Approximation of the ergodic measure of SDEs with singular drift by Euler-Maruyama scheme

We study the approximation of the ergodic measure of the following stochastic differential equation (SDE) on $\mathbb{R}^d$: \begin{eqnarray}\label{e:SDEE} d X_t &=& (b_1(X_t)+b_2(X_t)) d t+\sigma(X_t) d W_t, \end{eqnarray} where $W_t$ is a $d$-dimensional standard Brownian motion, and $b_1: \mathbb{R}^d \mapsto \mathbb{R}^d$, $b_2: \mathbb{R}^d \mapsto \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \mapsto \mathbb{R}^{d\times d}$ are the functions to be specified in Assumption 2.1 below. In particular, $b_1$ satisfies $b_1\in \mathbb{L}^\infty(\mathbb{R}^d)\cap \mathbb{L}^1(\mathbb{R}^d)$ or $b_1 \in \mathcal{C}_b^{\alpha}(\mathbb{R}^d)$ with $\alpha\in (0,1)$, which makes the standard numerical schemes not work or fail to give a good convergence rate. In order to overcome these two difficulties, we first apply a Zvonkin's transform to SDE and obtain a new SDE which has coefficients with nice properties and admits a unique ergodic measure $\widehat \mu$, then discretize the new equation by Euler-Maruyama scheme to approximate $\widehat \mu$, and finally use the inverse Zvonkin's transform to get an approximation of the ergodic measure of SDE, denoted by $\mu$. Our approximation method is inspired by Xie and Zhang [22]. The proof of our main result is based on the method of introducing a stationary Markov chain, a key ingredient in this method is establishing the regularity of a Poisson equation, which is done by combining the classical PDE local regularity and a nice extension trick introduced by Gurvich [10].