We present a way to use Stein's method in order to bound the Wasserstein
distance of order 2 between two measures ν and μ supported on
Rd such that μ is the reversible measure of a diffusion
process. In order to apply our result, we only require to have access to a
stochastic process (Xt)t≥0 such that Xt is drawn from ν for
any t>0. We then show that, whenever μ is the Gaussian measure
γ, one can use a slightly different approach to bound the Wasserstein
distances of order p≥1 between ν and γ under an additional
exchangeability assumption on the stochastic process (Xt)t≥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≥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