Among distances in the space of probability measures, optimal transport distances, also called Wasserstein distances, are particularly interesting in statistics and to assess algorithms performances. However, these distances are often difficult to study from a theoretical point of view in the multidimensional setting. Instead, one can consider a (possibly non-optimal) transport plan defined by a diffusion process interpolation between the two measures considered. By bounding the transport cost of this transport plan one can then bound the Wasserstein distance between the two measures. This method, which can be seen as a variant of Stein's approximation method, can be used to provide non-asymptotic bounds for the Central Limit Theorem and seems a promising tool to study steady-state diffusion approximation.
