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Non-selection of Lagrangian trajectories in the zero-noise limit for a class of stochastic regularizations

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Pith's one-line read Adding small noise to a rough incompressible flow does not select a unique trajectory in the zero-noise limit

desk verdict Strong negative result on vanishing-noise selection for Hölder drifts; main issue is an unproven and non-identical Proposition 5.2 that the final theorem depends on. read the letter →

arxiv 2606.07096 v2 pith:AEKAZ3UC submitted 2026-06-05 math.PR math.AP

Lucio Galeati , Filippo Giovagnini , Massimo Sorella This is my paper
classification math.PRmath.AP MSC 60H1076B0360H5060G22
keywords zero-noiselimitstochasticregularizationfractionalBrownianmotionstableLévyprocessLagrangiantrajectoriesnon-selectionmixingbyshearflowssewinglemma
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

This paper proves that adding a small random perturbation (fractional Brownian motion or stable Lévy noise) to a divergence-free, Hölder-continuous velocity field and then letting the noise intensity tend to zero does not, in general, select a unique trajectory of the underlying ordinary differential equation. For any Hölder exponent α strictly between a noise-dependent threshold α_W and 1, the authors construct an incompressible drift for which, on a set of initial data with measure arbitrarily close to 1, the trajectories obtained with even-indexed noise intensities and those with odd-indexed ones fail to meet, staying a positive distance apart with probability 1. The same construction shows that even the probability laws of the noisy trajectories do not converge to a single limit measure. This matters because vanishing-noise limits are a standard selection criterion for non-unique ODE flows, and the result rules out such selection under mere Hölder regularity and incompressibility, in any dimension d≥2. A companion PDE corollary shows that vanishing fractional viscosity also fails to select a unique solution to the transport equation.

What carries the argument

The drift u is built from alternating horizontal and vertical shear flows active on super-exponentially shrinking time intervals, producing a mixing mechanism on [0,1/2] and an unmixing one on [1/2,1] with a small 'swap' perturbation that makes even and odd smooth approximations u_q behave differently. Parameters a_q (spatial scales), ℓ_q (mollification), κ_q (noise intensity), and t_q (time cutoffs) are tuned so that the stochastic flow X^{κ_q} stays close to the deterministic flow of u_q (Propositions 5.1, 5.2), using pathwise estimates on regular regimes and the stochastic sewing lemma (Lemma 3.2) to control oscillatory integrals in the rough regime. The chessboard sets A_ε are designed s

What would settle it

Run the SDE on the constructed shear-flow drift with Brownian noise and the paper's even/odd noise intensities κ_q; if the two subsequences of X_1(x) converge to the same point for a set of x of positive measure with positive probability, then the parity property (4.26)–(4.27) and Proposition 6.1 cannot both hold.

Watch

Extended reading notes

Core claim

Theorem 1.2 establishes a divergence-free drift u∈C^α([0,1]×T²) such that for any ε>0 there is a set A_ε of initial data with |A_ε|≥1−ε, a vanishing sequence (κ_q), and c_ε>0 for which, with probability 1, the stochastic flows X^{κ_{2q}}_1 and X^{κ_{2q+1}}_1 stay at distance at least c_ε for Lebesgue-a.e. x∈A_ε. Moreover, for each fixed x∈A_ε one can extract subsequences along which X^{κ_{2q}}_1(x) and X^{κ_{2q+1}}_1(x) converge almost surely to two distinct deterministic points y≠y′. Consequently the vanishing-noise limit does not select a unique Lagrangian trajectory, and the laws Law(X^{κ_q}_1(x)) do not converge to a unique measure.

Load-bearing premise

The whole construction presupposes the cited strong well-posedness theorems for SDEs with Hölder drift driven by fBm or stable Lévy noise: the stochastic flow X^κ used in the conclusion exists only when the drift exponent α exceeds the threshold α_W; if those thresholds are not exactly as stated (e.g., for H>1/2 or small β), the interval (α_W,1) would shrink and the constructed u might not admit a well-defined vanishing-noise sequence.

Editorial extensions

If this is right

  • For any α∈(α_W,1) and any ε>0 there exists a divergence-free Hölder drift with non-selection on a set of initial data of measure at least 1−ε.
  • The probability laws Law(X^{κ_q}_1(x)) are tight but do not admit a unique limit for every x in that large set.
  • Selection in the sense of Regular Lagrangian Flows fails: no unique limit flow is selected by vanishing noise.
  • The set of drifts with the (1−ε)-non-selection property is dense in L^q([0,1];C^α) for q<∞ and α∈(α_W,1).
  • Vanishing fractional viscosity (order β∈(0,2]) fails to select a unique weak solution to the transport equation.

Reading between the lines

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

  • If the construction is stable under small perturbations, anomalous dissipation in passive scalar turbulence should be accompanied by exactly this kind of non-selection of Lagrangian trajectories; the paper's PDE corollary makes that link explicit for fractional viscosity.
  • The parity-dependent swap mechanism suggests a generic mechanism: any regularization that coarse-grains the drift at a threshold scale can produce different limits depending on how the cutoff is taken; one might test numerically whether the separation distance c_ε scales like the smallest shear scale ℓ_1.
  • A natural testable extension: run the same construction with noise intensities decaying at a different rate (e.g., κ_q ~ a_q^r) and check whether the two limit points y,y′ vary continuously with r; the paper's parameter choices are only one possible tuning.
  • The result suggests that for divergence-free Hölder drifts with α<1, the vanishing-noise limit may be generically multi-valued, in contrast to the Lipschitz or DiPerna–Lions regime where the limit is unique.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proves a non-selection result for Lagrangian trajectories in the zero-noise limit of SDEs with divergence-free Hölder drift on the two-dimensional torus. For any noise process covered by Assumption 1.1 (fractional Brownian motion or β-stable Lévy), any α in the regularizing window (α_W,1), and any ε>0, the authors construct a divergence-free drift u∈C^α, a large set A_ε of initial data, and a vanishing sequence (κ_q) such that, with probability one, the even and odd subsequences X^{κ_{2q}}_1(x) and X^{κ_{2q+1}}_1(x) stay separated by a positive constant for Lebesgue-a.e. x∈A_ε. The proof combines the deterministic alternating-shear construction of the drift with the stochastic sewing lemma to show that the stochastic flow at noise intensity κ_q is quantitatively close to the flow of the smooth approximation u_q. The paper also derives consequences for non-convergence of laws, for vanishing fractional viscosity limits of transport equations, and for an autonomous three-dimensional version.

Significance. If correct, the main theorem is a substantial advance: it shows that vanishing noise does not select a unique trajectory under merely Hölder and divergence-free assumptions on the drift, and it does so not just in law but almost surely simultaneously for a large set of initial data. The result is genuinely Lagrangian and covers a broad class of noises, including non-Markovian fBm and stable Lévy noise. The construction is explicit and the estimates are detailed; the use of the stochastic sewing lemma for a negative non-selection result is a notable methodological contribution. The deterministic mixing/unmixing construction and the Borel–Cantelli part of the proof are internally coherent, and the paper is careful to separate external well-posedness inputs from the new estimates. The central claim is therefore plausible, but one load-bearing stability statement in Section 5 is asserted without proof and is mis-stated; this must be corrected before the result can be considered established.

major comments (2)
  1. [§5, Proposition 5.2; §6, Eq. (6.2)] Proposition 5.2 is not proved, and the sentence “We omit the proof, as it is identical to the previous proposition” is not accurate. As written, the proposition compares X^{κ_q}_{1−t_q,t}(y) with X^q_{1−t_q,t}(x), i.e., it starts the deterministic flow from the same point x∈A at the later time 1−t_q. But the sets A_3 and A_4 in (4.20) control the images X^q_{1−t_p}(x), not the point x itself; a generic x∈A is not an admissible starting point at time 1−t_q. Moreover, the application in Proposition 6.1 uses the comparison with y close to X^q_{1−t_q}(x), not to x: in (6.2) the deterministic flow starts at X^q_{1−t_q}(x). Thus the stated proposition is not the statement used, and the proof cannot be identical to that of Proposition 5.1: a reverse induction over p≤q using A_3 and A_4 is required. Since (6.2) is the final link between the stochastic and deterministic flows and feeds directly i
  2. [§3, Lemma 3.2, Eq. (3.9)] In the β-stable case the displayed estimate (3.9) has a typo: the power of |t−s| is written as 1−θH, but the proof and condition (3.8) show it should be 1−θ/β. This is a local typo, but because the estimate is used quantitatively in Section 6 for the Borel–Cantelli sum, the displayed formula should be corrected.
minor comments (5)
  1. [§6, proof of Proposition 6.1] After the application of Proposition 5.2 the bound is written as (2/3)ℓ_q + 7q^2κ_q, but in the final lim sup line it becomes (2/3)ℓ_q + 6q^2κ_q. The constant should be made consistent.
  2. [§5, Proposition 5.1 proof, Step 1] In Step 1 the text says X_s(x)∈H_0 for s∈J_{0,1}; the notation H_0 is inconsistent with the surrounding indexing, which uses H_p with p≥1. This appears to be a typo for H_1.
  3. [§4.3, definition of A_3] The intersection defining A_3 runs over 0≤p≤q and uses V_p, whereas all earlier definitions of V_p and the estimates for A_3^c use p≥1. Please clarify whether V_0 is intended and, if so, define it explicitly.
  4. [§5, Proposition 5.2] The notation X^{κ_q}_{1−t_q,t}(y) is introduced without definition; the path X^{κ_q}(y) in (5.1) is defined from time 0 only. Please define explicitly the two-parameter stochastic and deterministic flows used in the statement.
  5. [§3, Lemma 3.2] The statement assumes f∈L∞([S,T];C^1_x) but the estimates involve C^{−θ}_x norms. For the applications this is harmless because f is smooth, but the hypotheses should be stated to match the conclusion, e.g., f∈L∞([S,T];C^1_x)∩L∞([S,T];C^{−θ}_x) or simply f∈L∞([S,T];C^1_x) with the right-hand side understood via the C^{−θ}_x norm.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the non-selection theorem is proved by explicit construction and estimates, not by assuming its conclusion. The only flagged issue is a non-circular omitted-proof gap in Proposition 5.2.

full rationale

The derivation of Theorem 1.2 does not reduce to its inputs by construction. The drift u is explicitly built in Section 4 from alternating shear flows and a swap term; the large set A is defined in (4.20) via the deterministic smooth flows X^q, not via the stochastic flow X^{κ_q}; and the crucial transfer step, Proposition 6.1, is proved from pathwise estimates (Proposition 5.1), the stochastic sewing estimates of Lemma 3.2, and Borel-Cantelli, without invoking the desired non-selection. The cited well-posedness results (Theorems 2.1-2.2, based on [33] and [53,15,54]) are external published theorems with explicit thresholds and are not equivalent to the zero-noise non-selection claim; although [33] involves the first author, it is independent support and is not load-bearing circularity. The construction is a refinement of [18] by the third author, but the parity-dependent flows and the stochastic transfer are new, so this is not renaming. The only in-scope flagged issue is non-circular: Section 5 states, before Proposition 5.2, "We omit the proof, as it is identical to the previous proposition." The skeptic is correct that the reflected estimate is not literally identical: Proposition 5.1 compares flows started at time 0 from x∈A, while the application in (6.2) requires comparing a stochastic flow started at time 1−t_q from X^{κ_q}_{1-t_q}(x) with a deterministic flow started from X^q_{1-t_q}(x). As printed, Proposition 5.2 is mis-quantified and the omitted reverse induction is a genuine proof gap affecting Proposition 6.1. This is a correctness risk, not a circular reduction, so it does not raise the circularity score beyond 1.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The theorem is an existence/counterexample result; its parameters (a_0, δ, γ, θ) are chosen by hand to satisfy explicit inequalities, not fitted to external data. No new physical entities are introduced. The strongest external input is the regularization-by-noise well-posedness theory, whose thresholds (α_W) define the regime of the construction.

free parameters (4)
  • a_0
    Chosen sufficiently small (in (4.5)-(4.6)) so that |A|≥1−ε and 100q²κ_q≤ℓ_q; controls the base scale of the chessboard construction.
  • δ
    Super-exponential decay rate in (a_q), with a_q/a_{q+1} ∈ [a_q^{-δ}, a_q^{-δ}+4]∩4N; taken <1/100 and then small enough so u∈C^α for the prescribed α (Lemma 4.1).
  • γ
    Mollification/time parameter in (4.2), chosen in (1/2,1) depending on the noise W (conditions (i)-(iii) in §4.1) so that θ/2−1+γ>0 and the Borel-Cantelli series converges.
  • θ
    Auxiliary exponent in Lemma 3.2, chosen from H (or β) and γ in the proof of Prop. 6.1; it tunes the bound between κ_q and ℓ_q.
assumptions (4)
  • domain assumption SDE well-posedness for fBm: if u∈L^∞([0,1];C^α) with α>1−1/(2H) (α≥0), the SDE (2.3) has a strong pathwise unique solution and stochastic flow of diffeomorphisms
    Invoked as Theorem 2.1, from [33]; required for the existence of X^κ.
  • domain assumption SDE well-posedness for isotropic β-stable Lévy processes: if α>1−β/2, the SDE has a strong pathwise unique solution and flow
    Invoked as Theorem 2.2, from [53,15,54]; also gives the PDE representation (2.10).
  • domain assumption Local nondeterminism property (2.1) for fBm, and Markov property (2.5) for stable processes, are used to derive the oscillatory-integral estimates in Lemma 3.2
    These are structural properties of the noise processes, cited to [33] and standard Lévy theory; they are the mechanism behind the stochastic estimates.
  • standard math Stochastic Sewing Lemma with shifts (Lemma 3.1) holds and yields the Riemann-sum limits used in Lemma 3.2
    Taken from [35, Lem. 2.2] and [33, Lem. 2.5]; used to control integrals of oscillatory functions against the noise.

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Pith. "Pith review of Non-selection of Lagrangian trajectories in the zero-noise limit for a class of stochastic regularizations." pith.science (2026). https://pith.science/paper/AEKAZ3UC

@misc{pith2026260607096,
  author       = {Pith},
  title        = {Pith review of: Non-selection of Lagrangian trajectories in the zero-noise limit for a class of stochastic regularizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AEKAZ3UC}},
  note         = {Machine review of arXiv:2606.07096}
}
abstract

We prove the lack of selection in the zero-noise limit for solutions to SDEs driven by a divergence-free, H\"older continuous vector field with exponent $\alpha\in(0,1)$, arbitrarily close to $1$ but fixed. The result applies to a broad class of regularizing additive noises, including fractional Brownian motion and stable L\'evy processes. The proof combines pathwise Lagrangian arguments, based on the analysis of the deterministic flows associated to mixing velocity fields, with probabilistic estimates coming from the stochastic sewing lemma. This allows to show that lack of selection happens simultaneously on a large set of initial data, whose complement has arbitrarily small Lebesgue measure.

Figures

Figures reproduced from arXiv: 2606.07096 by the authors.

Figure 1
Figure 1. The figure on the left shows the even chessboard set at scale a0. The figure in the center shows its image under the time-1 Lagrangian flow generated by the vector field v1,1. The figure on the right shows the image obtained by then applying the time￾1 Lagrangian flow generated by v1,2 to the previous image. The arrows indicate the velocity field v1,1 in the center figure and v1,2 in the right figure. Recall the def… view at source ↗
Figure 2
Figure 2. On the left, Fe is shown in green. On the right, Fo is shown in green. Recall the definition of C even a0 (resp. C odd a0 ), the even (resp. odd) chessboard of size a0 given in (4.7). Let us define the restricted chessboards (4.24) Fe := C even a0 ∩ V1 ∩ H1, Fo = C odd a0 ∩ V1 ∩ H1, see [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

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