arXiv subjects

Elias Hess-Childs

Publications and source records attributed to Elias Hess-Childs.

11 recordsLinked to original sources

arXiv2026

Effective Lagrangian regularity and the uniqueness threshold for random Hölder velocity fields

We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale finite range decomposition. The velocity fields we consider are only Hölder regular in space---$C^{α-}(\mathbb{T}^d)$ for some $α\in (0,1)$---thus the associated ODE and continuity equation are not a priori well-posed. However, above the critical threshold of $α= 1/2$, due to multiscale stochastic cancellations, we prove well-posedness is almost surely restored away from the zero level set of the velocity field. This threshold marks a genuine transition, as demonstrated by examples lying below the threshold that exhibit robust ill-posedness. We additionally provide effective regularity estimates below the critical threshold and prove analogous results in the related "refreshing" regime.

math.PR↗

arXiv2026

Sharp pathwise nonuniqueness for additive SDEs

We construct a family of velocity fields demonstrating the sharpness of the classical Zvonkin--Veretennikov--Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift $u$ and show that for any $α<0$, there exists a velocity field $u \in L^\infty_t C^α_x$ that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case $α\geq 0$, for which the existence of a unique strong solution is guaranteed. The velocity field construction is random, and the proof essentially uses central limit theorem scaling through the Berry--Esseen theorem. We also give natural extensions to non-Brownian driving noises, including nonuniqueness for arbitrary driving noises with certain Hölder regularities and an analogous sharpness of the strong well-posedness by noise regime for fractional Brownian motions.

math.PR↗

arXiv2026

Radon--Wasserstein Gradient Flows for Interacting-Particle Sampling in High Dimensions

Gradient flows of the Kullback--Leibler (KL) divergence, such as the Fokker--Planck equation and Stein Variational Gradient Descent, evolve a distribution toward a target density known only up to a normalizing constant. We introduce new gradient flows of the KL divergence with a remarkable combination of properties: they admit accurate interacting-particle approximations in high dimensions, and the per-step cost scales linearly in both the number of particles and the dimension. These gradient flows are based on new transportation-based Riemannian geometries on the space of probability measures: the Radon--Wasserstein geometry and the related Regularized Radon--Wasserstein (RRW) geometry. We define these geometries using the Radon transform so that the gradient-flow velocities depend only on one-dimensional projections. This yields interacting-particle-based algorithms whose per-step cost follows from efficient Fast Fourier Transform-based evaluation of the required 1D convolutions. We additionally provide numerical experiments that study the performance of the proposed algorithms and compare convergence behavior and quantization. Finally, we prove some theoretical results including well-posedness of the flows and long-time convergence guarantees for the RRW flow.

stat.ML↗

arXiv2026

Another look at regularity in transport-commutator estimates

We are interested in how regular a transport velocity field must be in order to control Riesz-type commutators. Estimates for these commutators play a central role in the analysis of the mean-field limit and fluctuations for systems of particles with pairwise Riesz interactions, which we start by reviewing. Our first new result shows that the usual $L^\infty$ assumption on the gradient of the velocity field cannot, in general, be relaxed to a BMO assumption. We construct counterexamples in all dimensions and all Riesz singularities $-2< s<d$, except for the one-dimensional logarithmic endpoint $s=0$. At this exceptional endpoint, such a relaxation is possible, a fact related to the classical Coifman-Rochberg-Weiss commutator bound for the Hilbert transform. Our second result identifies a trade-off between the singularity of the interaction potential and the required regularity of the velocity field. Roughly speaking, smoother (less singular) interactions require stronger velocity control if one wants a commutator estimate in the natural energy seminorm determined by the potential. We formulate this principle for a broad class of potentials and show that, in the sub-Coulomb Riesz regime, the velocity regularity appearing in the known commutator inequality is sharp. Despite these negative findings, we show as our third result that a defective commutator estimate holds for almost-Lipschitz transport fields. Such a defective estimate, which is a consequence of the celebrated Brezis-Wainger-Hansson inequality, allows us to prove rates of convergence when the mean-field density belongs to the scaling-critical Sobolev space.

math.AP↗

arXiv2025

A sharp commutator estimate for all Riesz modulated energies

We prove a functional inequality in any dimension controlling the derivative along a transport of the Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the third author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where this control is an essential ingredient. Previous work of the last two authors and Q.H. Nguyen arXiv:2107.02592 showed a similar functional inequality but with an additive $N$-dependent error (where $N$ is the number of particles, $\mathsf{d}$ the dimension, and $\mathsf{s}$ the inverse power of the Riesz potential) which was not sharp. In this paper, we obtain the optimal $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$ error, for all cases, including the sub-Coulomb case. Our method is conceptually simple and, like previous work, relies on the observation that the derivative along a transport of the modulated energy is the quadratic form of a commutator. Through a new potential truncation scheme based on a wavelet-type representation of the Riesz potential to handle its singularity, the proof reduces to averaging over a family of Kato-Ponce type estimates. The commutator estimate has applications to sharp rates of convergence for mean-field limits, quasi-neutral limits, and central limit theorems for the fluctuations of Coulomb/Riesz gases both at and out of thermal equilibrium. In particular, we show here for $\mathsf{s}<\mathsf{d}-2$ the expected $N^{\frac{\mathsf{s}}{\mathsf{d}}-1}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows. This complements the recent work arXiv:2407.15650 on the optimal rate for the (super-)Coulomb case $\mathsf{d}-2\le \mathsf{s}<\mathsf{d}$ and therefore resolves the entire potential Riesz case.

math.AP↗

arXiv2025

Turbulent and intermittent phenomena in a universal total anomalous dissipator

For all $α\in (0,1)$, we construct an explicit divergence-free vector field $V \in L^\infty([0,1],C^α(\mathbb{T}^2))$ that exhibits universal anomalous (total) dissipation, accelerating dissipation enhancement, Richardson dispersion, anomalous regularization, and spatial intermittency. Additionally, we demonstrate the sharpness of the intermittent Obukhov-Corrsin regime for certain parameter ranges.

math.AP↗

arXiv2025

Divergence-free drifts decrease concentration

We show that bounded divergence-free vector fields $u : [0,\infty) \times \mathbb{R}^d \to\mathbb{R}^d$ decrease the ''concentration'', quantified by the modulus of absolute continuity with respect to the Lebesgue measure, of solutions to the associated advection-diffusion equation when compared to solutions to the heat equation. In particular, for symmetric decreasing initial data, the solution to the advection-diffusion equation has (without a prefactor constant) larger variance, larger entropy, and smaller $L^p$ norms for all $p \in [1,\infty]$ than the solution to the heat equation. We also note that the same is not true on $\mathbb{T}^d$.

math.AP↗

arXiv2025

A universal total anomalous dissipator

For all $α\in(0,1)$, we construct an explicit divergence-free vector field $V\in L^\infty_tC^α_x \cap C^{\fracα{1-α}}_t L^\infty_x$ so that the solutions to the drift-diffusion equations $$\partial_tθ^κ-κΔθ^κ+V\cdot\nablaθ^κ=0$$ exhibit asymptotic total dissipation for all mean-zero initial data: $\lim_{κ\rightarrow 0}\|θ^κ(1,\cdot)\|_{L^2}=0$. Additionally, we give explicit rates in $κ$ and uniform dependence on initial data.

math.AP↗

arXiv2024

Large deviation principles for singular Riesz-type diffusive flows

We combine hydrodynamic and modulated energy techniques to study the large deviations of systems of particles with pairwise singular repulsive interactions and additive noise. Specifically, we examine periodic Riesz interactions indexed by parameter $\mathbf{s}\in[0,d-2)$ for $d\geq 3$ on the torus. When $\mathbf{s}\in(0,d-2)$, we establish a large deviation principle (LDP) upper bound and partial lower bound given sufficiently strong convergence of the initial conditions. When $\mathbf{s}=0$ (i.e., the interaction potential is logarithmic), we prove that a complete LDP holds. Additionally, we show a local LDP holds in the distance defined by the modulated energy.

math.PR↗

arXiv2024

Higher-order propagation of chaos in $L^2$ for interacting diffusions

In this paper, we study diffusions with bounded pairwise interaction. We show for the first time propagation of chaos on arbitrary time horizons in a stronger $L^2$-based distance, as opposed to the usual Wasserstein or relative entropy distances. The estimate is based on iterating inequalities derived from the BBGKY hierarchy and does not follow directly from bounds on the full $N$-particle density. This argument gives the optimal rate in $N$, showing the distance between the $j$-particle marginal density and the tensor product of the mean-field limit is $O(N^{-1})$. We use cluster expansions to give perturbative higher-order corrections to the mean-field limit. For an arbitrary order $i$, these provide ``low-dimensional'' approximations to the $j$-particle marginal density with error $O(N^{-(i+1)})$.

math.PR↗

arXiv2020

On the Strong Attraction Limit for a Class of Nonlocal Interaction Energies

This note concerns the problem of minimizing a certain family of non-local energy functionals over measures on $\mathbb{R}^n$, subject to a mass constraint, in a strong attraction limit. In these problems, the total energy is an integral over pair interactions of attractive-repulsive type. The interaction kernel is a sum of competing power law potentials with attractive powers $α\in (0, \infty)$ and repulsive powers associated with Riesz potentials. The strong attraction limit $α\to \infty$ is addressed via Gamma-convergence, and minimizers of the limit are characterized in terms of an isodiametric capacity problem. We also provide evidence for symmetry-breaking in high dimensions.

math.AP↗