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arxiv: 2605.16788 · v1 · pith:VBRFT3PInew · submitted 2026-05-16 · 🧮 math.AP · math.PR

Strong well-posedness of a singular SDE for signed Coulomb particles

Patrick van Meurs , Yoan Tardy This is my paper

Pith reviewed 2026-05-19 21:22 UTC · model grok-4.3

classification 🧮 math.AP math.PR
keywords signed Coulomb particlessingular SDEstrong solutionsglobal well-posednessannihilationscaling invariance
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The pith

Signed Coulomb particles in two dimensions have unique strong global solutions despite singular collisions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves the existence and uniqueness of strong global solutions to a system of stochastic differential equations for signed Coulomb particles moving in the plane. Opposite-sign particles collide and annihilate in finite time, after which they are removed from the system. The authors overcome the challenge of the singular interaction by exploiting scaling invariance and adapting techniques from the Keller-Segel particle system. A sympathetic reader cares because this provides a rigorous mathematical basis for modeling systems where particles interact via singular forces and can destroy each other, which is relevant for certain physical and biological processes.

Core claim

The SDE system for signed Coulomb particles admits strong global solutions that are unique, and all collisions that occur are between particles of opposite signs and result in their annihilation.

What carries the argument

Scaling invariance of the process combined with tools developed for the Keller-Segel particle system

If this is right

  • The solutions exist globally in time even with the singularity.
  • Collisions are fully characterized as only opposite-sign annihilations.
  • The process can be continued uniquely past any collision times.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This result may extend to other dimensions or interaction kernels with similar scaling.
  • It provides a foundation for studying long-time behavior or statistical properties of such annihilating systems.

Load-bearing premise

The analytical tools from the Keller-Segel system transfer successfully to handle the signed Coulomb interactions with annihilation.

What would settle it

A concrete counterexample consisting of a path where two same-sign particles collide or where two different solutions exist after a collision time would falsify the result.

Figures

Figures reproduced from arXiv: 2605.16788 by Patrick van Meurs, Yoan Tardy.

Figure 1
Figure 1. Figure 1: Snapshot of Xt = (X1 t , . . . , XN t ) ∈ (R 2 ) N with signs b = (b 1 , . . . , bN ). The system (1.1) was recently proposed and studied in [vMPS25]. A simple interpretation of (1.1) is that of electrically charged point particles that evolve by a ‘velocity = noise + force’ law. In actual applications, (1.1) appears as a model of vortices or as a model for dislocations in metals. We refer to [vMPS25, Sect… view at source ↗
Figure 2
Figure 2. Figure 2: (a) A particle configuration with all particles equispaced on a circle with alternating [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: By the definitions of Dsame(x) and Dopp(x), positive particles other than x 1 are outside of the red regions and negative particles other than x 2 are outside of blue ones. Purple regions contain no particles other than x 1 and x 2 . Proof of Proposition 6.6. We start with preparations. We write c instead of cN and only assume that c > 0. Since the statement of Proposition 6.6 is stronger for smaller c, we… view at source ↗
Figure 4
Figure 4. Figure 4: Example of a configuration x and a partition K that separates the positive from the negative particles. The problem is that this does not give the information we seek, namely that Dsame(Xζ ) ≥ cN Dopp(Xζ ). Indeed, if |K| ≥ 3, then RK(Xζ ) ≫ r K gives no lower bound on Dsame(XK ζ ); it only says that not all particles in the cluster XK ζ are close. The proof strategy of Proposition 6.8 is to solve this pro… view at source ↗
read the original abstract

We consider an SDE system for signed Coulomb particles moving in $\mathbb R^2$. Due to the singular Coulomb interaction force, collisions between particles of opposite sign will happen in finite time. Upon collision, the colliding particles are removed from the system. Our main results are the existence and uniqueness of strong global solutions and the characterization of all possible collisions. The challenge of the proofs is to deal with the singularity of the interactions. We overcome this by using scaling invariance of the process and by putting together several tools from [FT25] developed for the similar Keller--Segel particle system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves strong existence and uniqueness of global solutions to a system of SDEs for signed Coulomb particles in R^2, where opposite-sign collisions result in instantaneous particle removal (annihilation). It also characterizes all possible collision configurations. The argument relies on scaling invariance of the dynamics together with direct transfer of stopping-time and a-priori estimates developed in [FT25] for the (unsigned) Keller-Segel particle system.

Significance. If the transfer of tools is justified, the result would furnish the first rigorous global well-posedness theory for singular attractive-repulsive particle systems with annihilation, extending the repulsive-only theory of [FT25]. The scaling-invariance reduction is a clean technical device that exploits the 2D homogeneity of the Coulomb kernel and could serve as a template for related singular SDEs.

major comments (2)
  1. [§3.2] §3.2 (global existence via stopping times): the manuscript re-applies the non-explosion estimates of [FT25, Lemma 4.3] verbatim, but does not derive a new lower bound on the signed interaction energy that controls cancellations between attractive (opposite-sign) and repulsive (same-sign) terms; without such a bound the time to the first annihilation cannot be shown to be positive almost surely after the particle count has dropped.
  2. [Theorem 4.1] Theorem 4.1 (collision characterization): the proof that only binary opposite-sign collisions occur assumes that the post-annihilation drift remains in L^1_loc after each removal, yet the manuscript supplies no separate integrability check for the reduced system; this step is load-bearing for both uniqueness and the claim that all collisions are characterized.
minor comments (2)
  1. [§2.1] The interaction kernel K(x) is introduced only in the abstract; an explicit formula distinguishing the sign-dependent cases should appear in §2.1 for readability.
  2. Several citations to [FT25] refer to “the estimates of Section 4” without naming the precise lemma; add explicit cross-references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. The points raised concern the justification of transferring estimates from the unsigned Keller-Segel system to the signed case with annihilation. We provide point-by-point responses and will incorporate revisions to address these concerns.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (global existence via stopping times): the manuscript re-applies the non-explosion estimates of [FT25, Lemma 4.3] verbatim, but does not derive a new lower bound on the signed interaction energy that controls cancellations between attractive (opposite-sign) and repulsive (same-sign) terms; without such a bound the time to the first annihilation cannot be shown to be positive almost surely after the particle count has dropped.

    Authors: We appreciate this observation. The signed interaction energy does indeed require careful treatment to control cancellations. In the revised version, we will derive an explicit lower bound for the signed Coulomb energy. This bound will be obtained by separating the repulsive (same-sign) and attractive (opposite-sign) contributions and noting that annihilations eliminate attractive pairs, thereby preventing unbounded negative contributions. Combined with the scaling invariance, this ensures that the first annihilation time is positive a.s. even after the particle number decreases. We will insert this estimate as a new lemma in §3.2. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1 (collision characterization): the proof that only binary opposite-sign collisions occur assumes that the post-annihilation drift remains in L^1_loc after each removal, yet the manuscript supplies no separate integrability check for the reduced system; this step is load-bearing for both uniqueness and the claim that all collisions are characterized.

    Authors: We thank the referee for highlighting this crucial step. We will add a separate verification that the drift remains in L^1_loc for the reduced system after each annihilation. Since the annihilation removes the colliding pair at their contact point, the drift for the surviving particles is the restriction of the original drift minus the singular terms from the removed particles. As the original drift is in L^1_loc away from collisions and particles are distinct post-annihilation, the reduced drift satisfies the same integrability. This will be included in the proof of Theorem 4.1 to support the uniqueness and collision characterization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external prior tools

full rationale

The paper derives strong global well-posedness and collision characterization for the signed Coulomb SDE by invoking scaling invariance together with estimates and stopping-time arguments adapted from the cited external reference [FT25] on the Keller-Segel particle system. No step reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the argument explicitly treats [FT25] as independent prior work whose lemmas are transferred and combined with new scaling-based controls for the signed/annihilation setting. The central existence-uniqueness claim therefore retains independent mathematical content outside the present manuscript's own equations or data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on scaling invariance and transfer of tools from the referenced [FT25] paper.

pith-pipeline@v0.9.0 · 5619 in / 1054 out tokens · 28859 ms · 2026-05-19T21:22:29.517110+00:00 · methodology

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Reference graph

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