Convergence rate of Euler--Maruyama scheme to the invariant probability measure under total variation distance for the SDEs

Mathematics > Probability [Submitted on 7 May 2025 (v1), last revised 16 Jun 2026 (this version, v3)] Title:Convergence rate of Euler--Maruyama scheme to the invariant probability measure under total variation distance for the SDEs View PDF HTML (experimental)Abstract:This article shows the geometric decay rate of Euler-Maruyama scheme for 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 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. Submission history From: Yinna Ye [view email][v1] Wed, 7 May 2025 08:16:57 UTC (26 KB) [v2] Mon, 1 Dec 2025 10:14:37 UTC (22 KB) [v3] Tue, 16 Jun 2026 10:27:44 UTC (27 KB) Current browse context: math.PR References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.