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Metric viscosity solutions and distance-like functions on the Wasserstein space

Huajian Jiang, Xiaojun Cui

TL;DR

This work extends viscosity-solution theory for the eikonal equation to metric spaces, with a focus on the Wasserstein space $\mathcal{P}_p(\mathcal{X})$. It proves that on complete unbounded length spaces, metric viscosity solutions to $|\partial u|=1$ coincide with $dl_G$-functions, and introduces the strong variant requiring global negative gradient rays. It provides two constructive pathways to obtain strong metric viscosity solutions on $\mathcal{P}_p(\mathcal{X})$: (i) building from carefully chosen $dl_C$-functions that satisfy a (CS) condition, and (ii) transferring solutions from $\mathcal{X}$ via $\hat u(\omega)=\int u(x) \, d\omega$, yielding a robust family of solutions. A representation via Busemann functions is developed, and the roles of (CS) condition, co-rays, and non-branching are analyzed to illuminate the geometric-PDE interplay in optimal transport settings.

Abstract

Viscosity solutions to the eikonal equation |Du|g = 1, known to be exactly distance-like functions, on a non-compact complete Riemannian manifold (M,g) are crucial for understanding the underlying geometric and topological properties. In this work, we explore metric viscosity solutions, distance-like functions and their relationship on a metric space, especially on the Wasserstein space Pp(X) where X is a complete, separable, locally compact and non-compact geodesic space. Meanwhile, we provide two distinct ways to construct (strong) metric viscosity solutions on Pp(X) and study their properties.

Metric viscosity solutions and distance-like functions on the Wasserstein space

TL;DR

This work extends viscosity-solution theory for the eikonal equation to metric spaces, with a focus on the Wasserstein space . It proves that on complete unbounded length spaces, metric viscosity solutions to coincide with -functions, and introduces the strong variant requiring global negative gradient rays. It provides two constructive pathways to obtain strong metric viscosity solutions on : (i) building from carefully chosen -functions that satisfy a (CS) condition, and (ii) transferring solutions from via , yielding a robust family of solutions. A representation via Busemann functions is developed, and the roles of (CS) condition, co-rays, and non-branching are analyzed to illuminate the geometric-PDE interplay in optimal transport settings.

Abstract

Viscosity solutions to the eikonal equation |Du|g = 1, known to be exactly distance-like functions, on a non-compact complete Riemannian manifold (M,g) are crucial for understanding the underlying geometric and topological properties. In this work, we explore metric viscosity solutions, distance-like functions and their relationship on a metric space, especially on the Wasserstein space Pp(X) where X is a complete, separable, locally compact and non-compact geodesic space. Meanwhile, we provide two distinct ways to construct (strong) metric viscosity solutions on Pp(X) and study their properties.
Paper Structure (4 sections, 26 theorems, 103 equations)

This paper contains 4 sections, 26 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The relationship between metric viscosity solutions and distance-like functions
  4. Existence of metric viscosity solutions on the Wasserstein space

Key Result

Theorem 1.5

Let $(\mathcal{Y},d)$ be a complete unbounded length space and $u: \mathcal{Y} \rightarrow \mathbb{R}$ be a continuous function. Then the following several statements are equivalent:

Theorems & Definitions (69)

  • Definition 1.1: Metric Viscosity Solution
  • Definition 1.2: Dl$_{C}$-function
  • Definition 1.3: Dl$_{G}$-function
  • Remark 1.4
  • Theorem 1.5
  • Definition 1.6: $\varepsilon$-Negative Gradient Curve
  • Theorem 1.7
  • Definition 1.8: Strong Metric Viscosity Solution
  • Definition 1.9: (CS) Condition
  • Remark 1.10
  • ...and 59 more