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

Arthur Schichl

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.

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

Abstract

We define new differential structures on the Wasserstein spaces for and a general Riemannian manifold . 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 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.

Paper Structure

This paper contains 27 sections, 42 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Optimal transport
  5. Fundamental results
  6. Wasserstein embeddings
  7. Functionals on Wasserstein spaces
  8. Compact subsets of Wasserstein spaces
  9. Riemannian Geometry
  10. Extension of smooth vector--fields
  11. Smooth approximations of Lipschitz functions
  12. Parabolic structure on the Wasserstein space
  13. Tangent bundle and vector field probabilities
  14. The parabolic Wasserstein--flow equation
  15. Properties of solutions of the MDE
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $(M,g)$ be a Riemannian manifold, $x_0\in M$, and $1\leq p<\infty$. For a sequence $(\mu_k)_{k\in\mathbb N}$ in $\mathcal{P}_p(M)$ and $\mu\in\mathcal{P}_p(M)$ the following are equivalent:

Theorems & Definitions (110)

  • Theorem 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Theorem 2.3: (Monge-)Kantorovich duality
  • Proof 3
  • Proposition 2.4
  • Proof 4
  • Lemma 2.5
  • Proof 5
  • ...and 100 more