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arxiv: 2605.23529 · v1 · pith:HWKYOVFAnew · submitted 2026-05-22 · 🧮 math.PR

Generalized Bessel-Dunkl diffusions

Jacek Ma{\l}ecki This is my paper

Pith reviewed 2026-05-25 03:52 UTC · model grok-4.3

classification 🧮 math.PR
keywords Bessel-Dunkl diffusionsWeyl chambersroot systemsweak existencesingular repulsionparticle systemsmean-field convergenceDyson processes
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The pith

Generalized Bessel-Dunkl diffusions admit weak existence even with degenerate repulsion that vanishes on walls.

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

The paper develops a unified framework for Bessel-Dunkl type diffusions inside Weyl chambers of classical root systems. Coefficients may depend on time and configuration, and singular repulsion terms are allowed to vanish on chamber walls. Weak existence follows in both the strictly positive repulsion regime, where multiple collisions are ruled out at positive times, and the degenerate regime, where geometric conditions on the root system prevent boundary sticking. The results also cover non-explosion, uniqueness under Lipschitz conditions, and mean-field convergence for root systems of types A, B, and D. A sympathetic reader would care because the setup encompasses Dyson particle systems, radial Dunkl processes, Wishart systems, and non-colliding diffusions under one set of tools.

Core claim

Using the geometry of the underlying root system and a symmetric-polynomial approach, weak existence holds for these generalized diffusions in both the strictly positive repulsion and the degenerate regimes. In the strictly positive regime, positive repulsion prevents positive-time multiple collisions. In the degenerate case, geometric conditions exclude boundary sticking and allow recovery of the genuine singular equation from its interior form. Non-explosion holds under a radial linear-growth condition. Pathwise uniqueness and strong existence follow in the non-sticky class under Yamada-Watanabe type or locally Lipschitz assumptions. A mean-field convergence theorem is proved for systems 3

What carries the argument

Symmetric-polynomial approach combined with the geometry of classical root systems, which controls singular repulsion terms even when they vanish on chamber walls

If this is right

  • Positive repulsion prevents positive-time multiple collisions.
  • Geometric conditions on the root system exclude boundary sticking in the degenerate regime.
  • Non-explosion holds under a radial linear-growth condition.
  • Pathwise uniqueness and strong existence hold under Yamada-Watanabe type or locally Lipschitz assumptions in the non-sticky class.
  • Mean-field convergence holds for systems of types A_{N-1}, B_N, and D_N.

Where Pith is reading between the lines

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

  • The same geometric control may extend to other classes of singular SPDEs with root-system symmetry.
  • The mean-field result could be tested on eigenvalue processes with time-varying potentials in random matrix models.
  • Numerical checks of the no-sticking geometric conditions for specific root systems would provide direct validation.
  • The framework opens a route to infinite-particle limits by relaxing the finite-N assumption.

Load-bearing premise

The diffusion and drift coefficients permit the symmetric-polynomial approach with root system geometry controlling the singular repulsion terms, including when those terms vanish on chamber walls.

What would settle it

A numerical simulation of particles under strictly positive repulsion that produces multiple collisions at some positive time, or a simulation in the degenerate regime that exhibits boundary sticking when the geometric conditions are violated.

read the original abstract

We develop a general theory of Bessel-Dunkl type diffusions in Weyl chambers associated with classical root systems. The class considered here allows time-dependent and configuration-dependent diffusion and drift coefficients, as well as state-dependent singular repulsion coefficients which may vanish on the walls of the chamber. This includes, in a unified framework, Dyson-type logarithmic particle systems, radial Dunkl processes, squared Bessel and Wishart particle systems, and non-colliding diffusion models. Using the geometry of the underlying root system and a symmetric-polynomial approach, we establish weak existence in both the strictly positive repulsion and the degenerate regimes. In the strictly positive regime, we prove that positive repulsion prevents positive-time multiple collisions. In the degenerate case, we identify geometric conditions which exclude boundary sticking and allow one to recover the genuine singular equation from its interior form. We also prove non-explosion under a radial linear-growth condition, obtain pathwise uniqueness and strong existence in the non-sticky class under natural Yamada-Watanabe type or locally Lipschitz assumptions, and establish a mean-field convergence theorem for systems of types $A_{N-1}$, $B_N$, and $D_N$. Together, these results build the structural foundations for a systematic theory of generalized Bessel-Dunkl diffusions with non-constant and possibly degenerate coefficients.

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 / 1 minor

Summary. The paper develops a general theory of Bessel-Dunkl type diffusions in Weyl chambers associated with classical root systems. It allows time-dependent and configuration-dependent diffusion and drift coefficients, as well as state-dependent singular repulsion coefficients which may vanish on chamber walls. This unifies Dyson-type logarithmic particle systems, radial Dunkl processes, squared Bessel and Wishart particle systems, and non-colliding diffusion models. Using root-system geometry and a symmetric-polynomial approach, the authors establish weak existence in both strictly positive repulsion and degenerate regimes, prove that positive repulsion prevents positive-time multiple collisions, identify geometric conditions excluding boundary sticking in the degenerate case, prove non-explosion under radial linear-growth, obtain pathwise uniqueness and strong existence under Yamada-Watanabe or locally Lipschitz assumptions, and establish mean-field convergence for systems of types A_{N-1}, B_N, and D_N.

Significance. If the central claims hold, the work provides structural foundations for a systematic theory of generalized Bessel-Dunkl diffusions with non-constant and possibly degenerate coefficients. Strengths include the unified framework covering multiple models, the mean-field convergence theorem for three root-system types, and the explicit geometric conditions for the degenerate regime. The symmetric-polynomial approach grounded in root-system geometry is a notable technical contribution for controlling singular terms.

major comments (2)
  1. [Abstract, unified framework paragraph] Abstract and unified framework paragraph: the symmetric-polynomial approach is asserted to control singular repulsion terms (including when they vanish on walls) for arbitrary time-dependent diffusion and drift coefficients, but the infinitesimal generator applied to symmetric polynomials need not map the space into itself under explicit time dependence; this directly affects the comparison arguments used for weak existence and non-collision, and is load-bearing for both the strictly positive and degenerate regimes.
  2. [Existence proofs, degenerate regime] Existence proofs (degenerate regime): the geometric conditions identified to exclude boundary sticking must be shown to compensate for both vanishing repulsion and time-dependent coefficients; without an explicit verification that the generator preserves the necessary polynomial closure or comparison properties, the recovery of the genuine singular equation from its interior form remains at risk.
minor comments (1)
  1. Notation for the root systems and the precise definition of the symmetric polynomials used in the generator calculations could be stated more explicitly at the outset for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment below with clarifications on the applicability of the symmetric-polynomial approach to time-dependent coefficients.

read point-by-point responses

  1. Referee: [Abstract, unified framework paragraph] Abstract and unified framework paragraph: the symmetric-polynomial approach is asserted to control singular repulsion terms (including when they vanish on walls) for arbitrary time-dependent diffusion and drift coefficients, but the infinitesimal generator applied to symmetric polynomials need not map the space into itself under explicit time dependence; this directly affects the comparison arguments used for weak existence and non-collision, and is load-bearing for both the strictly positive and degenerate regimes.

    Authors: The coefficients are configuration-dependent but symmetric with respect to the underlying root system (i.e., Weyl-group invariant). Consequently, for any symmetric polynomial p the spatial action of the generator at each fixed time t produces another symmetric function, even though the coefficients depend explicitly on t. The resulting time-inhomogeneous martingale problem is handled via the integrated form, which preserves the comparison arguments used for weak existence and collision prevention. The symmetry preservation is independent of the explicit time dependence. revision: partial

  2. Referee: [Existence proofs, degenerate regime] Existence proofs (degenerate regime): the geometric conditions identified to exclude boundary sticking must be shown to compensate for both vanishing repulsion and time-dependent coefficients; without an explicit verification that the generator preserves the necessary polynomial closure or comparison properties, the recovery of the genuine singular equation from its interior form remains at risk.

    Authors: The geometric conditions are stated purely in terms of the root-system geometry and the vanishing order of the repulsion coefficients on the walls; they do not depend on the time dependence of the diffusion and drift coefficients. Because the symmetry preservation established above continues to hold, the comparison between the interior process and the candidate boundary behavior carries over directly to the time-inhomogeneous setting. The recovery of the singular equation therefore remains valid under the stated linear-growth assumption. revision: no

Circularity Check

0 steps flagged

No circularity: results rest on standard stochastic analysis applied to root-system geometry

full rationale

The paper develops weak existence, non-collision, and non-explosion results for generalized Bessel-Dunkl diffusions via the symmetric-polynomial approach and root-system geometry. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the abstract and described framework invoke standard comparison arguments and Yamada-Watanabe-type uniqueness without renaming known results or smuggling ansatzes. The time-dependent coefficients are handled within the stated assumptions on the generator, with no evidence that the mapping back into symmetric polynomials is presupposed rather than proved. This is the normal case of an independent derivation grounded in classical stochastic analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of classical root systems and stochastic differential equations without introducing new free parameters or postulated entities; all technical assumptions are drawn from domain knowledge in probability on Weyl chambers.

axioms (2)
  • standard math Geometry and reflection properties of classical root systems A_{N-1}, B_N, D_N and their Weyl chambers
    Invoked throughout to control the singular repulsion and to apply the symmetric-polynomial approach (abstract).
  • domain assumption Standard existence and uniqueness theory for SDEs with singular coefficients when repulsion is strictly positive
    Used as background for the strictly positive regime results.

pith-pipeline@v0.9.0 · 5747 in / 1467 out tokens · 24427 ms · 2026-05-25T03:52:27.923798+00:00 · methodology

discussion (0)

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