{"id":"1c47d0e9-7737-46cb-90d3-9dd64b8a7aa7","arxiv_id":"2606.07096","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For divergence-free Hölder drifts on T², the vanishing-noise limit of SDEs with fBm or stable-Lévy perturbations does not select a unique Lagrangian trajectory: even and odd subsequences of the stochastic flow separate on a large set of initial data.","lead":"This paper proves that adding a small regularizing noise (fractional Brownian motion or stable Lévy process) to a rough, divergence-free velocity field does not select a unique trajectory as the noise vanishes: even and odd subsequences of noise intensities converge to different endpoints. It shows that in two dimensions, vanishing-noise selection fails for every noise in this class, for Hölder exponents arbitrarily close to 1.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.2 is asserted without proof and is mis-quantified: it claims a stability bound for flows starting at time 1−t_q from arbitrary x∈A, but the deterministic comparison point must be X^q_{1−t_q}(x), not x; the omitted time-reversed argument is not identical to Proposition 5.1.","rationale":"I read the central construction and the main estimates (Lemmas 3.2, 4.1, 4.2, Propositions 5.1 and 6.1). The algebraic choices of γ, θ and δ are consistent, and the stochastic-sewing estimates are plausible; the displayed typos (e.g., |t−s|^{1−θH} in (3.9) and the u_q/u notation in Proposition 6.1) do not appear to invalidate the argument. The reader’s weakest assumption concerned the external well-posedness thresholds; that is a legitimate caution but not an internal gap. The more concrete load-bearing concern is the omitted proof of Proposition 5.2. It is used essentially to control the final time interval [1−t_q,1], and its statement as written is not a literal time-reversal of Proposition 5.1 because the good-set hypotheses A_3, A_4 apply to initial data at time 0, whereas the two-parameter flow in the conclusion requires a starting point at time 1−t_q. The intended corrected statement is likely true, but it is neither stated nor proved. I therefore keep the reader’s CONDITIONAL verdict unchanged, with the added condition that the proof of Proposition 5.2 be supplied or at least precisely formulated.","tokens_in":40067,"tokens_out":38971,"duration_ms":343141,"concrete_test":"Write out the proof of Proposition 5.2 in the corrected form: set B_q := X^q_{1−t_q}(A) and prove by reverse induction on p=q,q−1,...,1 that for every z∈B_q and every y with |y−z|≤(2/3)ℓ_q+4qκ_q∥ξ∥∞, the two flows stay within ℓ_q of each other at all times when the reflected shear of scale p is active. The induction step should use exactly the inclusions A_3⊂(X^q_{1−t_p})^{-1}(V_p) and A_4⊂(X^q_{1−t_p+2(t_{p+1}−t_p)/3})^{-1}(H_p). If the induction goes through, add the corrected statement to the paper; if it fails at some p, Theorem 1.2 collapses because (6.1) is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 6.1, which transfers the deterministic non-selection of X^q_1 to the stochastic flows X^{κ_q}_1, relies on Proposition 5.2 exactly at the last step: estimate (6.2) for t≥1−t_q. The paper gives no proof, saying only “We omit the proof, as it is identical to the previous proposition.” But the two propositions are not identical. Proposition 5.1 compares two flows both started at time 0 from an initial point x∈A. Proposition 5.2 must compare the stochastic flow started at time 1−t_q from y≈X^q_{1−t_q}(x) with the deterministic flow started at time 1−t_q from X^q_{1−t_q}(x). As written, however, the proposition says “for all x∈A” and then uses X^q_{1−t_q,t}(x), which would start the deterministic flow from the initial point x at time 1−t_q. That statement is false in general: a generic x∈A is not a good starting point at the later time, since the good sets A_3, A_4 were defined for trajectories emanating from x at time 0, not for arbitrary points at time 1−t_q. The intended statement (with x replaced by X^q_{1−t_q}(x)) is plausible and is what the application actually uses, but it requires a separate reverse induction over p≤q using A_3 and A_4, together with a measure estimate for the class of admissible starting points at time 1−t_q. Until that argument is supplied, the final comparison (6.1), and hence Theorem 1.2, has an unverified step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":40364,"tokens_out":13207,"duration_ms":121280,"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":[{"comment":"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","section":"§5, Proposition 5.2; §6, Eq. (6.2)"},{"comment":"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.","section":"§3, Lemma 3.2, Eq. (3.9)"}],"minor_comments":[{"comment":"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.","section":"§6, proof of Proposition 6.1"},{"comment":"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.","section":"§5, Proposition 5.1 proof, Step 1"},{"comment":"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.","section":"§4.3, definition of A_3"},{"comment":"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.","section":"§5, Proposition 5.2"},{"comment":"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.","section":"§3, Lemma 3.2"}],"recommendation":"major_revision","confidential_remarks":"The main concern is localized: Proposition 5.2 is both mis-stated and unproved, and it is exactly the step that transfers the deterministic non-selection to the stochastic flows. I do not think the construction is broken — the corrected statement with the deterministic flow started at X^q_{1−t_q}(x) is consistent with the definition of A_3/A_4 and with the way Proposition 6.1 uses it — but the omitted argument is not identical to Proposition 5.1 and should be supplied. I would ask for a complete proof of the corrected Proposition 5.2 before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper deserves a serious referee. It establishes a genuinely new negative result: for a divergence-free, Hölder continuous drift with exponent α arbitrarily close to 1, vanishing additive noise from a broad class of processes (fractional Brownian motion and stable Lévy) does not select a unique Lagrangian trajectory. The non-selection is simultaneous on a set of initial data of measure at least 1−ε, and the stochastic flows fail to be Cauchy there. That answers a question that had been open in d≥2, and it connects to spontaneous stochasticity, so the significance is real.\n\nThe paper does several things well. The construction is careful, with explicit parameters and a clean alternating-shear mechanism. The use of stochastic sewing estimates to compare stochastic and deterministic flows is clever and broad enough to cover non-Markovian and jump noises. The density corollary and the autonomous 3D version are welcome additions. I checked the main parameter choices and the Borel–Cantelli estimates in Section 6; they look internally consistent.\n\nThe soft spots are two. First, Lemma 3.2(3.9) contains a typo: the exponent should be 1−θ/β, not 1−θH, in the stable-Lévy case. That is cosmetic. Second, and more seriously, Proposition 5.2 is asserted without proof, with the remark that it is identical to Proposition 5.1. It is not. The time-reversed comparison must start the deterministic flow from X^q_{1−t_q}(x), not from an arbitrary x∈A; the sets A_3 and A_4 were defined for trajectories from x at time 0, so a separate argument is needed. As written, the proposition is mis-quantified, and the last step of Proposition 6.1—and hence Theorem 1.2—depends on it. The intended argument is plausible, but it is not in the manuscript. There is also a garbled line in the proof of Proposition 5.1, but that is minor.\n\nOverall: the central idea is sound and the result is significant if that gap can be closed. I would send it to a good referee and ask for a complete proof of Proposition 5.2 before acceptance. For my own work, I would cite the construction cautiously, and I'd bring the paper to the reading group.","headline":"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.","tokens_in":41026,"tokens_out":3566,"would_cite":true,"duration_ms":34253,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","76B03","60H50","60G22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding small noise to a rough incompressible flow does not select a unique trajectory in the zero-noise limit","keywords":["zero-noise limit","stochastic regularization","fractional Brownian motion","stable Lévy process","Lagrangian trajectories","non-selection","mixing by shear flows","stochastic sewing lemma"],"falsifier":"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.","tokens_in":39805,"feed_emoji":"🔀","tokens_out":4477,"duration_ms":43679,"temperature":0.7,"pith_summary":"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.","feed_headline":"Vanishing noise fails to select a unique path","feed_subtitle":"For Hölder divergence-free drifts, fBm or stable Lévy noise leaves two possible limits even as intensity goes to zero.","key_machinery":"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","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Zero-noise limit still leaves two possible paths","Even vanishing Lévy noise can't pick a unique path","Noise vanishes, ambiguity persists in trajectory limits","Hölder drift and vanishing noise: two limits, no selection","Zero-noise limit leaves multiple trajectories for Hölder drifts"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zero-noise limit still leaves two possible paths","Even vanishing Lévy noise can't pick a unique path","Noise vanishes, ambiguity persists in trajectory limits","Hölder drift and vanishing noise: two limits, no selection","Zero-noise limit leaves multiple trajectories for Hölder drifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000397,"raw_usage":{"total_tokens":1897,"prompt_tokens":704,"completion_tokens":1193,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":1115}},"tokens_in":448,"tokens_out":1193,"duration_ms":8813,"temperature":1.0,"reasoning_tokens":1115,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T12:10:53.021894+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}