(x_t)_{t\geqslant 0}$ satisfying for any $q>0$.

Figures (8)

  • Figure 1: Functions with outside-of-sphere dissipativity or dissipativity at infinity.
  • Figure 2: Trajectories of $\mathbb{E} \cos(\sqrt{x_t^2+y_t^2})$: Left---fixed step size with different initial conditions; Right---fixed initial condition with different step sizes.
  • Figure 3: Strong error between the numerical solution and the reference solution at terminal time $T = 32$, starting from initial condition $(x_0, y_0) = (1, 0.5)$, based on 1000 sample paths, plotted as a function of step size $h \in \{2^{-10}, 2^{-11}, 2^{-12}, 2^{-13}, 2^{-14}\}$.
  • Figure 4: Joint density plots of the numerical solutions starting from the initial condition $(x_0, y_0) = (1, 0.5)$, using 8000 samples and step sizes $h \in \{2^{-7}, 2^{-8}, 2^{-9}, 2^{-10}\}$.
  • Figure 5: Empirical cumulative marginal density plots of the numerical solutions starting from the initial condition $(x_0, y_0) = (1, 0.5)$, using 8000 samples and step sizes $h \in \{2^{-7}, 2^{-8}, 2^{-9}, 2^{-10}\}$.
  • ...and 3 more figures

Theorems & Definitions (51)

  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • Remark 2.8
  • Remark 3.1
  • Lemma 3.2
  • proof
  • ...and 41 more