  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.
