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

TL;DR

This paper introduces a new, more detailed differential structure on Wasserstein spaces for p > 2, enabling the analysis of degenerate second-order PDEs with measure-dependent coefficients and establishing solution existence, uniqueness, and a generalized CLT.

Contribution

It defines a finer differential structure on Wasserstein spaces and constructs smooth solutions to complex PDEs, extending classical theories and proving a generalized CLT.

Findings

Explicit construction of smooth solutions as limits of approximation series

Proof of solution uniqueness under certain conditions

Establishment of a generalized Central Limit Theorem

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.