Evolution equations driven by dissipative operators in Wasserstein spaces

In this talk we present new results framing into the recent theory of Measure Differential Equations introduced by B. Piccoli (Rutgers University-Camden). The state space where these evolution equations are set is the Wasserstein space of probability measures, hence tools of Optimal Transport are essential. The key point here is that the vector field itself maps into the space of probability measures lying on the tangent bundle, in a way compatible with the projection on the state space. We give a stronger definition of solution which indeed “selects” only one of the (not unique) solutions in the sense of Piccoli. In addition to uniqueness, we are also able to prove stability results. To do so, we borrow ideas from the theory of evolution equations driven by dissipative operators on Hilbert spaces, giving a notion of solution in terms of a so called Evolution Variational Inequality.

This is a joint work with G. Savaré (Bocconi University) and G. E. Sodini (TUM-IAS).