Non-linear degenerate parabolic flow equations and a finer differential structure on Wasserstein spaces

Abstract

We define new differential structures on the Wasserstein spaces $\mathcal{W}_p(M)$ for $p > 2$ and a general Riemannian manifold $(M,g)$. We consider a very general and possibly degenerate second order partial differential flow equation with measure dependent coefficients to expand the notion of smooth curves and to ensure that the new differential structure is finer than the classical one. Under weak assumptions, we explicitly construct smooth solutions as uniform limits of Average Flow Approximation Series (a variant of explicit Euler--scheme approximations) in $\mathcal{W}_p(M)$ and, thus, prove a generalzed version of the Central Limit Theorem. Under slightly stronger assumptions, we prove that smooth solutions of our newly introduced flow--equation are unique.