Entropic regularization of incompressible optimal transport

A lot of attention has been devoted to the entropic regularization of optimal transport in recent years for its links with the theory of large deviations and for its formidable efficiency in terms of numerical computations. The purpose of this talk will be to introduce and to study the zero noise limit of the incompressible version of this regularized problem. With this aim in view, I will present an elementary way to modify an absolutely continuous curve with values in the Wasserstein space to get an admissible curve for the dynamical entropic optimal transport. This construction allows to recover in a variational way two famous theorems:

• the displacement convexity of the entropy along the optimal transport (due to McCann),
• the Gamma-convergence of the dynamical entropic optimal transport towards the classical optimal transport (due to Léonard).

Besides, our modification preserves incompressibility, so these proofs adapt transparently to the incompressible case, and we get as new results these theorems in that setting. Doing so, we extend a result by Lavenant for the convexity of the entropy, and a result by Benamou-Carlier-Nenna for the Gamma-convergence.

This is a joint work with L. Monsaingeon (Lisbon and Nancy University).