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On the equivalence of static and dynamic weak optimal transport

Bohdan Bulanyi

Abstract

We show that there is a PDE formulation in terms of Fokker-Planck equations for weak optimal transport problems. The main novelty is that we introduce a minimization problem involving Fokker-Planck equations in the extended sense of measure-valued solutions and prove that it is equal to the associated weak transport problem.

On the equivalence of static and dynamic weak optimal transport

Abstract

We show that there is a PDE formulation in terms of Fokker-Planck equations for weak optimal transport problems. The main novelty is that we introduce a minimization problem involving Fokker-Planck equations in the extended sense of measure-valued solutions and prove that it is equal to the associated weak transport problem.
Paper Structure (19 sections, 30 theorems, 146 equations)

This paper contains 19 sections, 30 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Main results
  4. Structure of the paper
  5. Preliminaries
  6. Conventions and Notation
  7. Fokker-Planck equation for general measures
  8. Subharmonic functions
  9. Fenchel conjugate
  10. A PDE constrained optimization problem
  11. Existence and duality
  12. Lower semicontinuity, convexity and subadditivity
  13. Associated weak transport problem
  14. Duality for $F$ in terms of invariant functions under $G$-transform
  15. $F$ versus $H$
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $(\varrho, \lambda)$ solve 1.1, $\mu=w*-\lim_{t\searrow0}\varrho_{t}$ and $\nu=w*-\lim_{t\nearrow 1}\varrho_{t}$. If $\psi \in C^{2}_{b}(\mathbb{R}^{d})$ is convex, the function $t \in [0,1] \mapsto \int_{\mathbb{R}^{d}}\psi \mathop{} d\varrho_{t}$ is nondecreasing. In particular, $\int_{\mathbb

Theorems & Definitions (77)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Remark 3.1
  • ...and 67 more