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

Key Points

• The research aims to analyze the convergence rate of the Euler–Maruyama scheme toward its invariant probability measure.
• Studied the existence and uniqueness of the invariant probability measure.
• Examined uniform geometric ergodicity using non-atomic Markov chains.
• Constructed a split Markov chain to find equivalent conditions for ergodicity.
• Demonstrated geometric decay rate of the Euler–Maruyama scheme under total variation distance.
• Established conditions for uniform geometric ergodicity of the Markov chain.
• Provided a framework for understanding the convergence of the drifted chain.