On blow up NLS with a multiplicative noise

Chenjie Fan; Junzhe Wang

arxiv: 2601.19330 · v2 · submitted 2026-01-27 · 🧮 math.AP · math.PR

On blow up NLS with a multiplicative noise

Chenjie Fan , Junzhe Wang This is my paper

Pith reviewed 2026-05-16 11:06 UTC · model grok-4.3

classification 🧮 math.AP math.PR

keywords nonlinear Schrödinger equationmultiplicative noiseblow-uplarge deviationsstochastic partial differential equations

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The pith

For the nonlinear Schrödinger equation with multiplicative noise, the probability of blow-up in arbitrarily short times is small and admits a large-deviation upper bound.

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

This paper studies the nonlinear Schrödinger equation driven by multiplicative noise. Earlier work established that under nondegenerate conditions the noise can cause blow-up to occur in any short time interval with positive probability. The authors show that this probability is in fact quite small by deriving a large-deviation-type upper bound. A reader would care because the bound quantifies how rarely the noise produces extremely rapid singularities, sharpening the picture of noise effects on blow-up.

Core claim

Under the nondegenerate conditions on the multiplicative noise, the probability that blow-up occurs in arbitrarily short time is small, and this is quantified by a large deviation type upper bound.

What carries the argument

Large-deviation upper bound on the probability of short-time blow-up for the stochastic nonlinear Schrödinger equation.

If this is right

Blow-up can still occur in short times but such events carry an exponentially small probability controlled by the large-deviation rate.
The same nondegenerate conditions that make noise accelerate blow-up also make rapid blow-up rare.
The large-deviation estimate supplies a quantitative refinement of the positive-probability result from the cited earlier work.

Where Pith is reading between the lines

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

The bound suggests that typical blow-up times remain bounded away from zero even though the probability of arbitrarily small times is positive.
The method may extend to other stochastic dispersive equations where noise is known to promote singularities.
Direct Monte-Carlo sampling of many solution trajectories could test the sharpness of the upper bound for specific choices of noise.

Load-bearing premise

The nondegenerate conditions on the multiplicative noise remain in force and allow application of large-deviation estimates.

What would settle it

A concrete numerical simulation or exact calculation showing that the probability of blow-up in a short time interval exceeds the derived large-deviation upper bound for some admissible noise and initial data would falsify the claim.

read the original abstract

It is of significant interest to understand whether a noise will speed up or prevent blow up. Under certain nondegenerate conditions, \cite{dD2005Blowup} proved a multiplicative noise will speed up blow up of NLS, in the sense that, blow up can happen in any short time with positive probability. We prove that such probability is indeed quite small, and provide a large deviation type upper bound.

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.

Desk Editor's Note private letter to a colleague

This paper turns the known positive probability of fast blow-up under multiplicative noise into an explicit large-deviation upper bound.

read the letter

The central result here is a large-deviation upper bound showing that the probability of blow-up in arbitrarily short time for the stochastic NLS is exponentially small. It builds directly on the 2005 result that such blow-up occurs with positive probability under nondegenerate multiplicative noise, and the new contribution is the quantitative rate obtained by applying large-deviation principles to the controlled equation. The argument reduces the stochastic problem to a deterministic controlled system and shows the rate function stays positive on the set of paths that blow up quickly, which is a clean and natural extension. The setup follows the existing nondegeneracy conditions without introducing new assumptions, and the reduction step looks standard for this area. One minor soft spot is that the bound itself may not be sharp or easy to evaluate in concrete examples, and its usefulness will depend on how well the nondegeneracy carries over to specific noise terms; if the noise is near the boundary of the condition, the rate could become large and less informative. The paper does not provide lower bounds or numerical checks, so the estimate stands as an upper bound only. This work is aimed at people working on stochastic dispersive equations and blow-up phenomena who already know the 2005 result and want a more precise handle on the probabilities. It is the kind of incremental quantitative step that fits well in the literature and deserves a serious referee to check the large-deviation estimates and error controls in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript considers the stochastic nonlinear Schrödinger equation (NLS) with multiplicative noise. Building on dD2005Blowup, which established that under nondegenerate conditions on the noise the solution can blow up in arbitrarily short time with positive probability, the paper derives a large-deviation upper bound showing that this probability is small. The argument invokes an LDP for the controlled deterministic equation and shows the associated rate function is strictly positive on the set of paths that blow up rapidly.

Significance. If the derivation holds, the result supplies a quantitative large-deviation rate that complements the qualitative existence statement in the cited work. It demonstrates that rapid blow-up events, while possible, are exponentially rare, which is useful for understanding the typical behavior of the stochastic NLS and for designing numerical experiments that sample rare events. The approach reuses the existing nondegeneracy hypotheses without introducing new parameters or ad-hoc assumptions.

major comments (2)

[Abstract and §2 (main result)] The abstract states that a large-deviation upper bound is provided, but the manuscript does not explicitly identify the function space or topology in which the LDP is applied (e.g., C([0,T];H^1) or a weaker space). Without this, it is impossible to verify that the rate function is indeed strictly positive on the blow-up set, as required for the upper bound to be nontrivial.
[§3 (derivation of the upper bound)] The reduction from the stochastic equation to the controlled deterministic system (presumably via the LDP for the driving noise) requires explicit control on the remainder terms when the control is chosen to produce a blow-up path. The manuscript should state the precise error estimate or cite the exact theorem from the large-deviation literature that justifies passing to the limit.

minor comments (2)

[References] The citation dD2005Blowup should be expanded with full bibliographic details (title, journal, year, pages) in the reference list.
[§1 (introduction)] Notation for the multiplicative noise coefficient and the precise definition of blow-up (e.g., ||u(t)||_{H^1} → ∞ as t → T) should be introduced at the beginning of §1 rather than assumed from the cited paper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments. We address each major point below and have revised the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

Referee: [Abstract and §2 (main result)] The abstract states that a large-deviation upper bound is provided, but the manuscript does not explicitly identify the function space or topology in which the LDP is applied (e.g., C([0,T];H^1) or a weaker space). Without this, it is impossible to verify that the rate function is indeed strictly positive on the blow-up set, as required for the upper bound to be nontrivial.

Authors: We agree that explicit identification of the function space strengthens the presentation. In the revised manuscript we state that the large-deviation principle holds in C([0,T];H^1) equipped with the uniform topology. Under the nondegeneracy assumptions inherited from dD2005Blowup, the associated rate function is strictly positive on the set of paths whose H^1-norm blows up in finite time; a short argument establishing this positivity is now included in §2. revision: yes
Referee: [§3 (derivation of the upper bound)] The reduction from the stochastic equation to the controlled deterministic system (presumably via the LDP for the driving noise) requires explicit control on the remainder terms when the control is chosen to produce a blow-up path. The manuscript should state the precise error estimate or cite the exact theorem from the large-deviation literature that justifies passing to the limit.

Authors: We thank the referee for highlighting this point. The revised §3 now cites the precise large-deviation theorem for the multiplicative noise (the version of the LDP for controlled stochastic PDEs used in the proof) and supplies the explicit error bound between the controlled deterministic solution and the stochastic solution when the control is chosen to approximate a rapid blow-up path. This makes the passage to the limit fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation applies large-deviation principles to the controlled deterministic NLS obtained from the stochastic equation, showing that the rate function is strictly positive on paths that blow up in arbitrarily short time. This step uses the nondegeneracy hypotheses from the external reference dD2005Blowup as an independent input and does not reduce any quantity to a fitted parameter, self-definition, or prior result by the same authors. No load-bearing self-citation, ansatz smuggling, or renaming of known results occurs; the upper bound is a direct consequence of the LDP and the positivity of the rate function on the target set.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument relies on standard large-deviation principles and nondegeneracy assumptions from stochastic PDE theory without introducing new free parameters or invented entities.

axioms (1)

domain assumption Nondegenerate conditions on the multiplicative noise as stated in dD2005Blowup
Required to invoke the large-deviation principle for the stochastic NLS flow.

pith-pipeline@v0.9.0 · 5347 in / 1120 out tokens · 28029 ms · 2026-05-16T11:06:58.149154+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that such probability is indeed quite small, and provide a large deviation type upper bound... ln P(u blows up in [0,T]) ≲ ||u0||_H1,ϕ T^{-β}
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Under certain nondegenerate conditions, [11] proved a multiplicative noise will speed up blow up of NLS

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