The dynamical formulation of optimal transport between two probability measures $\mu_0,\mu_1$ on a (sub)Riemannian manifold $M$, aims at minimizing the square integral of a Borel family of vector fields $ \int_0^1\int_M||v_t||^2d\mu_t dt, $ where the narrowly continuous curve of probabilities $\mu_t$ and $v_t$ must respect the continuity equation. The equivalence between this Benamou-Brenier formulation and the Kantorovich formulation of optimal transport, is well known in Riemannian context, but still open in sub-Riemannian manifolds (in the SR case, $v_t$ is a family of horizontal vector fields). I will present some recent advancements in this problem and a joint work (with Giovanna Citti and Andrea Pinamonti), proving the equivalence under general regularity assumptions in the case of a sub-Riemannian manifold with no non-trivial abnormal geodesics. The key idea is the formulation of a relaxed version of the dynamical problem that hinges the other two versions, and allows to prove the equivalence of the Kantorovich formulation with the relaxed and the original Benamou-Brenier formulation.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006