A Lagrangian approach to totally dissipative evolutions in Wasserstein spaces

Understand

We introduce and study the class of totally dissipative multivalued probability vector fields (MPVF) $\boldsymbol{\mathrm F}$ on the Wasserstein space $(\mathcal{P}_2(\mathsf{X}),W_2)$ of Euclidean or Hilbertian probability measures.

• We show that such class of MPVFs is in one to one correspondence with law-invariant dissipative operators in a Hilbert space $L^2(\Omega,\mathcal{B},\mathbb{P};\mathsf{X})$ of random variables, preserving a natural maximality property.
• This allows us to import in the Wasserstein framework many of the powerful tools from the theory of maximal dissipative operators in Hilbert spaces, deriving existence, uniqueness, stability, and approximation results for the flow generated by a maximal totally dissipative MPVF and the equivalence of its Eulerian and Lagrangian characterizations.
• We will show that demicontinuous single-valued probability vector fields satisfying a metric dissipativity condition are in fact totally dissipative.