🔢 mathematics

Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance

This paper establishes quantitative stability estimates for optimal transport maps and Brenier potentials under various regularity assumptions by linearizing the Monge-Ampère equation, and further derives an explicit formula for the second variation of the quadratic Monge-Kantorovich distance.

Original authors: F. -U. Caja-Lopez, Matias G. Delgadino, Jun Kitagawa

Published 2026-05-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: F. -U. Caja-Lopez, Matias G. Delgadino, Jun Kitagawa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have two piles of sand. One pile is your starting point (the "source"), and the other is your destination (the "target"). Your goal is to move every grain of sand from the source to the target in the most efficient way possible, minimizing the total distance traveled. In mathematics, this is called Optimal Transport.

The "map" that tells you exactly where each grain of sand should go is called an Optimal Transport Map. If you have a perfect, smooth map, you know exactly where to move the sand. But what happens if you slightly change the shape of the sand piles? Does your map change drastically, or does it stay mostly the same?

This paper, written by Caja-Lopez, Delgadino, and Kitagawa, is all about stability. It asks: If I nudge the source or target sand piles just a tiny bit, how much does the optimal moving map wiggle?

Here is a breakdown of their findings using simple analogies:

1. The "Rubber Sheet" Analogy

Think of the optimal transport map as a rubber sheet stretched over a landscape. The landscape is defined by the density of the sand (how thick or thin the piles are).

The Problem: If you pinch the landscape slightly (change the sand density), the rubber sheet stretches or shifts.
The Question: Is the shift proportional to the pinch? If I pinch the sand by 1%, does the map move by 1% (Lipschitz stability), or does it jump wildly by 100%?

2. The Main Discovery: "Smoothness Matters"

The authors found that the answer depends heavily on how "smooth" the sand piles are.

The Ideal Scenario (Smooth Sand): If the sand piles are perfectly smooth and don't have any holes or sharp spikes (mathematically, they are "non-degenerate" and "Hölder continuous"), then the map is very stable.
- The Result: If you change the sand piles by a small amount, the map changes by a proportional small amount. It's like a well-oiled machine: a small input gives a small, predictable output.
- The Analogy: Imagine pushing a smooth, heavy ball on a flat floor. A small push moves it a small, predictable distance.
The Rough Scenario (Chunky Sand): If the sand piles are rough, have gaps, or are made of discrete chunks (like individual grains rather than a smooth mound), the map can be very unstable.
- The Result: The authors show that if the sand isn't smooth, you can't guarantee that a small change in the sand leads to a small change in the map. The map might jump around unpredictably.
- The Analogy: Imagine trying to push a pile of jagged rocks. A tiny nudge might cause the whole pile to shift unexpectedly or not move at all until a critical point is reached.

3. The "Second Variation" (The Bounce)

The paper also calculates something called the Second Variation of the distance between the two piles.

The Analogy: Think of the "distance" between your two sand piles as the height of a hill. The "First Variation" tells you which way is downhill (the direction to move). The "Second Variation" tells you how steep the hill is or how much it curves.
The Discovery: The authors derived a precise formula for this "curvature." They showed that the distance between the piles behaves in a very specific, predictable way when you wiggle the data. It's like having a precise formula for how much a trampoline bounces back when you push down on it.

4. How They Did It: The "Linearization" Trick

To prove these things, the authors used a mathematical technique called linearization.

The Metaphor: Imagine you are trying to predict the path of a rollercoaster that twists and turns wildly (the non-linear equation). It's too hard to solve directly. So, you zoom in on a tiny, tiny section of the track. From that close-up view, the track looks like a straight line.
The Application: They "zoomed in" on the complex equations governing the sand movement (the Monge-Ampère equation) to turn them into simpler, straight-line equations. By solving the simple version, they could predict how the complex version would behave when the sand piles were slightly changed.

Summary of the "Rules" They Found

Smooth Sand = Stable Map: If your source and target densities are smooth and don't vanish (there's always some sand everywhere), the map is Lipschitz stable. This means the error in the map is directly proportional to the error in the sand piles.
Rough Sand = Unstable Map: If the sand is rough or has holes, this nice, proportional relationship breaks down.
The Formula: They provided a new, explicit formula to calculate exactly how the "distance" between two sand configurations changes when you wiggle the configuration slightly.

In a nutshell: This paper proves that if you are moving smooth, continuous fluids (or sand), your transportation plan is robust and predictable. Small changes in the plan lead to small changes in the outcome. However, if the material is rough or patchy, the plan becomes fragile and unpredictable. The authors provided the mathematical "rulebook" to quantify exactly how stable these plans are.

Technical Summary: Stability of Optimal Transport Maps and Second Variation of the 2-Monge–Kantorovich Distance

1. Problem Statement

The paper addresses the quantitative stability of optimal transport maps and potentials in the context of the quadratic cost ( $p=2$ ). Specifically, it investigates how the optimal transport map $T = \nabla \phi$ and the associated Brenier potential $\phi$ change when the source density $f$ and target density $g$ are perturbed.

While qualitative stability (convergence in measure or uniform convergence under strong assumptions) is well-established via compactness arguments, the authors focus on deriving quantitative estimates. These estimates bound the distance between optimal maps/potentials corresponding to different densities in terms of the distance between the densities themselves (measured in $L^p$ , $L^2$ , or 2-Monge–Kantorovich distance). The work specifically targets settings where densities are non-degenerate (bounded away from zero) and possess Hölder regularity, on uniformly convex domains.

2. Methodology

The core technical strategy involves linearizing the Monge–Ampère equation along smooth interpolations of the data. The methodology proceeds as follows:

Interpolation: Given two pairs of densities $(f_0, g_0)$ $(f_{0}, g_{0})$ and $(f_1, g_1)$ $(f_{1}, g_{1})$ , the authors construct a smooth curve of densities $\{(f_t, g_t)\}_{t \in [0,1]}$ ${(f_{t}, g_{t})}_{t \in [0, 1]}$ .
- For Theorem 1.1, they utilize 2-Monge–Kantorovich geodesics (Benamou–Brenier formulation).
- For Theorems 1.2 and 1.3, they use linear interpolation in $L^p$ spaces.
Differentiation of Potentials: They define a curve of Brenier potentials $\phi_t$ corresponding to $(f_t, g_t)$ . The difference $\phi_1 - \phi_0$ is expressed as the integral of its time derivative: $\phi_1 - \phi_0 = \int_0^1 \partial_t \phi_t \, dt$ .
Linearized Equation: The derivative $\xi_t = \partial_t \phi_t$ is characterized as the unique solution to a linearized Monge–Ampère equation derived in divergence form:
$-\text{div}\left[ f_t (D^2 \phi_t)^{-1} \nabla \xi_t \right] = \left( \frac{\partial_t g_t(\nabla \phi_t)}{g_t(\nabla \phi_t)} - \frac{\partial_t f_t}{f_t} \right) f_t \quad \text{in } \Omega,$
subject to Neumann boundary conditions.
Regularity and Ellipticity: A crucial step is establishing that the coefficient matrix $(D^2 \phi_t)^{-1}$ $(D^{2} ϕ_{t})^{- 1}$ remains uniformly elliptic and that the solution $\xi_t$ $ξ_{t}$ retains sufficient regularity (e.g., $C^{1,\alpha}$ $C^{1, α}$ ) along the interpolation path. This relies on:
- Lemma 1.7: A compactness argument proving that if densities are $C^{0,\beta}$ and non-degenerate, the Brenier potential is $C^{2,\alpha}$ up to the boundary.
- Implicit Function Theorem: Used to prove the smoothness of the map $t \mapsto \phi_t$ in appropriate Banach spaces (Theorem 1.5).
Estimation: By estimating $\xi_t$ using elliptic regularity theory (Schauder estimates, $L^\infty$ bounds) and integrating over $t$ , the authors derive the final stability bounds.

3. Key Contributions and Results

A. Quantitative Stability Estimates

The paper establishes three main stability theorems under the assumption of non-degenerate, Hölder continuous densities on uniformly convex domains with smooth boundaries:

Theorem 1.1 (Lipschitz $L^2$ Stability):
For Hölder continuous densities, the authors prove Lipschitz continuity of the optimal transport maps in the $L^2$ norm with respect to the 2-Monge–Kantorovich distance ( $d_2$ ) between densities:
$\|\nabla \phi_1 - \nabla \phi_0\|_{L^2} \leq C [d_2(f_0, f_1) + d_2(g_0, g_1)].$
This improves upon previous results (e.g., Berman [20]) which often involved dimension-dependent exponents or required stronger regularity. The authors note this Lipschitz behavior is a special feature of regular, non-degenerate measures.
Theorem 1.2 (Stability without Continuity Assumption):
Removing the Hölder continuity assumption but retaining $L^2$ integrability and non-degeneracy, the authors obtain a Lipschitz stability estimate for the Brenier potentials in the $L^2$ norm:
$\|\phi_0 - \phi_1\|_{L^2} \leq C (\|f_0 - f_1\|_{L^2} + \|g_0 - g_1\|_{L^2}).$
They also derive a corresponding estimate for the gradients with a $1/3$ power exponent. The authors conjecture that the $L^2$ bound on potentials could potentially be improved to an $L^\infty$ bound if a specific Sobolev inequality for the linearized operator could be proven.
Theorem 1.3 (Uniform $C^{1,\alpha}$ Stability):
In a uniform metric setting, the authors prove that for $C^{0,\beta}$ densities, the optimal maps are stable in the $C^{1,\alpha}$ norm with respect to $L^p$ distances of the densities (where $p = d/(1-\alpha)$ ):
$\|\phi_1 - \phi_0\|_{C^{1,\alpha}} \leq C (\|f_1 - f_0\|_{L^p} + \|g_1 - g_0\|_{L^p}).$
Unlike previous interior estimates (e.g., [28, 29]), this result is global, applies to perturbations around non-identity maps, and does not suffer from dimension-decaying exponents. The authors emphasize that the lower bound condition ( $g_i \geq a > 0$ ) is fundamental; without it, even in 1D, no Hölder stability exponent $\eta < 1$ exists.

B. Second Variation of the Distance

Theorem 1.4: The authors derive an explicit formula for the second variation of the squared 2-Monge–Kantorovich distance along a perturbation of densities.
$\frac{1}{2} \frac{d^2}{dt^2}\bigg|_{t=0} [d_2(f_t, g_t)^2] = \int_\Omega \langle (D^2 \phi_0)^{-1} \nabla \xi, \nabla \xi \rangle f \, dx,$
where $\xi$ is the solution to the linearized equation. This formula quantifies the linear convexity of the distance and extends formal calculations previously found in [30] (which were restricted to perturbations around the identity) to general regular optimal transport maps.

C. Technical Tool: Time Differentiability

Theorem 1.5 & Corollary 1.6: A significant portion of the work is dedicated to proving that the Brenier potential $\phi_t$ is differentiable with respect to the interpolation parameter $t$ . The authors characterize the derivative $\partial_t \phi_t$ as the weak solution to the linearized Monge–Ampère equation in divergence form. This result is of independent interest for the analysis of time-dependent optimal transport problems.

4. Significance and Claims

The authors position their work as a significant advancement in the quantitative stability of optimal transport, a currently active research area.

Improvement on Exponents: The paper claims to provide Lipschitz (exponent 1) stability estimates in settings where previous literature often yielded sub-linear exponents (e.g., $1/2$ , $1/5$ , or dimension-dependent decaying exponents).
Global vs. Interior: Theorem 1.3 is highlighted as a global estimate, contrasting with prior results that were often restricted to interior estimates or specific geometric settings (e.g., codimension-one surfaces).
Explicit Second Variation: The derivation of the second variation formula is presented as a tool that explicitly quantifies the convexity of the Monge–Kantorovich distance, extending the theoretical understanding beyond the identity map.
Limitations: The authors are modest about the scope, explicitly stating that their results rely on non-degenerate densities (bounded away from zero) and Hölder regularity. They acknowledge that for discrete measures or vanishing densities, the Lipschitz nature of the estimates breaks down, and the optimal exponents may be lower (conjectured to be $1/2$ for discrete targets). They do not claim to resolve the stability for general rough measures or singular targets.

In summary, the paper provides a rigorous framework for linearizing the Monge–Ampère equation in divergence form to obtain sharp, quantitative stability bounds for optimal transport maps and potentials under regularity and non-degeneracy assumptions, while also providing a new formula for the second variation of the transport distance.