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Entropy
Volume 28
Issue 6
10.3390/e28060687
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Convergence Rate of Euler–Maruyama Scheme to the Invariant Probability Measure Under Total Variation Distance for the SDEs

by
Yuke Wang
1 and
Yinna Ye
2,*
1
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
2
Department of Applied Mathematics, School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China
*
Author to whom correspondence should be addressed.
Entropy 2026, 28(6), 687; https://doi.org/10.3390/e28060687 (registering DOI)
Submission received: 22 May 2026 / Revised: 12 June 2026 / Accepted: 12 June 2026 / Published: 14 June 2026
(This article belongs to the Special Issue Convergence Rates for Markov Chains)
Versions Notes

Abstract

This article shows the geometric decay rate of the Euler–Maruyama scheme for a one-dimensional stochastic differential equation towards its invariant probability measure under total variation distance. Firstly, the existence and uniqueness of invariant probability measure and the uniform geometric ergodicity of the chain are studied through the introduction of non-atomic Markov chains. Secondly, the equivalent conditions for uniform geometric ergodicity of the chain are discovered by constructing a split Markov chain based on the original Euler–Maruyama scheme.
Keywords: Euler–Maruyama scheme; invariant probability measure; total variation distance; uniform geometric ergodicity; langevin monte carlo; Markov chain Monte Carlo Euler–Maruyama scheme; invariant probability measure; total variation distance; uniform geometric ergodicity; langevin monte carlo; Markov chain Monte Carlo

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MDPI and ACS Style

Wang, Y.; Ye, Y. Convergence Rate of Euler–Maruyama Scheme to the Invariant Probability Measure Under Total Variation Distance for the SDEs. Entropy 2026, 28, 687. https://doi.org/10.3390/e28060687

AMA Style

Wang Y, Ye Y. Convergence Rate of Euler–Maruyama Scheme to the Invariant Probability Measure Under Total Variation Distance for the SDEs. Entropy. 2026; 28(6):687. https://doi.org/10.3390/e28060687

Chicago/Turabian Style

Wang, Yuke, and Yinna Ye. 2026. "Convergence Rate of Euler–Maruyama Scheme to the Invariant Probability Measure Under Total Variation Distance for the SDEs" Entropy 28, no. 6: 687. https://doi.org/10.3390/e28060687

APA Style

Wang, Y., & Ye, Y. (2026). Convergence Rate of Euler–Maruyama Scheme to the Invariant Probability Measure Under Total Variation Distance for the SDEs. Entropy, 28(6), 687. https://doi.org/10.3390/e28060687

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