We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure $\rho$ and a target measure $\mu$, where the underlying kernel is given by a Coulomb potential. For $L^\infty$ target densities $\mu$, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove that the density $\rho_t$ belongs to $L^\infty$ for any $t>0$. We also show that the H\"older norm can grow exponentially in time. On the flat torus ${\mathbb{T}}^\mathsf{d}$, we prove a global metric PL inequality for every finite-Coulomb-energy source and nearly uniform target. For general bounded, uniformly positive targets, we prove exponential decay of the squared MMD without requiring a lower bound on the initial data, using a defective PL inequality. We also prove that the usual PL inequality may fail when the target vanishes only at one point and that, when $\mathsf{d}\ge2$, no PL constant can hold uniformly over all targets satisfying a prescribed lower bound. On ${\mathbb{R}}^\mathsf{d}$, for $\mathsf{d}\ge2$, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence. On the unrestricted whole-space class, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold. Finally, in every dimension and in both spatial settings, we prove that every Lagrangian critical point coincides with the target when $(\rho-\mu)^+$ is absolutely continuous. In dimension two, the energy supplies uniform tightness. This implies that if our constructed solutions have finite energy at some positive time, then they converge to the target narrowly and strongly in negative-order Sobolev spaces.
We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels $K(z)=-|z|^q$, $0<q<2$. In dimensions $d\ge2$, the corresponding energies are not displacement semiconvex, so standard Wasserstein-gradient-flow theory does not apply. When $d+q-2>0$, we prove global well-posedness on $\mathbb{R}^d$ for probability densities in subcritical $L^p$ spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint $d=q=1$. For the associated $N$-particle system, we prove global noncollision and fixed-$N$ convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as $N\to\infty$ on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For $1\le q<2$, every continuum solution in our class has a narrowly relatively compact orbit, every $\omega$-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For $0<q<1$, the same conclusions hold under uniform-in-time moment and subcritical $L^p$ bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order $q$, except when $0<q<1$ and $d\in\{1,3\}$. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on $\mathbb{R}^d$, and that global Polyak--\L ojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.
Matthew Rosenzweig, D. Slepčev, Lihan Wang· 1 citation
Wasserstein gradient flows are intimately connected with evolution partial differential equations and diffusion processes. We take the first step in developing such connections for inner product Gromov--Wasserstein (IGW) gradient flows by studying the IGW gradient flow of the relative entropy $\mathsf{H}(\cdot\|\gamma)$ with respect to the standard Gaussian measure $\gamma$. We first show that $\mathsf{H}(\cdot\|\gamma)$ fails to be $\lambda$-convex along generalized or modified generalized IGW geodesics for any $\lambda \in \mathbb{R}$, and therefore falls outside the scope of the existing IGW gradient flow theory from Zhang et al. (2026). We bridge this gap by establishing a suitable \emph{local} convexity estimate that enables the construction of the gradient flow and its extension to the infinite time horizon. We then obtain increasingly explicit representations of the resulting dynamics. Starting from a partial integro-differential equation, we derive a nonlinear Fokker--Planck equation and show that its second-moment dynamics decouple from the law as they satisfy an autonomous matrix ODE. This reduces the IGW dynamics to a linear, time-inhomogeneous Fokker--Planck equation, yielding a probabilistic representation as the time-marginal flow of a linear stochastic differential equation resembling the Ornstein--Uhlenbeck process. Finally, we study its asymptotic behavior by establishing exponential convergence of the flow to $\gamma$ in relative entropy.
Venkatkrishna Karumanchi, Z. Goldfeld, Kengo Kato et al.· 0 citations
We introduce a unified framework to establish the high-dimensional asymptotic behavior of maximal functions associated with radial log-concave probability densities, encompassing the maximal heat semigroup, Hardy-Littlewood maximal function over Euclidean balls, and, additionally, maximal spherical means. Namely, for any $p \in (1, \infty)$, we prove that the $L^p(\mathbb{R}^d)$ operator norms of these maximal operators all converge as the dimension $d \to \infty$ to a single, universal limit $\lambda(p)$. Furthermore, by proving that the $L^p$ operator norms for the heat semigroup $\mathcal G_*^d$ are monotonically non-decreasing in the dimension, we provide explicit quantitative bounds on the universal limit, showing that $\frac{2}{5}\frac{p}{p-1} \le \|\mathcal{G}_*^1\|_{L^p(\mathbb{R}) \to L^p(\mathbb{R})} \le \lambda(p) \le \frac{p}{p-1}$. We also prove an extremality property: among all symmetric convex bodies in high dimensions, the maximal operator associated with the Euclidean ball achieves the asymptotically minimal $L^p$ operator norm. Our main results are established via a general transference principle that allows us to control maximal functions via Fourier multiplier symbols. To estimate these symbols uniformly across log-concave densities, we import variance type bounds and thin-shell type concentration of measure results, which are novel tools in the study of maximal functions. In particular, to prove the extremality property, we require a variance type bound for general log concave measures established in a recent series of breakthroughs in high dimensional convex geometry.
We develop two Brownian stopping methods: one uses hitting probabilities and capacity, the other moments of weighted exit distributions. On a closed $m$-dimensional Riemannian manifold, $m\geq 2$, the first gives low-energy spectral inequalities for equilibrium measures supported on sets of zero volume. Combining scales gives heat observability and null controllability from a full-support measure carried by a dense set of Hausdorff dimension $m-2$, minimal under a bounded $H^1$-trace condition. For every nondecreasing unbounded $\Phi$ with $\Phi(4E)\leq D\Phi(E)$, the measure can be chosen so that the optimal spectral constant is comparable to $\Phi(E)$ and the small-time control cost to $T^{-1/2}\Phi(T^{-1})^{1/2}$. Squaring the Feynman-Kac martingale doubles the potential: for radial $V$, the boundary second moment is log-convex in $\log r$ if $r^2V(r)$ is nondecreasing, wherever the regular solution of $(\tfrac12\Delta+2V)g=0$ stays positive; the factor $2$ is sharp when $V(0)>0$. On small geodesic balls, weighted second moments give an almost-monotone frequency and a weighted three-radius inequality; bounds for its logarithmic derivative give doubling estimates, which, after a ground-state transform, feed the Remez theorem of Logunov-Malinnikova. The normalised first-moment exit law gives an averaged boundary-variance identity for Laplace eigenfunctions. At $r=c\lambda^{-1/2}$, outside a set carrying $O(c^{2/m})$ of the $\varphi^2$-mass, the component of $\{\varphi(x)\varphi>0\}\cap B(x,r)$ through $x$ contains a concentric ball of nearly full radius, fills nearly all of $B(x,r)$, and nearly matches its first Dirichlet eigenvalue. First-moment stopping also controls positive superlevel components. Reflected Brownian motion and local time give boundary-to-collar $L^\infty$ estimates for Steklov eigenfunctions on bounded Lipschitz domains, and an $L^2$ estimate for $C^2$ domains.
We establish the sharp logarithmic order $(\log N)^{-1/2}$ for the expected $p$-Wasserstein distance, induced by the supremum norm, between the empirical law of $N$ independent copies of a continuous It\^o process and their common path law. We only assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $\rho$-moment for some $\rho>p\geq1$. Under this assumption, we use an adaptive random time interval partition argument, which leads to a $(\log n)^{-1/2}$ functional quantization rate. A general transfer principle then converts the quantization estimate into a mean estimate and nonasymptotic deviation bounds for equal-weight empirical laws. Applications include empirical path-law estimates for path-dependent SDEs and a path-space propagation-of-chaos estimate for path-dependent McKean--Vlasov interacting particle systems.
We establish quantitative mean-field convergence and propagation of chaos for repulsive Coulomb gradient flows at the bounded-density regularity of the limiting equation. The argument couples the dissipative modulated-energy identity with the normalized quadratic transport cost of the full $N$-particle law. The remaining negative mean-square force-error term is used through an exact completion of squares after mollification: the non-Lipschitz remainder is absorbed by this negative term, while a sharp first-order commutator estimate is applied to the mollified Lipschitz field. For the Coulomb equation, the sharp $L^\infty$ decay gives the density envelope $m(t)=\|\rho_0\|_{L^\infty}/(1+t\|\rho_0\|_{L^\infty})$. A density-adapted transport weight and mollification scale $m(t)^{-1/d}$ yield an Osgood comparison. Thus, for every $d\ge2$ and $\rho_0\in\mathcal P_2(\mathbb R^d)\cap L^\infty(\mathbb R^d)$, we obtain quantitative comparison with the global bounded-density Coulomb solution on every prescribed finite interval. For tensorized initial data, the normalized squared Wasserstein distance of the full $N$-particle law, the expected modulated energy, and the time-integrated mean-square force error are bounded by $N^{-2\gamma_{T,d}/d}$ for $d\ge3$ and $((1+\log N)/N)^{\gamma_{T,2}}$ for $d=2$, where $\gamma_{T,d}=(1+T\|\rho_0\|_{L^\infty})^{-c_d}$. For $d-2<s<d$, we also prove Riesz weak--strong stability for prescribed reference solutions in $L^\infty(0,T;B^{s-d+2}_{\infty,q})$, with Gronwall, Bihari, and Osgood comparisons according to $q$, together with uniqueness in the stated Besov class. Finally, an outlier construction separates modulated-energy convergence and Kac chaos from normalized Wasserstein convergence of the full $N$-particle law.
Ning Jiang, Zhengyang Qiao, Juntao Wu et al.· 0 citations