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

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.