Wasserstein Convergence Rate for Empirical Measures of Markov Chains

We consider a Markov chain on $\mathbb{R}^d$ with invariant measure $\mu$. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to the $1$-Wasserstein distance. The main result of this article is a new upper bound for the expected Wasserstein distance, which is proved by combining the Kantorovich dual formula with a Fourier expansion. In addition, we show how concentration inequalities around the mean can be obtained.