Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Here I use the notation given in the question above, and the statements made assume the domain of the SDE problem is unbounded, which seems to be the context of the question.

Assume the following conditions hold.

1. (Regularity) Every derivative of $b(x)$ and $\sigma(x)$ exists and is bounded, and $\sigma(x)$ itself is bounded.
2. (Ellipticity) The matrix $\sigma(x) \sigma(x)^T$ is positive definite for all $x \in \mathbb{R}^d$.
3. (Weak Dissipativity) There exist $\beta >0$ and $\alpha \ge 0$ such that $$ b(x)^T x \le - \beta |x|^2 + \alpha $$ for all $x \in \mathbb{R}^d$.

Let $h$ be the time step size. Under these assumptions Talay and Tubaro 1990 prove that there exists $C>0$ such that $$ | \mathbb{E}_{\mu} f - \mathbb{E}_{\mu_h} f| \le C h $$ for all smooth functions $f$ with at most polynomial growth at infinity and for $h$ sufficiently small.

In other words, the weak accuracy of the Euler-Maruyama scheme at finite-time can be extended to infinite-time horizons. The proof uses an expansion of the global error of the Euler-Maruyama scheme that is commonly referred to as a Talay-Tubaro expansion. (Caveat: when the derivative of the drift is not bounded, the moments of the Euler-Maruyama scheme may diverge on finite-time intervals.)

I am not aware of a similar estimate for the TV distance: $$ \| \mu - \mu_h \|_{\text{TV}} = \sup\left\{ \left| \int_{\mathbb{R}^d} f \; d \mu - \int_{\mathbb{R}^d} f \; d \mu_h \right| ~~\text{s.t.}~~ \begin{array}{c} f: \mathbb{R}^d \to \mathbb{R} \;, \\ |f(x)| \le 1 ~\text{for all $x \in \mathbb{R}^d$} \end{array} \right\} \;. $$ The issue with making the desired statement in the TV distance is that the supremum in this definition is taken over functions which are not necessarily smooth, and so, the derivatives of $\mathbb{E}_x f(X_t)$ (where $\mathbb{E}_x$ denotes expectation conditional on $X_0=x$) may become unbounded if $t$ is small enough. A finite-time, TV error estimate for Euler-Maruyama is available in Lemma 4.2 of Bou-Rabee and Hairer 2013.