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arxiv: 2401.09970 · v2 · submitted 2024-01-18 · 🧮 math.PR · math.CA

Zero noise limit for singular ODE regularized by fractional noise

{\L}ukasz M\k{a}dry , Paul Gassiat This is my paper

Pith reviewed 2026-05-24 04:15 UTC · model grok-4.3

classification 🧮 math.PR math.CA
keywords singular ODEfractional noisezero noise limitextremal solutionssubexponential estimatesnon-Markovian processpower singularityconvergence
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The pith

As the intensity of fractional noise goes to zero, solutions to singular ODEs converge to the extremal deterministic solutions that exit the origin instantly.

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

The paper examines a scalar ordinary differential equation with a power-law singularity at the origin, smoothed by additive fractional noise. It proves that in the limit of vanishing noise intensity, the random solutions approach the special deterministic solutions that leave the origin immediately rather than staying there. This convergence is quantified using subexponential probability estimates. The result extends known findings for standard Brownian noise to the fractional case, where the noise lacks the Markov property, making the analysis more challenging.

Core claim

In the zero-noise limit, the solution to the regularized singular ODE converges to the extremal solutions of the unregularized ODE, which exit the origin instantly, and this convergence is accompanied by subexponential probability estimates. The methods adapt a dynamical approach to the non-Markovian fractional setting using large-time analysis techniques.

What carries the argument

Dynamical analysis combined with large-time estimates for fractional stochastic differential equations, adapted to the non-Markovian singular drift case.

If this is right

  • The convergence holds without the Markov property for the driving noise.
  • Deviation probabilities satisfy subexponential bounds rather than exponential ones.
  • The result applies to power singularities without further restrictions on the exponent.
  • The zero-noise limit selects the extremal solutions that leave the origin right away.

Where Pith is reading between the lines

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

  • The same limit selection may occur for other non-Markovian noises besides fractional Brownian motion.
  • The methods could apply to zero-noise limits in higher-dimensional singular ODEs.
  • Subexponential bounds might improve error control in numerical simulations of these systems.
  • The instant-exit behavior could connect to regularization phenomena in related singular stochastic models.

Load-bearing premise

The dynamical approach and large-time fractional techniques can be adapted to the singular non-Markovian setting without needing extra conditions on the Hurst parameter or the singularity power.

What would settle it

A counterexample where for some Hurst index the solution remains at the origin with positive probability even as noise intensity approaches zero, or where the deviation probabilities fail to be subexponential.

read the original abstract

We consider scalar ODE with a power singularity at the origin, regularized by an additive fractional noise. We show that, as the intensity in front of the noise goes to $0$, the solution converges to the extremal solutions to the ODE (which exit the origin instantly), and we quantify this convergence with subexponential probability estimates. This extends classical results of Bafico and Baldi in the Brownian case. The main difficulty lies in the absence of the Markov property for the system. Our methods combine a dynamical approach due to Delarue and Flandoli, with techniques from the large time analysis of fractional SDE (due in particular to Panloup and Richard).

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

0 major / 2 minor

Summary. The paper considers scalar ODEs with a power-law singularity at the origin, regularized by additive fractional noise of intensity ε. It proves that as ε → 0 the solutions converge to the extremal (instant-exit) solutions of the deterministic ODE, with quantitative subexponential probability estimates. The argument extends the Bafico–Baldi Brownian result by combining the Delarue–Flandoli dynamical approach with large-time fractional-SDE techniques of Panloup–Richard, the chief technical obstacle being the lack of the Markov property.

Significance. If the proofs are complete, the result supplies a non-Markovian extension of classical zero-noise limits together with explicit tail estimates; the successful adaptation of the cited dynamical and large-time methods to fractional noise constitutes a concrete technical contribution.

minor comments (2)
  1. [Abstract] Abstract: the admissible range of the Hurst index H and the singularity exponent should be stated explicitly, as the applicability of the Panloup–Richard large-time estimates may depend on these parameters.
  2. [Introduction] The manuscript should include a short remark confirming that no hidden Markovian reduction is used when transferring the Delarue–Flandoli exit-time controls to the fractional setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our work. The report correctly identifies the main contribution as a non-Markovian extension of the Bafico–Baldi zero-noise limit together with quantitative tail estimates, obtained by combining the Delarue–Flandoli dynamical approach with large-time techniques for fractional SDEs. We are pleased that the referee views the adaptation to fractional noise as a concrete technical contribution. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper claims a zero-noise limit result for a singular ODE driven by fractional noise, extending Bafico-Baldi via external techniques of Delarue-Flandoli (dynamical approach) and Panloup-Richard (large-time fractional SDE analysis). No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the non-Markovian adaptation is presented as a direct methodological extension without internal renaming or uniqueness imported from the authors' prior work. The derivation is self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The result rests on standard properties of fractional noise and on two external techniques whose applicability to the singular case is asserted but not detailed here.

axioms (1)
  • domain assumption Fractional noise satisfies the usual Hurst-parameter regularity and covariance assumptions used in Panloup-Richard large-time analysis.
    Invoked implicitly when combining the cited fractional-SDE techniques with the singular ODE.

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discussion (0)

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Reference graph

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