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We introduce and study the class of totally dissipative multivaluedprobability vector fields (MPVF) $\boldsymbol{\mathrm F}$ on the Wassersteinspace $(\mathcal{P}_2(\mathsf{X}),W_2)$ of Euclidean or Hilbertian probabilitymeasures. We show that such class of MPVFs is in one to one correspondence withlaw-invariant dissipative operators in a Hilbert space$L^2(\Omega,\mathcal{B},\mathbb{P};\mathsf{X})$ of random variables, preservinga natural maximality property. This allows us to import in the Wassersteinframework many of the powerful tools from the theory of maximal dissipativeoperators in Hilbert spaces, deriving existence, uniqueness, stability, andapproximation results for the flow generated by a maximal totally dissipativeMPVF and the equivalence of its Eulerian and Lagrangian characterizations. Wewill show that demicontinuous single-valued probability vector fieldssatisfying a metric dissipativity condition are in fact totally dissipative.Starting from a sufficiently rich set of discrete measures, we will also showhow to recover a unique maximal totally dissipative version of a MPVF, provingthat its flow provides a general mean field characterization of the asymptoticlimits of the corresponding family of discrete particle systems.Such anapproach also reveals new interesting structural properties for gradient flowsof displacement convex functionals with a core of discrete measures dense inenergy.

A Lagrangian approach to totally dissipative evolutions in Wasserstein spaces