  We present a way to use Stein's method in order to bound the Wasserstein
distance of order $2$ between two measures $\nu$ and $\mu$ supported on
$\mathbb{R}^d$ such that $\mu$ is the reversible measure of a diffusion
process. In order to apply our result, we only require to have access to a
stochastic process $(X_t)_{t \geq 0}$ such that $X_t$ is drawn from $\nu$ for
any $t > 0$. We then show that, whenever $\mu$ is the Gaussian measure
$\gamma$, one can use a slightly different approach to bound the Wasserstein
distances of order $p \geq 1$ between $\nu$ and $\gamma$ under an additional
exchangeability assumption on the stochastic process $(X_t)_{t \geq 0}$. Using
our results, we are able to obtain convergence rates for the multi-dimensional
Central Limit Theorem in terms of Wasserstein distances of order $p \geq 2$.
Our results can also provide bounds for steady-state diffusion approximation,
allowing us to tackle two problems appearing in the field of data analysis by
giving a quantitative convergence result for invariant measures of random walks
on random geometric graphs and by providing quantitative guarantees for a Monte
Carlo sampling algorithm.
