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Optimal transport in the frame of abstract Lax-Oleinik operator revisited

Wei Cheng, Jiahui Hong, Tianqi Shi

Abstract

This is our first paper on the extension of our recent work on the Lax-Oleinik commutators and its applications to the intrinsic approach of propagation of singularities of the viscosity solutions of Hamilton-Jacobi equations. We reformulate Kantorovich-Rubinstein duality theorem in the theory of optimal transport in terms of abstract Lax-Oleinik operators, and analyze the relevant optimal transport problem in the case the cost function $c(x,y)=h(t_1,t_2,x,y)$ is the fundamental solution of Hamilton-Jacobi equation. For further applications to the problem of cut locus and propagation of singularities in optimal transport, we introduce corresponding random Lax-Oleinik operators. We also study the problem of singularities for $c$-concave functions and its dynamical implication when $c$ is the fundamental solution with $t_2-t_1\ll1$ and $t_2-t_1<\infty$, and $c$ is the Peierls' barrier respectively.

Optimal transport in the frame of abstract Lax-Oleinik operator revisited

Abstract

This is our first paper on the extension of our recent work on the Lax-Oleinik commutators and its applications to the intrinsic approach of propagation of singularities of the viscosity solutions of Hamilton-Jacobi equations. We reformulate Kantorovich-Rubinstein duality theorem in the theory of optimal transport in terms of abstract Lax-Oleinik operators, and analyze the relevant optimal transport problem in the case the cost function is the fundamental solution of Hamilton-Jacobi equation. For further applications to the problem of cut locus and propagation of singularities in optimal transport, we introduce corresponding random Lax-Oleinik operators. We also study the problem of singularities for -concave functions and its dynamical implication when is the fundamental solution with and , and is the Peierls' barrier respectively.
Paper Structure (18 sections, 30 theorems, 116 equations)

This paper contains 18 sections, 30 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Singularities of $c$-concave functions
  3. An alternative formulation of Kantorovich-Rubinstein duality
  4. Random Lax-Oleinik operators
  5. Preliminaries
  6. Semiconcave functions
  7. Fundamental solutions and Lax-Oleinik representation
  8. Optimal transport and singularities of $c$-concave function
  9. Abstract Lax-Oleinik commutators
  10. Singularities of $c$-concave functions
  11. $c(x,y)=h(t_1,t_2,x,y)$ with $t_2-t_1\ll1$
  12. $c(x,y)=h(t_1,t_2,x,y)$ with finite $t_2-t_1$
  13. $c(x,y)=h(x,y)$ with $h$ the Peierls' barrier
  14. Application of abstract Lax-Oleinik operators to optimal transport
  15. An alternative formulation of Kantorovich-Rubinstein duality
  16. ...and 3 more sections

Key Result

Proposition 2.1

A function $\phi:\Omega\to\mathbb{R}$ is a semiconcave function with constant $C\geqslant0$ if and only if there exists a family of $C^2$-functions $\{\phi_i\}$ with $D^2\phi_i\leqslant CI$ such that

Theorems & Definitions (67)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • ...and 57 more