In this talk we present well posedness of Measure Differential Equations, i.e. evolution equations in the Wasserstein space of probability measures driven by dissipative probability vector fields. We take inspiration from the theory of \emph{dissipative operators} in Hilbert spaces and of Wasserstein gradient flows of geodesically convex functionals. Our approach is based on a measure-theoretic version of the Explicit Euler scheme, for which we prove convergence results with optimal error estimates. We finally characterize the limit solutions by a suitable Evolution Variational Inequality and compare this notion with the weaker barycentric/distributional one introduced by Piccoli. (This is a joint work with G. Savaré (Bocconi University) and G. E. Sodini (TUM-IAS).)

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