arXiv subjects

Guillaume Carlier

Publications and source records attributed to Guillaume Carlier.

At least 19 recordsLinked to original sources

arXiv2026

Nonlinear kinetic Fokker-Planck equations as gradient flows of the free energy

We consider nonhomogeneous kinetic equations that involve a free transport operator and a diffusion of porous medium type acting on velocities. The main novelty is a gradient flow interpretation of dynamics driven by an interplay of conservative and dissipative effects. We rely on a notion of discrepancy adapted to a phase space of positions and velocities, built upon second-order characteristics obeying Newton's laws. The equation appears as the steepest descent of the free energy functional. We also prove that approximate solutions constructed with an implicit Euler scheme converge to a solution of the kinetic equation. Thus, we generalise to a family of nonlinear kinetic equations the celebrated JKO scheme in mass transport theory. Most of our results are new even in the case of the linear Vlasov-Fokker-Planck equation.

math.AP↗

arXiv2026

The fundamental solution of a nonlinear kinetic Fokker-Planck equation

This paper is devoted to a fundamental solution of a nonlinear kinetic equation involving a porous medium or fast diffusion operator acting on velocities. Such a nonlinearity has interesting scaling properties, which result in a self-similar behaviour of the fundamental solution. Here fundamental solution means a Dirac distribution initial datum which moreover governs the large time asymptotics of a large class of solutions. Using a self-similar change of variables, the equation becomes a nonlinear kinetic Fokker-Planck equation with harmonic confinement and the intermediate asymptotics regime is transformed into a stability property of a special stationary solution, which attracts the solutions for large times. In the homogeneous case (pure nonlinear diffusion), the problem is reduced to a classical nonlinear diffusion equation with Barenblatt-Pattle self-similar profiles. Unexpectedly, this beautiful structure is preserved at kinetic level, with remarkable consequences for relative entropy estimates, detailed intermediate asymptotics and nonlinear diffusion limits in adapted functional spaces.

math.AP↗

arXiv2026

Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization

We consider weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglböck and Juillet.

math.OC↗

arXiv2025

Sharp comparisons between sliced and standard $1$-Wasserstein distances

Sliced Wasserstein distances are widely used in practice as a computationally efficient alternative to Wasserstein distances in high dimensions. In this paper, motivated by theoretical foundations of this alternative, we prove quantitative estimates between the sliced $1$-Wasserstein distance and the $1$-Wasserstein distance. We construct a concrete example to demonstrate the exponents in the estimate is sharp. We also provide a general analysis for the case where slicing involves projections onto $k$-planes and not just lines.

math.ST↗

arXiv2025

On a class of adversarial classification problems which admit a continuous solution

We consider a class of adversarial classification problems in the form of zero-sum games between a classifier and an adversary. The latter is able to corrupt data, at the expense of some optimal transport cost. We show that quite general assumptions on the loss functions of the classifier and the transport cost functions of the adversary ensure the existence of a Nash equilibrium with a continuous (or even Lipschitz) classifier's strategy. We also consider a softmax-like regularization of this problem and present numerical results for this regularization.

math.OC↗

arXiv2025

Monge-Kantorovich interpolation with constraints and application to a parking problem

We consider optimal transport problems where the cost for transporting a given probability measure $μ_0$ to another one $μ_1$ consists of two parts: the first one measures the transportation from $μ_0$ to an intermediate (pivot) measure $μ$ to be determined (and subject to various constraints), and the second one measures the transportation from $μ$ to $μ_1$. This leads to Monge-Kantorovich interpolation problems under constraints for which we establish various properties of the optimal pivot measures $μ$. Considering the more general situation where only some part of the mass uses the intermediate stop leads to a mathematical model for the optimal location of a parking region around a city. Numerical simulations, based on entropic regularization, are presented both for the optimal parking regions and for Monge-Kantorovich constrained interpolation problems.

math.OC↗

arXiv2025

Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced by Bigot, Cazelles and Papadakis (2019) as a regularization of Wasserstein barycenters first presented by Agueh and Carlier (2011). After characterizing these barycenters in terms of a system of Monge-Ampère equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters.

math.AP↗

arXiv2025

Wasserstein medians: robustness, PDE characterization and numerics

We investigate the notion of Wasserstein median as an alternative to the Wasserstein barycenter, which has become popular but may be sensitive to outliers. In terms of robustness to corrupted data, we indeed show that Wasserstein medians have a breakdown point of approximately $\frac{1}{2}$. We give explicit constructions of Wasserstein medians in dimension one which enable us to obtain $L^p$ estimates (which do not hold in higher dimensions). We also address dual and multimarginal reformulations. In convex subsets of $\mathbb{R}^d$, we connect Wasserstein medians to a minimal (multi) flow problem à la Beckmann and a system of PDEs of Monge-Kantorovich-type, for which we propose a $p$-Laplacian approximation. Our analysis eventually leads to a new numerical method to compute Wasserstein medians, which is based on a Douglas-Rachford scheme applied to the minimal flow formulation of the problem.

math.OC↗

arXiv2024

Well-posedness and convergence of entropic approximation of semi-geostrophic equations

We prove existence and uniqueness of solutions for an entropic version of the semi-geostrophic equations. We also establish convergence to a weak solution of the semi-geostrophic equations as the entropic parameter vanishes. Convergence is also proved for discretizations that can be computed numerically in practice as shown recently in [6].

math.AP↗

arXiv2024

On Optimal Transport Maps Between 1 /d-Concave Densities

In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$. Our focus is on a broader category of densities, specifically those that are $\nicefrac{1}{d}$-concave and can be represented as $V^{-d}$, where $V$ is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli's theorem.

math.AP↗

arXiv2024

Displacement smoothness of entropic optimal transport

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schrödinger map. We prove that when the cost function is $\mathcal{C}^{k+1}$ with $k\in \mathbb{N}^*$ then this map is Lipschitz continuous from the $L^2$-Wasserstein space to the space of $\mathcal{C}^k$ functions. Our result holds on compact domains and covers the multi-marginal case. We also include regularity results under negative Sobolev metrics weaker than Wasserstein under stronger smoothness assumptions on the cost. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.

math.OC↗

arXiv2024

Quantitative Stability of the Pushforward Operation by an Optimal Transport Map

We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.

math.OC↗

arXiv2023

Quantitative Stability of Barycenters in the Wasserstein Space

Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability measures of interest are often not accessible in their entirety and the practitioner may have to deal with statistical or computational approximations instead. In this article, we quantify the effect of such approximations on the corresponding barycenters. We show that Wasserstein barycenters depend in a H{ö}lder-continuous way on their marginals under relatively mild assumptions. Our proof relies on recent estimates that quantify the strong convexity of the dual quadratic optimal transport problem and a new result that allows to control the modulus of continuity of the push-forward operation under a (not necessarily smooth) optimal transport map.

math.NA↗

arXiv2023

Entropic approximation of $\infty$-optimal transport problems

We propose an entropic approximation approach for optimal transportation problems with a supremal cost. We establish $Γ$-convergence for suitably chosen parameters for the entropic penalization and that this procedure selects $\infty$-cyclically monotone plans at the limit. We also present some numerical illustrations performed with Sinkhorn's algorithm.

math.AP↗

arXiv2022

A simple city equilibrium model with an application to teleworking

We propose a simple semi-discrete spatial model where rents, wages and the density of population in a city can be deduced from free-mobility and equilibrium conditions on the labour and residential housing markets. We prove existence and (under stronger assumptions) uniqueness of the equilibrium. We extend our model to the case where teleworking is introduced. We present numerical simulations which shed light on the effect of teleworking on the structure of the city at equilibrium.

math.OC↗

arXiv2022

A mean field model for the interactions between firms on the markets of their inputs

We consider an economy made of competing firms which are heterogeneous in their capital and use several inputs for producing goods. Their consumption policy is fixed rationally by maximizing a utility and their capital cannot fall below a given threshold (state constraint). We aim at modeling the interactions between firms on the markets of the different inputs on the long term. The stationary equlibria are described by a system of coupled non-linear differential equations: a Hamilton-Jacobi equation describing the optimal control problem of a single atomistic firm; a continuity equation describing the distribution of the individual state variable (the capital) in the population of firms; the equilibria on the markets of the production factors. We prove the existence of equilibria under suitable assumptions.

math.AP↗

arXiv2022

Convergence rate of general entropic optimal transport costs

We investigate the convergence rate of the optimal entropic cost $v_\varepsilon$ to the optimal transport cost as the noise parameter $\varepsilon \downarrow 0$. We show that for a large class of cost functions $c$ on $\mathbb{R}^d\times \mathbb{R}^d$ (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and $L^{\infty}$ marginals, one has $v_\varepsilon-v_0= \frac{d}{2} \varepsilon \log(1/\varepsilon)+ O(\varepsilon)$. Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on $c$, i.e. invertibility of $\nabla_{xy}^2 c(x,y)$, we get the lower bound by establishing a quadratic detachment of the duality gap in $d$ dimensions thanks to Minty's trick.

math.OC↗

arXiv2021

Convex geometry of finite exchangeable laws and de Finetti style representation with universal correlated corrections

We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If $(Z_1,...,Z_N)$ is a finitely exchangeable sequence of $N$ random variables taking values in some Polish space $X$, we show that the law $μ_k$ of the first $k$ components has a representation of the form $μ_k=\int_{{\mathcal P}_{\frac{1}{N}}(X)} F_{N,k}(λ) \, \mbox{d} α(λ)$ for some probability measure $α$ on the set of $1/N$-quantized probability measures on $X$ and certain universal polynomials $F_{N,k}$. The latter consist of a leading term $N^{k-1}\! /{\small \prod_{j=1}^{k-1}(N\! -\! j)\, λ^{\otimes k}}$ and a finite, exponentially decaying series of correlated corrections of order $N^{-j}$ ($j=1,...,k$). The $F_{N,k}(λ)$ are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals $λ$. Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis-Freedman between finite and infinite exchangeable laws.

math.PR↗