[Paper Review] On a solution to the Monge transport problem on the real line arising from the strictly concave case

Nicolas Juillet|arXiv (Cornell University)|Jul 1, 2019

Geometric Analysis and Curvature Flows12 references4 citations

TL;DR

This paper introduces the 'excursion coupling' as a unique solution to the Monge optimal transport problem on the real line for the critical case $p=1$, derived as the limit of $L^p$ transport problems with $p\to 1^{-}$. It characterizes this solution via monotonicity of transport routes, minimization of $L^q$-type secondary costs, and a geometric construction based on the difference of cumulative distribution functions, resolving non-uniqueness in the classical $L^1$ case.

ABSTRACT

It is well-known that the optimal transport problem on the real line for the classical distance cost may not have a unique solution. In this paper we recover uniqueness by considering the transport problems where the costs are a power smaller than one of the distance, and letting this parameter tend to one. A complete construction of this solution that we call excursion coupling is given. This is reminiscent to the one in the convex case. It is also characterized as the solution of secondary transport problems. Moreover, a combinatoric/geometric characterization of the routes used for this transport plan is provided.

Motivation & Objective

• To resolve non-uniqueness in the Monge optimal transport problem on the real line for $p=1$, where classical $L^1$ cost fails to yield a unique solution.
• To construct a canonical solution—called the excursion coupling—by taking the limit of $L^p$ transport problems as $p \to 1^{-}$, i.e., from the strictly concave regime.
• To characterize this solution via monotonicity of transport routes, minimizing a secondary $L^q$ cost, and a geometric construction based on the difference of cumulative distribution functions.
• To establish connections between this solution and stochastic processes, such as Brownian motion excursions and local time theory, suggesting broader applicability in geometric and probabilistic transport.

Proposed method

• Define the $L^p$ transport problem on $\mathbf{R}$ with cost $|y - x|^p$, and consider the limit as $p \to 1^{-}$ to recover uniqueness.
• Construct the excursion coupling as the limit of optimal plans for $p_n < 1$ with $p_n \to 1$, ensuring convergence in the weak topology of measures.
• Characterize the solution via monotonicity: transport routes (arches) do not cross, do not connect, and have consistent orientation when nested.
• Use a secondary optimization problem: the excursion coupling minimizes $\iint |y - x|^q \, d\pi(x,y)$ over all $L^1$-optimal plans for $q \in (0,1)$.
• Geometrically define the coupling using the function $F^* = (F_\mu - F_\nu)^*$, where $F_\mu, F_\nu$ are cumulative distribution functions, and identify transport routes as level sets of this function.
• Establish a probabilistic interpretation: the excursion coupling corresponds to embedding a Brownian motion with random time shift, minimizing $\mathbb{E}[\varphi(T)]$ for concave $\varphi$, and relate to local times via Meyer–Tanaka formula.

Experimental results

Research questions

• RQ1 Can a unique solution be recovered for the $L^1$ Monge transport problem on $\mathbf{R}$, which is known to lack uniqueness?
• RQ2 What is the limiting behavior of $L^p$ optimal transport plans as $p \to 1^{-}$, and does this limit yield a canonical solution?
• RQ3 How can the excursion coupling be characterized geometrically and combinatorially in terms of non-crossing, non-connecting, and consistently oriented transport routes?
• RQ4 Is the excursion coupling the unique minimizer of a secondary cost $\iint |y - x|^q \, d\pi(x,y)$ among all $L^1$-optimal plans for $q \in (0,1)$?
• RQ5 Can the excursion coupling be interpreted as a random process on a tree constructed from the difference of cumulative distribution functions, with the length measure as the natural probability measure?

Key findings

• The excursion coupling is the unique limit of $L^p$-optimal transport plans as $p \to 1^{-}$, resolving non-uniqueness in the $L^1$ case.
• The solution is characterized as the unique $L^1$-optimal plan that minimizes the secondary cost $\iint |y - x|^q \, d\pi(x,y)$ for any $q \in (0,1)$.
• Transport routes in the excursion coupling are non-crossing, non-connecting, and consistently oriented when nested, ensuring monotonicity.
• The coupling arises from the function $F^* = (F_\mu - F_\nu)^*$, where transport occurs along level sets of this function, interpreted as excursions.
• The solution corresponds to a probabilistic embedding of Brownian motion with random time shift, minimizing $\mathbb{E}[\varphi(T)]$ for concave $\varphi$, linking to stochastic calculus.
• The construction generalizes the quantile coupling (which arises as $p \to 1^{+}$) and suggests a duality between $p \to 1^{-}$ and $p \to 1^{+}$ limits in optimal transport.