Section: New Results

Applications of optimal transport

Participants : Aurélien Alfonsi, Benjamin Jourdain, Arturo Kohatsu-Higa.

A. Alfonsi and B. Jourdain study the Wasserstein distance between two probability measures in dimension $n$ sharing the same copula $C$. The image of the probability measure $dC$ by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension $n=1$. In dimension $n>1$, it turns out that for cost functions equal to the $p$-th power of the $L^q$ norm, this coupling is optimal only when $p=q$ i.e. when the cost function may be decomposed as the sum of coordinate-wise costs.

As another application of optimal transport, they are working with A. Kohatsu-Higa on the uniform in time estimation of the Wasserstein distance between the time-marginals of an elliptic diffusion and its Euler scheme. To generalize in higher dimension the result that they obtained previously in dimension one using the optimality of the explicit inverse transform, they compute the derivative of the Wasserstein distance with respect to the time variable thanks to the theory developped by Ambrosio Gigli and Savare. The abstract properties of the optimal coupling between the time marginals then enable them to estimate this time derivative.