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PDE aspects of the dynamical optimal transport in the Lorentzian setting

Nicola Gigli, Felix Rott, Matteo Zanardini

TL;DR

This work extends the core link between Wasserstein geometry and PDEs to Lorentzian spacetimes by replacing the continuity equation with a causal continuity inequality, establishing a Lorentzian Benamou–Brenier formula. It introduces a Lorentzian Hopf–Lax semigroup and Kuwada duality to connect PDE and metric transport structures, and proves the BB formula via lifting of causal paths and duality arguments on globally hyperbolic spacetimes. The framework relies on causal couplings, the $p$-Lorentz–Wasserstein distance with $0<p<1$, and the notion of compact emerald neighborhoods to manage support and regularity. Overall, the paper provides a rigorous bridge between Lorentzian geometry, causal transport, and dynamic optimal transport, yielding a Lorentzian analogue of the classical BB formula with potential PDE applications in spacetime settings.

Abstract

One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable `continuity inequality', to which we shall refer to as `causal continuity inequality'. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou--Brenier formula.

PDE aspects of the dynamical optimal transport in the Lorentzian setting

TL;DR

This work extends the core link between Wasserstein geometry and PDEs to Lorentzian spacetimes by replacing the continuity equation with a causal continuity inequality, establishing a Lorentzian Benamou–Brenier formula. It introduces a Lorentzian Hopf–Lax semigroup and Kuwada duality to connect PDE and metric transport structures, and proves the BB formula via lifting of causal paths and duality arguments on globally hyperbolic spacetimes. The framework relies on causal couplings, the -Lorentz–Wasserstein distance with , and the notion of compact emerald neighborhoods to manage support and regularity. Overall, the paper provides a rigorous bridge between Lorentzian geometry, causal transport, and dynamic optimal transport, yielding a Lorentzian analogue of the classical BB formula with potential PDE applications in spacetime settings.

Abstract

One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable `continuity inequality', to which we shall refer to as `causal continuity inequality'. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou--Brenier formula.
Paper Structure (9 sections, 18 theorems, 98 equations)

This paper contains 9 sections, 18 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lorentzian optimal transport
  4. The Kantorovich Duality
  5. The causal continuity inequality
  6. Analysis on globally hyperbolic spacetimes
  7. A Hopf--Lax type formula
  8. The Kuwada Duality
  9. The Benamou--Brenier formula on spacetimes

Key Result

Theorem 1.1

Let $M$ be a smooth complete Riemannian manifold, $p\in(1,+\infty)$ and $(\mu_t)\subset \mathscr{P}_p(M)$ be a weakly continuous curve. Then:

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Global hyperbolicity
  • Definition 2.2: Causal/steep functions and causal vector fields
  • Definition 2.3: Asymptotic steepness constant
  • Definition 2.4: Causal speed
  • Definition 2.5: Causal transport plans
  • Definition 2.6: $p$-Lorentz--Wasserstein distance
  • Theorem 2.7: Lifting paths of measures to measures on paths
  • Definition 2.8: $c_p$-concave and transform
  • ...and 35 more