Wasserstein gradient flows for Coulomb discrepancies

Version 2 adds a global metric PL inequality near the uniform target and a rigidity theorem for Lagrangian critical points.

\(\newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\T}{\mathbb{T}} \newcommand{\N}{\mathbb{N}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cP}{\mathcal{P}} \newcommand{\dd}{\mathop{}\!\mathrm{d}} \newcommand{\eps}{\varepsilon} \newcommand{\grad}{\nabla} \newcommand{\supp}{\operatorname{supp}} \newcommand{\coloneqq}{\mathrel{:=}} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\inner}[2]{\left\langle #1,#2\right\rangle}\)

Antonin Chodron de Courcel and I have posted version 2 of our paper Wasserstein gradient flows for Coulomb discrepancies. Since I did not write about the first version when it appeared, I want to use this post both to introduce the problem and to describe the main additions in the new version.

Coulomb discrepancy as a transport energy

The Maximum Mean Discrepancy (MMD) uses a kernel to measure the difference between two probability distributions. In our setting, the kernel is the Coulomb potential \(\mathsf g\), and, whenever this energy is finite, the resulting squared discrepancy is the negative-order Sobolev energy

\[ \mathsf{MMD}^{2}(\rho,\mu) :=\frac12\int_{\Omega}\mathsf g\ast(\rho-\mu)\,d(\rho-\mu) =\frac12\|\rho-\mu\|_{\dot H^{-1}}^{2}. \]

We fix a target distribution \(\mu\) and evolve a source distribution \(\rho\) by the Wasserstein gradient flow of this functional. The resulting equation is a nonlocal transport equation in which \(\rho\) moves under the Coulomb field generated by the discrepancy \(\rho-\mu\). It can be viewed as a distribution-matching dynamics, but also as the mean-field evolution of particles interacting through Coulomb forces in a fixed background of opposite charge.

The Coulomb structure makes the problem unusually explicit: the discrepancy potential solves a Poisson equation, so the energy is directly related to the field it generates. At the same time, this does not make convergence automatic. The source density acts as the mobility in the energy dissipation, so vacuum regions can weaken the usual coercivity argument. On the whole space, mass can also begin arbitrarily far from the target. Much of the paper concerns how these two effects shape the long-time behavior.

What the first version established

We first develop a Cauchy theory that allows rough initial data. When the target has a bounded density, we construct global weak solutions starting from an arbitrary Borel probability measure. In particular, the source may initially contain atoms. Coulomb repulsion produces an instantaneous \(L^\infty\) regularization, which we call ultracontractivity by analogy with the \(L^p\)-to-\(L^\infty\) smoothing of classical diffusion semigroups [1, 2]. At every positive time, the solution has a bounded density, with an estimate independent of the initial data. Uniqueness is recovered when the initial density is bounded. This smoothing effect does not extend to higher regularity: even for smooth initial data and targets, the Hölder seminorm of the solution can grow exponentially in time.

On the flat torus, we prove exponential decay of the squared MMD toward any bounded, uniformly positive target without assuming that the initial source is bounded away from zero. The point is that the standard Polyak–Łojasiewicz (PL) argument sees the vacuum in the evolving density. We replace it with a defective PL inequality whose defect explicitly accounts for this vacuum and can still be controlled along the flow.

Independent and overlapping results were obtained by Lénaïc Chizat, Maria Colombo, Roberto Colombo, and Xavier Fernández-Real [3] in a broader study of Sobolev and Riesz kernel discrepancies on the torus. At the Coulomb endpoint, they prove global weak-* convergence to an arbitrary bounded target from bounded initial densities and exponential energy decay for uniformly positive targets even when the initial density has vacuum regions. Their exponential-convergence argument fills those regions through a maximum principle and then applies the usual PL inequality; our argument instead retains the vacuum as an explicit defect, and our construction can start from an arbitrary Borel probability measure before entering the bounded-density class at positive times.

On Euclidean space, the answer depends strongly on geometry. For radial data, if the target has connected support, is bounded above and below on that support, and contains the support of the source, we obtain a PL inequality and exponential convergence. On the unrestricted whole-space class, there is a simple but decisive obstruction: a localized source placed a distance \(D\) from the target retains a fixed fraction of its initial squared MMD for times of order \(D\). Thus neither a global PL inequality nor a multiplicative convergence rate uniform over all initial data can hold in that setting.

What is new in version 2

The first main addition is a genuinely global-in-the-source metric PL inequality on the torus. It applies to every source with finite Coulomb energy when the target is uniform or sufficiently close to uniform, in the sense that if \(m\) and \(M\) are the essential lower and upper bounds of the target density, then the coercivity constant is positive when \(m-\frac{d-1}{d}M>0\). For the uniform target, the constant is \(2/d\). Beyond finite energy, no lower bound, smoothness, or proximity assumption is imposed on the source. Formulating the result using the descending Wasserstein metric slope is important here, since the Coulomb field need not have an intrinsic value on the support of a singular source. The inequality yields exponential convergence for the gradient flow after any positive time.

The proof begins with the uniform target, rewriting the dissipation as a Dirichlet term minus a cubic correction; a variational barrier argument controls this correction by evaluating the Euler–Lagrange equation at a minimum and using the Hessian trace inequality. Heat regularization, normalization, and a short-time flow argument then extend the estimate from smooth positive sources to the metric-slope statement for arbitrary finite-energy sources and to the near-uniform targets above.

The second addition concerns critical points and planar long-time behavior. We prove, in every dimension and on both the torus and Euclidean space, that a Lagrangian critical point must coincide with the target whenever the positive part \((\rho-\mu)^+\) of the discrepancy is absolutely continuous. In two dimensions, finite Coulomb energy also supplies uniform tightness through a logarithmic-capacity estimate. Combining this with dissipation, local compactness of the Coulomb fields, and the critical-point rigidity theorem gives qualitative convergence to the target for every solution constructed in the paper whose Coulomb energy is finite at some positive time. The convergence is narrow, weak-* in \(L^\infty\), and strong in \(H^{-\alpha}(\mathbb R^2)\) for every \(\alpha>0\).

The rigidity proof uses a fine locality theorem of Ambrosio, Ponce, and Rodiac [4, Theorem 1.1]: in dimensions \(d\geq 2\), if the distributional Laplacian of a locally integrable function is a locally finite signed measure, then its absolutely continuous part vanishes almost everywhere on every vector level set of the gradient. Writing \((\rho-\mu)^+=f\,dx\), criticality gives \(\nabla h=0\) almost everywhere on \(\{f>0\}\), while the Poisson equation says that the absolutely continuous part of \(\Delta h\) equals \(-f\) there. Applying their theorem to the level set \(\{\nabla h=0\}\) forces \(f=0\); in one dimension, the same conclusion follows from the analogous \(BV\) locality statement [5, Theorem 6.3]. Equality of the total masses then gives \(\rho=\mu\).

This last conclusion deliberately stops short of convergence in the Coulomb MMD itself, and it does not provide a rate. A vanishing amount of mass at an increasingly remote scale can disappear in the stated topologies while retaining nonzero Coulomb energy. Determining whether the dynamics can actually produce this mechanism remains open.

Looking ahead

Stay tuned as well for a forthcoming joint paper with Dejan Slepčev and Lihan Wang on Wasserstein gradient flows for MMDs generated by energy kernels, which we plan to post to arXiv this week. It studies analogous questions in a complementary kernel regime, with several overlapping results and some shared ideas and techniques. I will discuss that work in a separate blog post once it appears.

The Coulomb paper is available on arXiv, with the PDF available here.

References

1. E. B. Davies and B. Simon, “Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians,” Journal of Functional Analysis 59 (1984), no. 2, 335–395. doi:10.1016/0022-1236(84)90076-4.

2. E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, 1989. doi:10.1017/CBO9780511566158.

3. L. Chizat, M. Colombo, R. Colombo, and X. Fernández-Real, “Quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancies,” arXiv:2603.01977 (2026). arXiv.

4. L. Ambrosio, A. C. Ponce, and R. Rodiac, “Critical weak-\(L^p\) differentiability of singular integrals,” Revista Matemática Iberoamericana 36 (2020), no. 7, 2033–2072. doi:10.4171/RMI/1190.

5. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised ed., Textbooks in Mathematics, CRC Press, Boca Raton, 2015.