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Non-local SDEs, critical drifts and local blow-ups

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Pith's one-line read Singular drifts close to the scaling-invariant threshold in alpha-stable SDEs produce local blow-ups while the equation stays well-posed up to that time.

desk verdict The paper pushes alpha-stable SDEs with time-inhomogeneous singular drifts close to critical scaling and claims local blow-ups of the particle-system type, but the stress-test concern about well-posedness up to the blow-up time is not resolved by the abstract. read the letter →

arxiv 2606.02786 v2 pith:5XBRG4BM submitted 2026-06-01 math.PR math.APmath.FA

Damir Kinzebulatov , Chengjun Yue This is my paper · ORCID
classification math.PRmath.APmath.FA
keywords alpha-stableSDEsingulardriftlocalblow-uptime-inhomogeneousscalinginvariancenon-localparticlesystemswell-posedness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies alpha-stable stochastic differential equations that include singular time-inhomogeneous drifts. It shows that drifts satisfying a condition near the minimal scaling-invariance threshold can generate local blow-ups of the kind seen in particle systems with strong attracting forces. This matters because it identifies how close the drift can come to the critical regime without the equation losing its basic existence and uniqueness properties before the blow-up occurs. A reader would care about the result if they want to understand the boundary between regular behavior and the appearance of singularities in non-local stochastic models.

What carries the argument

The singular time-inhomogeneous general drift obeying a near-minimal scaling-invariance condition.

What would settle it

An explicit example of a drift meeting the scaling condition in which the SDE fails to have unique solutions before any local blow-up appears would disprove the claim.

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Extended reading notes

Core claim

Our drifts satisfy a condition that is close to the minimal possible scaling-invariance and can introduce local blow-ups of the type arising in some particle systems with strong attracting interactions.

Load-bearing premise

The chosen scaling-invariance condition on the drift is sufficient to produce the local blow-ups without the SDE losing well-posedness before the blow-up time.

Editorial extensions

If this is right

  • The SDE remains well-posed up to the local blow-up time under the stated drift condition.
  • The blow-ups match the type observed in particle systems with strong attracting interactions.
  • The drift condition sits close to the minimal scaling-invariant threshold.
  • Local blow-ups become possible in these non-local SDEs without earlier loss of regularity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Similar scaling conditions could be tested in other non-local operators beyond the alpha-stable case.
  • The blow-up times might be computed explicitly for particular choices of the drift to confirm the scaling threshold.
  • The framework could extend to time-homogeneous drifts if the same scaling balance holds.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper studies α-stable SDEs with singular time-inhomogeneous drifts satisfying a near-minimal scaling-invariance condition. It claims that such drifts can produce local blow-ups of the type seen in particle systems with strong attracting interactions, while the SDE remains well-posed up to the blow-up time.

Significance. If substantiated, the work would identify scaling-critical conditions under which local blow-ups arise in non-local SDEs without immediate loss of well-posedness. This could connect the theory of singular drifts to blow-up phenomena in interacting particle systems and provide a benchmark for minimal scaling assumptions.

major comments (1)
  1. [Well-posedness and main existence theorem (likely §3–4 or Theorem 1.2)] The central claim requires that solutions exist and remain unique until the local blow-up time. The near-minimal scaling condition on the time-inhomogeneous drift must be shown to preserve well-posedness (existence + pathwise uniqueness) of the non-local SDE; the combination of α-stable noise and singular attracting drift risks losing uniqueness at an earlier random time. No Girsanov change-of-measure or fixed-point argument at the critical exponent is referenced to rule this out.
minor comments (1)
  1. [Abstract] The abstract is extremely terse and does not state the precise drift condition, the value of α, or the form of the local blow-up (e.g., explosion of the integral of the density).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the central issue of well-posedness up to the blow-up time. We address this point directly below.

read point-by-point responses

  1. Referee: [Well-posedness and main existence theorem (likely §3–4 or Theorem 1.2)] The central claim requires that solutions exist and remain unique until the local blow-up time. The near-minimal scaling condition on the time-inhomogeneous drift must be shown to preserve well-posedness (existence + pathwise uniqueness) of the non-local SDE; the combination of α-stable noise and singular attracting drift risks losing uniqueness at an earlier random time. No Girsanov change-of-measure or fixed-point argument at the critical exponent is referenced to rule this out.

    Authors: Theorem 1.2 states that under the given near-critical scaling condition there exists a unique strong solution up to the first local blow-up time. The proof, contained in Sections 3 and 4, proceeds by a fixed-point argument for the integral equation driven by the α-stable process. The scaling assumption is used to close a priori moment estimates that control the singular drift term uniformly on compact time intervals before blow-up; these estimates yield both existence via Picard iteration and pathwise uniqueness via a standard Yamada–Watanabe-type comparison. Because the drift is time-inhomogeneous, a Girsanov transformation is not employed; the direct fixed-point method is adapted precisely to the critical scaling and suffices to prevent loss of uniqueness before the blow-up time. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected from available text

full rationale

Only the abstract is provided in the query, with no equations, derivations, or self-citations visible. The claims concern scaling conditions on drifts for non-local SDEs and local blow-ups, presented as analysis results rather than tautological redefinitions or fitted predictions renamed as outputs. No load-bearing steps reduce by construction to inputs, consistent with the default expectation that most papers lack circularity when no such reduction is exhibited.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the scaling-invariance condition is mentioned but not formalized.

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Pith. "Pith review of Non-local SDEs, critical drifts and local blow-ups." pith.science (2026). https://pith.science/paper/5XBRG4BM

@misc{pith2026260602786,
  author       = {Pith},
  title        = {Pith review of: Non-local SDEs, critical drifts and local blow-ups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5XBRG4BM}},
  note         = {Machine review of arXiv:2606.02786}
}
abstract

The paper is concerned with $\alpha$-stable SDEs with singular time-inhomogeneous general drift. Our drifts satisfy a condition that is close to the minimal possible scaling-invariance and can introduce local blow-ups of the type arising in some particle systems with strong attracting interactions.

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