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arxiv: 2508.21658 · v4 · submitted 2025-08-29 · 🧮 math.PR

Infinite-dimensional stochastic differential equations for Coulomb random point fields

Hirofumi Osada , Shota Osada This is my paper

Pith reviewed 2026-05-18 20:30 UTC · model grok-4.3

classification 🧮 math.PR
keywords Coulomb random point fieldsinfinite-dimensional SDEsCoulomb interacting Brownian motionspathwise uniquenesslogarithmic derivativespoint processesinfinite-particle systems
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The pith

Coulomb interacting Brownian motions exist as unique strong solutions to infinite-dimensional SDEs in every dimension d ≥ 2 and any β > 0.

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

The paper constructs solutions to infinite-dimensional stochastic differential equations that describe the motion of infinitely many particles interacting via Coulomb repulsion. It proves these equations have strong solutions with pathwise uniqueness for any dimension two or higher and any positive inverse temperature. This extends earlier results limited to two dimensions and a specific temperature tied to random matrices. A sympathetic reader would care because it supplies a rigorous dynamical description for point processes that model physical systems like plasmas or electrons, where particles repel logarithmically. The construction also shows these infinite systems emerge as limits of finite-particle approximations with reflecting boundaries.

Core claim

We construct the Coulomb interacting Brownian motions for all spatial dimensions d ≥ 2 and inverse temperatures β > 0 by solving the associated infinite-dimensional stochastic differential equations. These ISDEs admit strong solutions and pathwise uniqueness holds. The labeled dynamics form an R^{dN}-valued diffusion while the unlabeled process is reversible with respect to the Coulomb random point field. The dynamics are identified as the path-space limit of finite-particle systems using approximation schemes involving finite-domain reflecting boundaries and N-particle systems. The method relies on explicit computation of the logarithmic derivatives of the Coulomb random point fields as the

What carries the argument

Infinite-dimensional stochastic differential equations whose drift terms come from the logarithmic derivatives of Coulomb random point fields, supplying the infinite-particle interaction.

If this is right

  • The dynamics arise as the path-space limit of finite-domain systems with reflecting boundary conditions.
  • The dynamics also arise as the limit of N-particle systems driven by ordinary SDEs.
  • The unlabeled version of the process is a reversible diffusion with respect to the underlying Coulomb random point field.
  • Pathwise uniqueness holds for the infinite-particle ISDEs.

Where Pith is reading between the lines

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

  • The stochastic-analysis method using logarithmic derivatives may extend to other long-range interaction kernels beyond the Coulomb case.
  • Finite-particle simulations with reflecting boundaries could now be used to approximate equilibrium statistics of the infinite system in dimensions higher than two.
  • The existence of the diffusion opens questions about its long-time behavior and possible convergence to the Coulomb point field for general β.

Load-bearing premise

The explicit computation of the logarithmic derivatives of the Coulomb random point fields is valid and supplies the correct infinite-particle drift term.

What would settle it

A numerical or analytic check showing that sample paths from the constructed infinite-particle process fail to satisfy the ISDE with the logarithmic-derivative drift in dimension three for some β > 0 would falsify the claim.

read the original abstract

We study the infinite-dimensional stochastic differential equations (ISDEs) of infinite-particle systems associated with Coulomb random point fields. The stochastic dynamics described by these ISDEs are referred to as Coulomb interacting Brownian motions. In all spatial dimensions $ d \ge 2 $ and for all inverse temperatures $ \beta > 0 $, we construct the Coulomb interacting Brownian motions. We prove that the ISDEs admit strong solutions and that pathwise uniqueness holds. The resulting labeled dynamics form an $ \RdN $-valued diffusion, possibly without an invariant measure, while the corresponding unlabeled process is a reversible diffusion with respect to the underlying Coulomb random point field. Moreover, we identify the infinite-particle stochastic dynamics as the limit in path space of finite-particle systems driven by stochastic differential equations. This identification is achieved through two approximation schemes: finite-domain systems with reflecting boundary conditions and $ N $-particle systems. Although the $ N $-particle approximation is more fundamental, its justification relies crucially on the finite-domain approximation together with the uniqueness of solutions to the ISDEs. Previously, only the case $ d = 2 $ and $ \beta = 2 $, known as the Ginibre interacting Brownian motion, was understood through random matrix theory and determinantal random point fields. Extending this result beyond the determinantal setting has remained a major difficulty. We introduce a new, conceptually clear method based on stochastic analysis of infinite-particle systems with long-range interactions that yields a rigorous construction of Coulomb interacting Brownian motions. A key ingredient is an explicit computation of the logarithmic derivatives of Coulomb random point fields.

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 manuscript constructs infinite-dimensional SDEs (ISDEs) for Coulomb interacting Brownian motions associated with Coulomb random point fields μ_{d,β} in all dimensions d ≥ 2 and all inverse temperatures β > 0. It claims to prove strong existence and pathwise uniqueness of solutions, shows that the labeled process is an R^{dN}-valued diffusion (possibly without invariant measure), establishes reversibility of the unlabeled process w.r.t. the point field, and identifies the dynamics as path-space limits of two approximation schemes: finite-domain systems with reflecting boundary conditions and N-particle systems. The construction relies on an explicit computation of the logarithmic derivatives of the Coulomb point fields to supply the infinite-particle drift term, extending beyond the previously known determinantal Ginibre case (d=2, β=2).

Significance. If the central claims hold, the work provides a rigorous stochastic-analysis framework for infinite-particle systems with long-range Coulomb interactions outside the determinantal setting. This extends the scope of infinite-dimensional diffusions and interacting point processes in statistical mechanics, with the approximation results offering concrete links to finite-N simulations. The method is conceptually clear and could serve as a template for other long-range kernels.

major comments (2)
  1. [Section deriving the logarithmic derivative / drift term (near the statement of the main existence theorem)] The explicit computation of the logarithmic derivatives of μ_{d,β} (the key ingredient supplying the drift) is load-bearing for both existence and pathwise uniqueness. For d > 2 or β ≠ 2 the field is non-determinantal, so the limit from finite-volume or finite-N configurations must be controlled by new estimates on the interaction kernel and configuration discrepancies; the manuscript should supply these estimates (with explicit error bounds) in the section deriving the drift term, as any uncontrolled error would render the ISDE ill-posed.
  2. [Section on N-particle and finite-domain approximations] The identification of the infinite-particle dynamics as the limit of the N-particle systems is stated to rely on the finite-domain approximation together with uniqueness; however, the path-space convergence argument requires quantitative control on the discrepancy between the finite-domain reflecting process and the infinite-volume process (e.g., via tightness or moment bounds uniform in the domain size). Without such controls the approximation claim is incomplete.
minor comments (2)
  1. [Introduction / notation section] Notation for the infinite-particle configuration space and the labeling map should be introduced earlier and used consistently; the transition from unlabeled point field to labeled R^{dN}-valued process is not immediately clear on first reading.
  2. [Abstract] The abstract asserts reversibility of the unlabeled process but does not state the precise form of the generator or the Dirichlet form; a brief display of the associated Dirichlet form would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [Section deriving the logarithmic derivative / drift term (near the statement of the main existence theorem)] The explicit computation of the logarithmic derivatives of μ_{d,β} (the key ingredient supplying the drift) is load-bearing for both existence and pathwise uniqueness. For d > 2 or β ≠ 2 the field is non-determinantal, so the limit from finite-volume or finite-N configurations must be controlled by new estimates on the interaction kernel and configuration discrepancies; the manuscript should supply these estimates (with explicit error bounds) in the section deriving the drift term, as any uncontrolled error would render the ISDE ill-posed.

    Authors: We appreciate the referee's emphasis on this foundational step. The explicit computation of the logarithmic derivatives appears in the section immediately preceding the main existence theorem and proceeds from the integral representation of the Coulomb kernel together with the Palm version of the point field μ_{d,β}. For the non-determinantal regimes the required control on configuration discrepancies follows from the uniform moment bounds on the Coulomb point process that are already available in the literature and are independent of d and β. These bounds guarantee that the finite-volume and finite-N approximations converge to the infinite-volume drift term. To strengthen the presentation we will insert a new lemma containing explicit error estimates (with constants depending only on d, β and the time horizon) in the revised version. revision: partial

  2. Referee: [Section on N-particle and finite-domain approximations] The identification of the infinite-particle dynamics as the limit of the N-particle systems is stated to rely on the finite-domain approximation together with uniqueness; however, the path-space convergence argument requires quantitative control on the discrepancy between the finite-domain reflecting process and the infinite-volume process (e.g., via tightness or moment bounds uniform in the domain size). Without such controls the approximation claim is incomplete.

    Authors: We agree that making the quantitative aspects explicit improves clarity. The finite-domain reflecting processes are constructed so that their laws are reversible with respect to the restricted Coulomb measure; this reversibility supplies uniform second-moment bounds on particle displacements that are independent of domain size. Pathwise uniqueness of the ISDE then identifies every limit point with the unique solution. In the revision we will add a tightness lemma that records these uniform moment bounds and the resulting compactness in the space of continuous paths, thereby furnishing the requested quantitative control on the approximation error. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the ISDE construction

full rationale

The derivation constructs strong solutions and pathwise uniqueness for the infinite-dimensional SDEs associated with Coulomb random point fields in all d ≥ 2 and β > 0. It relies on an explicit computation of logarithmic derivatives (obtained via controlled limits from finite-volume and finite-N approximations) as an independent input, followed by application of standard stochastic analysis tools for long-range interactions. The identification of infinite dynamics as limits of finite-particle systems uses the already-established uniqueness but does not reduce the central existence/uniqueness claims to a self-referential definition or fitted parameter. No load-bearing self-citation chain, ansatz smuggling, or renaming of known results appears; the work extends the determinantal Ginibre case with new estimates rather than assuming its own conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the existence and regularity of Coulomb random point fields for general d and β, plus the validity of an explicit logarithmic-derivative computation that is presented as the central new step; no free parameters or new invented entities are introduced.

axioms (2)
  • domain assumption Coulomb random point fields exist and are sufficiently regular for all d ≥ 2 and β > 0
    The entire construction is built on these fields; their existence is presupposed before the logarithmic derivatives are computed.
  • ad hoc to paper The logarithmic derivatives of the point fields can be computed explicitly and yield the correct drift
    This computation is singled out in the abstract as the key ingredient that enables the infinite-particle SDE.

pith-pipeline@v0.9.0 · 5811 in / 1600 out tokens · 56651 ms · 2026-05-18T20:30:38.169570+00:00 · methodology

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