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Quantitative Diffusive Limits for Singular Nonlocal Transport

Andrea Agazzi, Giuseppe Bruno, Federico Pasqualotto, Philippe Rigollet

arXiv:2609.11837v1math.APstat.ML

TL;DR

Rigorous convergence to heat flow had remained largely open for diffusion–velocity methods, which transport particles deterministically but require regularization to define velocities on empirical measures. The paper analyzes a regularization scheme and a weak–strong modulated energy argument, proving a uniform-in-time heat-flow bound and particle convergence estimates in one dimension.

  • Problem

    Rigorous convergence to heat flow had remained largely open for diffusion–velocity methods, which require smoothing because the velocity is undefined on empirical measures.

    Source: §1 · p. 2, §2.2 · p. 8

  • Method

    The paper studies a regularization scheme for heat-flow approximation and compares particle and continuum evolutions using a weak–strong modulated energy estimate.

    Source: §1 · p. 2, §4.2 · p. 28

  • Results

    W1(µ, µb) ≤ Cb^2, and for d = 1 the paper proves uniform-in-time propagation-of-chaos estimates polynomial in b and N.

    Source: Main results · pp. 2–3

  • Takeaways & Limitations

    Smoothing the density before evaluating the velocity makes diffusion-velocity dynamics well-defined for empirical particle measures.

    Source: §1 · p. 2

    The dynamics lack a Wasserstein gradient-flow interpretation, and the particle convergence analysis is limited to d = 1.

    Source: Main results · p. 3, §2.2 · p. 7, §2.3 · p. 10

Abstract

from arXiv · show

We study the nonlocal continuity equation \[ \partial_t\mu_b =\operatorname{div}\!\left( \mu_b\nabla\log\bigl((I-b^2\Delta)^{-1}\mu_b\bigr) \right) \] on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as $b \to 0$, its global solution converges to heat flow $\mu(t)$ at the sharp, uniform-in-time rate \[ \sup_{t\ge0}\|\mu_b(t)-\mu(t)\|_{L^1}\le Cb^2. \] The key estimate is the uniform dissipation of a $b$-weighted higher-order resolvent energy, which yields exponential relaxation despite the absence of a Wasserstein gradient-flow structure. On the circle, we also analyze the corresponding deterministic $N$-particle dynamics. A weak--strong modulated energy argument gives \[ \mathbb E\!\left[ \sup_{t\ge0}W_1(\mu_b^N(t),\mu_b(t)) \right] \le C(Nb)^{-1/2} \] for iid initialization. Consequently, the choice $b\asymp N^{-1/5}$ approximates heat flow uniformly in time at rate $N^{-2/5}$.

1 Introduction

The paper studies a resolvent-regularized transport equation as a deterministic diffusion-velocity method, asking whether its heat-flow approximation remains accurate uniformly in time and for finite particle systems. It establishes quantitative heat-flow convergence and particle approximation results, with further relevance to transformer signal propagation.

1 Introduction

  • Diffusion-velocity methods recast diffusion as deterministic particle transport along a density-dependent velocity, but the logarithmic gradient is undefined for empirical measures.Regularizing the density before evaluating the velocity addresses this singularity; the paper studies a new scheme with uniform approximation guarantees for heat flow.Source: §1 · p. 2
  • The studied equation uses a resolvent operator K_b=(I-b^2Δ)^-1, whose formal approximation to the identity motivates convergence to heat flow but does not itself establish accuracy at arbitrarily large times.The paper also asks whether the approximation persists when the evolving density is replaced by finitely many particles.Source: §1 · p. 2

Main results

  • The continuum dynamics converge quantitatively and uniformly in time to heat flow, with a W1 error bounded by Cb^2 and the estimate also holding in L1.The comparison uses the same smooth initial condition for both evolutions.Source: Main results · p. 2
  • On the circle, the deterministic N-particle system admits quantitative, uniform-in-time propagation-of-chaos estimates in W1 that are polynomial in b and N.Together with the continuum limit, these results justify the particle system as a heat-flow approximation when b tends to zero and Nb tends to infinity.Source: Main results · p. 3
  • The paper’s results extend to an external noninteracting gradient field and the corresponding Fokker–Planck equation, under the stated regularity and confinement conditions.The broader setting described includes R^d with sufficiently regular, confining drift, or T^d without drift.Source: Main results · p. 3
  • For spherical domains, the results rigorously establish the heat equation as a diffusion limit of a class of transformer signal-propagation dynamics.The connection concerns a line of work where the heat equation has been proposed as an effective description in certain limiting hyperparameter regimes.Source: Main results · pp. 3–4

2 Main results

The paper studies a resolvent-regularized continuity equation on a closed connected Riemannian manifold and proves quantitative convergence to heat flow. It also establishes a uniform-in-time particle approximation result on the circle, using a modulated energy adapted to the nonlinear velocity field.

2.1 Setup

  • For b>0, Kb=(I−b²∆)⁻¹ nonlocally regularizes density and approaches identity as b↓0; its kernel is singular on the diagonal for d≥2, while in d=1 it is bounded with a cusp.For smooth, strictly positive probability-density initial data and every b>0, the Cauchy problem admits a unique global classical solution.Source: §2.1 · pp. 4–6

2.2 Quantitative convergence to the heat flow

  • O(b^2) in L1 uniformly in time for the nonlocal evolution toward heat flow; on S1, the particle-to-continuum W1 bound is of order (Nb)^-1/2.The paper also combines these limits when b→0 and Nb→∞.Source: §2.2 · p. 6, §2.3 · pp. 10–11
  • Uniform-in-b exponential decay estimates for the resolvent energy, combined with Duhamel’s formula, yield the uniform-in-time L1 error of order b^2.The b^2 rate is sharp: the paper gives a matching lower-bound example on S1.Source: §2.2 · pp. 6–7
  • The regularization appears only inside the logarithm, avoiding an outer convolution; this enables a direct comparison with heat flow but removes the Wasserstein gradient-flow structure and its variational arguments.For empirical measures, evaluating KbµN requires one interaction sum, whereas an outer convolution adds an ambient-space integral to approximate.Source: §2.2 · p. 7

2.3 Quantitative Propagation of Chaos for d = 1

  • On S1, a weak–strong modulated energy estimate handles the nonlinear density-dependent velocity and closes without truncating the cusp kernel or removing self-interactions.The integrated decay of log Kbµb keeps the estimate uniform in time rather than exponentially growing in T.Source: §2.3 · p. 10
  • The particle result is proved in dimension one; extending it to higher dimensions is left for future work because the kernel has a Coulomb-type singularity and would require adapting truncation estimates.The combined heat-flow limit applies when b=bN→0 and NbN→∞.Source: §2.3 · pp. 10–11

3 Convergence to the heat equation

This section establishes global-in-time convergence of the nonlocal solution to heat flow using Duhamel comparison, uniform-in-b energy decay, and regularity estimates. It also shows that the b^2 convergence rate is sharp in general.

3 Convergence to the heat equation

  • The heat-flow comparison yields the stated O(b^2) convergence rate, using uniform decay and regularity estimates for the smoothed density.The comparison applies when the smoothed density has a positive lower bound and the required exponential decay estimates hold.Source: §3 · pp. 12–13

3.3 Proof of Theorem 3.2

  • For initial data satisfying Assumption 1, the higher-order resolvent energy decays exponentially uniformly in b and provides the regularity bounds needed for heat comparison.The theorem’s constants depend only on the manifold and initial data, for b below a positive threshold.Source: §3 · pp. 12–13

3.1 Proof of Proposition 3.1

  • The Duhamel comparison bounds the L1 difference from heat flow uniformly in time, conditional on uniform-in-b dissipation and a positive lower bound for the smoothed density.The proof rewrites the difference equation with a b^2-scaled forcing term and applies heat-semigroup smoothing.Source: §3.1 · pp. 13–14

3.2.1 Kernel and Regularity estimates

  • The Bessel–Yukawa kernel has a Sobolev regularity threshold at its pole and is strictly positive when applied to any nonzero finite nonnegative measure.The kernel’s Sobolev threshold follows from the regularity of a point mass and the resolvent isomorphism; positivity uses the heat-kernel representation on a closed connected manifold.Source: §3.2.1 · pp. 15–16

3.2.2 Commutator and Remainder Estimates

  • The energy argument uses manifold commutator estimates and a nonlinear remainder bound under positive lower and finite upper bounds on the density.The commutator estimates control differentiated divergence terms, while the remainder bound handles the logarithmic nonlinearity.Source: §3.2.2 · pp. 16–18

3.2.3 Energy and Sobolev estimates

  • The higher-order energy and dissipation are compared with Sobolev norms, and an L1–L∞ interpolation estimate supports the bootstrap.The norm comparisons are stated for zero-mean functions and 0 < b ≤ 1.Source: §3.2.3 · pp. 18–20

3.3 Proof of Theorem 3.2

  • The heat flow stays bounded below by c0, supporting the comparison used to close the bootstrap.The bound holds for all t ≥ 0 and x ∈ M.Source: §3.3 · p. 23
  • Continuity closes the bootstrap, so the estimates proved on [0, T*) hold globally for all t ≥ 0.The argument concludes that T* = ∞.Source: §3.3 · p. 24

3.4 Sharpness of b2 rate

  • On M = S1 ≃ R/2πZ, the b^2 rate cannot be improved in general; the cited setup gives t0 > 0, b1 > 0, and c > 0 for every 0 < b ≤ b1.Source: §3.4 · pp. 24–25

4 Proof of the one-dimensional mean-field limit

The circle case avoids kernel truncation and uses particle well-posedness plus a weak–strong modulated-energy estimate to establish uniform-in-time stability.

4 Proof of the one-dimensional mean-field limit

  • In dimension one, the manifold is a circle and the cusp-singular kernel permits the particle dynamics and modulated-energy estimate to proceed without truncation.The particle derivative at the cusp uses the arithmetic mean of the two one-sided limits, corresponding to K′b(0) = 0.Source: §4 · pp. 25–26

4.1 Well-posedness of the particle dynamics

  • Distinct initial particle positions yield a unique global C1 solution, and trajectories remain distinct.The noncollision result supports continuation of the particle dynamics; iid initialization from the nonatomic initial law gives distinct positions almost surely.Source: §4.1 · pp. 26–27

4.2 Modulated Energy estimate

  • The weak–strong modulated-energy estimate compares particle and continuum evolutions uniformly in time, with constants uniform in b, N, t, and the initial configuration.Its possible growth depends only on the continuum solution, whose coefficients are controlled by decay estimates.Source: §4.2 · pp. 28, 30

A Kernel Estimates

The appendix collects kernel and velocity estimates and extends an H−1–W2 comparison to arbitrary smooth, closed, connected manifolds.

A Kernel Estimates

  • The distance bounds for Kb have dimension-dependent regimes: constant in d = 1, logarithmic in d = 2, and r^(2-d) in d ≥ 3.For d = 1, Kb and its first derivative are bounded.Source: A Kernel Estimates · pp. 40–41
  • The H−1–W2 comparison is extended from manifolds with nonnegative Ricci curvature to arbitrary smooth, closed, connected manifolds, at the cost of a geometry-dependent constant.The result is stated for bounded probability densities.Source: A Kernel Estimates · p. 45
  • The appendix also states a logarithmic bound for Hölder functions and a divergence bound for velocities generated by bounded probability densities.It includes fixed-b Schauder estimates for Kbρ and V[ρ].Source: A Kernel Estimates · pp. 42, 43, 46, 48

B Well-posedness of the Nonlocal PDE

The appendix establishes local bounded well-posedness and rules out finite-time breakdown, yielding global solutions. It also propagates Hölder regularity and gives smoothness for smooth initial data.

B Well-posedness of the Nonlocal PDE

  • The local theory gives a unique maximal bounded solution, with mass preservation and a continuation criterion.The construction adapts the Yudovich scheme using the geodesic distance on the manifold.Source: B Well-posedness of the Nonlocal PDE · pp. 48–50
  • The maximum principle and approximation argument establish global existence for bounded probability densities, with mass preserved.For smooth strictly positive data, the maximum principle rules out the L∞ blow-up alternative.Source: B Well-posedness of the Nonlocal PDE · pp. 53–54
  • Solutions propagate C^{k,α} regularity globally, and smooth initial data yield solutions smooth in space and time.The bootstrap controls each higher spatial regularity level before time regularity follows from the equation.Source: B Well-posedness of the Nonlocal PDE · pp. 50, 53, 54

C Heat-Flow Preliminaries

The preliminaries supply heat-flow bounds used later: positivity and upper bounds persist, mean-zero deviations decay, and the heat semigroup satisfies smoothing estimates.

C Heat-Flow Preliminaries

  • The heat flow preserves the initial positive lower bound and the initial L∞ upper bound for all time.The proof uses the parabolic maximum principle.Source: C Heat-Flow Preliminaries · p. 55
  • Mean-zero deviations from the conserved average decay exponentially in H^s, as stated by the spectral estimate.The decay uses the positive spectral gap λ1 of −∆ on the mean-zero subspace.Source: C Heat-Flow Preliminaries · p. 55
  • The preliminaries also state heat-semigroup smoothing estimates for bounded measurable data.The supplied lemma identifies the heat semigroup as P_t = e^{t∆}.Source: C Heat-Flow Preliminaries · p. 56

D Entropy Sketch for Uniform Convergence

Under the appendix’s Ric ≥ 0 assumption, the entropy comparison bounds the relative entropy between the regularized solution and heat flow by order b². A uniform L1 consequence follows, while the extension to bounded initial data is only qualitative.

D Entropy Sketch for Uniform Convergence

E Numerical comparison of kernels

The numerical comparison tests Bessel–Yukawa and Gaussian particle kernels against heat flow on the circle. For the reported parameters, the Laplace dynamics have smaller errors, while Gaussian dynamics retain visible oscillations.

E Numerical comparison of kernels

  • For these parameters, Laplace dynamics yield smaller errors than Gaussian dynamics and approach an evenly spaced configuration, with W1 close to π/(2N), optimal for approximating the uniform measure using N equally weighted particles.The simulations evolve up to t = 10; Gaussian dynamics retain visible density oscillations and a larger error over the simulated interval.Source: E Numerical comparison of kernels · p. 59
  • Gaussian dynamics retain visible density oscillations and a larger error over the simulated time interval.The systems start from the same particle configuration, and their first Fourier coefficients agree.Source: E Numerical comparison of kernels · p. 59
  • Figure 1 shows Bessel–Yukawa and Gaussian histograms against exact heat flow, with kernel profiles and histogram L1 and circular W1 errors below.Its dashed line marks π/(2N).Source: Figure 1 · p. 60
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