Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme

In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and H{\"o}lder continuous with exponent $\gamma$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C \left(1+\mathbf{1}\_{\gamma=1} \sqrt{\ln(N)}\right)N^{-\gamma}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio, Gigli and Savar{\'e} to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.