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

TL;DR

This paper introduces a Truncated Euler-Maruyama scheme for nonlinear ergodic SDEs, proving its numerical ergodicity, convergence rates, and invariant measure approximation with validation through numerical experiments.

Contribution

It presents an explicit TEM scheme with proven ergodicity and convergence rates, combining truncation and coupling methods for invariant measure approximation.

Findings

Proves numerical ergodicity of the TEM scheme in $L^p$-Wasserstein distance.

Establishes a uniform-in-time $1/2$-order convergence rate in moments.

Derives a $1/2$-order convergence rate for invariant measures in $L^1$-Wasserstein distance.

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.