Existence and stability of flows associated to vector fields in metric measure spaces - part 1

As the developments of the synthetic theories for lower bounds on Ricci curvature clearly illustrates, the connections between Eulerian notions (Gamma-calculus, Bakry-Emery theory etc.) and Lagrangian notions (Lott-Villani and Sturm theory) play an important role. In my lectures I will cover this topic, providing strong links between vector fields (derivations, in the metric measure setting) and solutions to the ODE. I will first cover the case of a single metric measure structure and then the case of measured Gromov-Hausdorff convergence. An important class of examples is provided by the case of gradient vector fields: indeed, thanks to the entropic formulation of the heat flow, one can obtain Mosco convergence and then weak and strong convergence of the associated derivations. As a byproduct one obtains also convergence of the ODE solutions, after an isometric embedding in a common metric space.

The lectures will be mostly based on joint papers with Dario Trevisan (Analysis & PDE, 2014) and with Dario Trevisan and Federico Stra (JFA, 2017).