Abstract.

The DiPerna-Lions theory provides a good axiomatization to the notion of well-posedness for the ODE associated to a nonsmooth vector field, e.g. having a Sobolev but not Lipschitz regularity.

I will illustrate, with a reasonable number of details, my (relatively) recent work with D.Trevisan which extends this picture, so to speak, also to the case when even the ambient space is not so smooth. First I will revisit the Di Perna-Lions theory, then I will explain how in this broader context the notions ``Sobolev vector field'' and ''solution to the ODE'' have to be properly understood. For the first one we achieve the goal using Gamma-calculus tools (and so the natural context will be Dirichlet forms, Markov semigroups, Carr\'e du Champ operators) for the second one we use ideas coming from Optimal Transport and Geometric Measure Theory, the so-called superposition principle.