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

Yuke Wang, Yinna Ye

TL;DR

This work analyzes the Euler–Maruyama scheme for a one-dimensional SDE and proves that the Markov chain it induces converges to its invariant measure in total variation at a geometric rate. By establishing existence and uniqueness of the invariant measure and a uniform drift condition, the authors demonstrate uniform geometric ergodicity with a rate that is independent of the step size $\eta$ (for small $\eta$). A novel splitting construction yields an associated split Markov kernel with an atom, enabling equivalent GE criteria and a clearer pathway to the uniform convergence results. The approach hinges on non-atomic Markov chain theory, small sets, and drift arguments, and connects to broader MCMC contexts via the split-chain framework. Overall, the paper provides rigorous conditions under which EM schemes converge rapidly to their invariant measures in TV distance, with explicit constants and a step-size-robust rate.

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. It turns out that this convergence rate is independent with the step size under total variation distance.

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

TL;DR

This work analyzes the Euler–Maruyama scheme for a one-dimensional SDE and proves that the Markov chain it induces converges to its invariant measure in total variation at a geometric rate. By establishing existence and uniqueness of the invariant measure and a uniform drift condition, the authors demonstrate uniform geometric ergodicity with a rate that is independent of the step size (for small ). A novel splitting construction yields an associated split Markov kernel with an atom, enabling equivalent GE criteria and a clearer pathway to the uniform convergence results. The approach hinges on non-atomic Markov chain theory, small sets, and drift arguments, and connects to broader MCMC contexts via the split-chain framework. Overall, the paper provides rigorous conditions under which EM schemes converge rapidly to their invariant measures in TV distance, with explicit constants and a step-size-robust rate.

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. It turns out that this convergence rate is independent with the step size under total variation distance.
Paper Structure (9 sections, 21 theorems, 68 equations)

This paper contains 9 sections, 21 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Properties about the Markov kernel $P_\eta$
  4. Existence and uniqueness of invariant probability measure $\pi_\eta$
  5. Splitting construction and split Markov kernel $\check{\mathbb{P}}_\eta$
  6. Uniform geometric ergodicity of the kernel $P_{\eta}$
  7. Proofs of main results
  8. Some properties related to small sets, positive and geometrically ergodic kernel
  9. Some auxiliary result

Key Result

Theorem 2.1

Under Assumption assump1, there exists a constant $\eta_{0}\in(0,1)$ depending only on $K_1$ and $L$, such that for any $\eta\in (0,\,\eta_{0}]$, the Markov kernel $P_{\eta}$ has a unique invariant probability measure $\pi_{\eta}$. Furthermore, there exist constants $\delta>1$, $\tau<\infty$, $\beta and for any initial probability measure $\xi$ on $(\mathbb{R},\,\mathcal{B}(\mathbb{R}))$,

Theorems & Definitions (37)

  • Theorem 2.1
  • Corollary 2.1
  • Theorem 2.2
  • Lemma 3.1: Corollary 9.2.14, DMPS18
  • Lemma 3.2: Theorem 9.3.6, DMPS18
  • Lemma 3.3: Proposition 9.4.5, DMPS18
  • Lemma 3.4: Lemma 9.4.7 (ii), DMPS18
  • Lemma 3.5: Theorem 9.4.10, DMPS18
  • Remark 3.1
  • Lemma 3.6: Theorem 10.1.2, DMPS18
  • ...and 27 more