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Effective Lagrangian regularity and the uniqueness threshold for random H\"older velocity fields

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Pith's one-line read For random Hölder velocity fields assembled from independent scales, bare regularity above 1/2 almost surely restores uniqueness of trajectories away from the zero set, while examples below the threshold show robust ill-posedness.

desk verdict A strong, honest paper proving a sharp well-posedness threshold at alpha=1/2 for random Holder velocity fields; deserves serious peer review. read the letter →

arxiv 2608.05931 v1 pith:SNZZM7Z6 submitted 2026-08-06 math.PR math.APmath.CA

Maria Colombo , Elias Hess-Childs , Keefer Rowan This is my paper
classification math.PRmath.APmath.CA MSC 60H1060G6035F25
keywords Lagrangianwell-posednessHöldervelocityfieldscriticalthresholdalpha=1/2stochasticcancellationseffectiveregularitycontinuityequationuniquenessmultiscalefiniterangedecompositionsweepingeffect
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 studies which random, spatially Hölder-rough velocity fields give well-posed ODE flows and transport equations. It claims that for autonomous fields built from independent finite-range scales, bare regularity $\alpha > 1/2$ is enough to restore almost-sure uniqueness of trajectories away from the zero level set, because large scales sweep particles over small scales and generate stochastic cancellations. Below $\alpha = 1/2$, the paper constructs divergence-free examples with nonunique solutions on a positive measure set of initial data and nonunique bounded solutions to the continuity equation, so the threshold is sharp. It also proves enhanced separation estimates below the threshold and analogous results for time-decorrelated fields with threshold $\alpha + \beta/2 > 1$. A fair reader would take away that typical random rough fields behave qualitatively better than deterministic Peano-type counterexamples, at least away from stagnation points.

What carries the argument

The load-bearing mechanism is the sweeping effect: large-scale velocity components advect particles over small-scale fluctuations, so integrating a small-scale field along a trajectory produces central-limit-type cancellations. To make this rigorous, the authors reparameterize the ODE in the coordinate $r$ along $\eta = U(y)/|U(y)|$, transforming the autonomous problem into a finite-range-in-time problem for the transverse variable, with a regularized field $V^{\delta,y}$ whose denominator $\phi_\delta(\eta \cdot U)$ is bounded below by $\delta$. The estimates are carried by an abstract averaging lemma splitting averaged scale fields into a $C^{1/2}$-in-time martingale part and a finite-variation remainder, sharp martingale central-limit bounds, and nonlinear Young integral stability propositions that convert time-space regularity of the averaged field into ODE stability.

What would settle it

Take a Gaussian power-law field on $\mathbb{T}^2$ with exponent $\alpha = 0.6$ (above threshold), sample many realizations, and integrate pairs of particles starting at distance $2^{-j}$ away from the zero set. If for a positive fraction of realizations the trajectories separate to order 1 by time 1, the claimed almost-sure bi-Lipschitz estimate of Theorem 1.2 would fail; conversely, verifying separation that stays bounded by $C 2^{-j}$ uniformly in $j$ would support the predicted threshold.

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Extended reading notes

Core claim

The central claim is that $\alpha = 1/2$ is a genuine transition for Lagrangian uniqueness in multiscale finite-range random velocity fields. For $\alpha \in (1/2,1)$ and $p > d/(\alpha-1/2)$, Theorem 1.2 asserts that almost surely the ODE admits a unique Lipschitz flow away from the random zero level set, with a uniform bi-Lipschitz stability estimate; Corollary 1.4 upgrades this to almost-everywhere uniqueness for the ODE and uniqueness for the continuity equation under a divergence-free and small-zero-set hypothesis. Below the threshold, Theorem 1.6 provides $C^{\alpha-}$ multiscale divergence-free fields with empty zero set for which nonuniqueness occurs for a positive measure set of initial data and for the continuity equation. The source of well-posedness is an effective Lagrangian regularity larger than the bare regularity, produced by stochastic cancellations when large scales sweep trajectories across small scales.

Load-bearing premise

The argument requires trajectories to stay in a region where the denominator $\eta \cdot U$ is bounded away from zero, so the large-scale field actually sweeps particles across small scales; if the zero set traps particles or has positive measure, no stochastic cancellation is obtained and no uniqueness is claimed.

Editorial extensions

If this is right

  • For any $\alpha \in (1/2,1)$, almost surely every pair of integral curves staying away from the thickened zero set satisfies a bi-Lipschitz estimate uniform in the curves, so initial closeness persists for all time up to the zero set.
  • If in addition the field is divergence-free and the zero set is small in the sense of (1.6), then almost surely the ODE has a unique solution for almost every initial point and the continuity equation has a unique bounded solution for every bounded initial datum.
  • Below $\alpha=1/2$, the paper's examples show instantaneous nonuniqueness for a positive measure set of initial data and nonunique bounded solutions to the continuity equation, even with divergence-free fields and empty zero set.
  • Even below the threshold, trajectory separation is bounded by $t^{(1-\alpha)/(1-2\alpha)-\varepsilon}$, which improves on the deterministic Bihari--LaSalle rate $t^{1/(1-\alpha)}$.
  • In the time-decorrelated refreshing regime, the same tools give almost sure ODE well-posedness and Lipschitz flow maps when $\alpha + \beta/2 > 1$, with threshold examples for nonuniqueness below it.

Reading between the lines

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

  • The zero-set restriction likely cannot be removed: the paper's own sweeping mechanism fails at stagnation points, and it conjectures ill-posedness there for all $\alpha \in (0,1)$; if true, uniqueness above the threshold is essential rather than global.
  • The paper's heuristic points to an effective regularity of $2\alpha$ below threshold; if matched by a lower bound, the two-point separation exponent $1/(1-2\alpha)$ would mark a sharp change in dispersion at $\alpha=1/2$ for the natural Gaussian power-law fields.
  • A numerically testable extension: sample the Gaussian power-law fields of Section 6 and integrate pairs of particles starting near but off the zero set; above $\alpha=1/2$ the prediction is separation bounded linearly in the initial distance, while below it separation should accelerate like a power law.
  • Since the sweeping regime behaves like the refreshing regime with $\beta=1$, the threshold $\alpha+\beta/2>1$ suggests that any explicit temporal decorrelation with effective rate near $1$ should reproduce the same well-posedness phenomenon, a route the paper's transformed coordinates nearly make explicit.
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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

0 major / 4 minor

Summary. The paper studies the ODE, flow maps, and continuity equation associated to autonomous random velocity fields admitting a multiscale finite range decomposition, together with a time-refreshing analog. The main theorem (Theorem 1.2) states that for spatial regularity exponent α > 1/2, almost surely the ODE admits a unique Lipschitz flow away from the random zero level set of the velocity field, with a uniform bi-Lipschitz stability estimate. Corollary 1.4 lifts this to a.e. uniqueness and uniqueness for the continuity equation under additional divergence-free and small-zero-set hypotheses. Below α < 1/2, Theorem 1.6 constructs divergence-free examples with empty zero level set that exhibit robust ODE and transport nonuniqueness, while Theorem 1.7 gives quantitative quasi-stability estimates below the threshold. The refreshing regime has an analogous threshold α + β/2 = 1, with well-posedness above and ill-posedness examples below. The proofs combine a sharp abstract averaging lemma, martingale central limit theorem bounds, nonlinear Young integration theory, a coordinate transform converting spatial sweeping into a refreshing-type problem, and a CLT-based construction for the ill-posedness examples.

Significance. If the proof is correct, this is a substantial advance: it identifies a sharp, dimension-independent uniqueness threshold α = 1/2 for a natural class of random Hölder velocity fields, and it supports the threshold with both positive and negative results. The paper is also methodologically valuable, introducing an effective Lagrangian regularity notion and showing how stochastic cancellations intrinsic to the field can be converted into ODE well-posedness. Strengths include the clear heuristic derivation of the threshold, the carefully structured proof with an explicit technical overview, the abstract averaging lemma that cleanly isolates the stochastic cancellation mechanism, and the concrete Gaussian and Poisson examples satisfying the hypotheses. The authors are explicit about the limitations: well-posedness is proved away from the zero level set, the effective regularity exponents below threshold are non-sharp, and the sweeping ill-posedness constructions require d ≥ 4.

minor comments (4)
  1. [Theorem 1.2] The quantifier structure in the statement could be made more precise: the phrase "almost surely" appears before "for all ε > 0", but the subsequent existence of the random constant M_ε is naturally read as holding for each fixed ε on a full-probability set that may depend on ε. A short clarification, for example by saying that for each ε > 0 there is a full-probability set on which M_ε is finite, and that the uniqueness statement follows by countable compact exhaustion away from Z, would remove ambiguity.
  2. [Section 10.1.3] When the conditional probability measure after conditioning on (u_k)_{k≠j} and u_j(y) is introduced, the notation L^p_{ω} is used before the measure itself is fully fixed. Please define the conditional measure P and the associated norm explicitly at the first use in Section 10.1.3.
  3. [Appendix A] Proposition 7.5 is a central tool and its proof is deferred to Appendix A. The final version should make the appendix self-contained, state precisely which result in [GG22, Theorem 4.8] is being adapted, and explicitly check that its hypotheses hold for the non-smooth velocity fields and solution curves in Section 10.
  4. [Sections 9.2 and 10.2] The finite-range truncation arguments in Proposition 9.4 and Proposition 10.1 are nearly identical in structure. A brief cross-reference or a single abstract truncation lemma stated once and specialized to the time and space settings would reduce repetition and make the paper easier to read.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the well-posedness threshold is proved from the multiscale hypotheses via an independent averaging argument, and the cited prior works are not load-bearing self-citations.

full rationale

The paper's central claim, Theorem 1.2, is not derived from its own conclusion. The alpha=1/2 threshold is first introduced as an explicit heuristic in Section 2.1 using a CLT-type scaling, and the rigorous proof in Sections 7-10 does not assume that threshold; it establishes the averaged-field estimates from the stated finite-range independence, centering, and moment assumptions, using external tools such as nonlinear Young integration from [GG22] and sharp martingale CLT bounds from [dlPnG99, Lat97]. These external results are independent of the present paper's conclusions. The transformed velocity field V^{delta,y} in (10.2) and the decomposition (10.26) are constructed inside the proof, not imposed as an ansatz from prior self-cited work. The approximation steps (Propositions 9.4 and 10.1) are proved in the paper. The only self-citations, [HCR26] and [HCR25a], appear as contextual comparisons or background examples, not as load-bearing justifications of the uniqueness theorems; Section 12 gives a self-contained construction. No fitted parameters are renamed as predictions, no known result is merely relabeled, and the zero-level-set limitation is explicitly acknowledged rather than hidden. Therefore the derivation chain is self-contained and no circular step can be exhibited from the paper's own equations or citation structure.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted constants and no new physical entities. The central claim rests on the finite-range multiscale random model, the regularity/moment assumptions, and several standard external results from martingale theory, nonlinear Young integration, and geometric measure theory. The zero-set condition is an explicit additional hypothesis for the PDE corollary.

assumptions (7)
  • domain assumption Assumption 1.1: U is an infinite sum of independent centered finite-range random fields, each with range of dependence 2^-j (u_0 unrestricted).
    This defines the model of random autonomous fields; the entire sweeping effect and all main theorems depend on it.
  • domain assumption Scale-by-scale regularity bound (1.3): max_{n=0,1,2} sup_j 2^{(alpha-n)j} ||∇^n u_j||_{L^p_omega C^0_x} < infinity, with p > d/(alpha-1/2) in Theorem 1.2.
    Gives U in C^{alpha-} and supplies moment control; used in Sections 9 and 10 for stochastic averaging and approximation arguments.
  • domain assumption Assumption 4.1: temporal finite-range dependence in the refreshing case, with decorrelation time 2^-beta j.
    Defines the refreshing regime and determines the threshold alpha + beta/2 = 1.
  • standard math Nonlinear Young integral results, Proposition 7.5 and Proposition 7.6, based on [GG22, Theorem 4.8].
    Used as a black box (with proof in Appendix A) to convert effective regularity of averaged fields into ODE stability estimates.
  • standard math Sharp martingale CLT bounds, Theorem 8.1 and Theorem 8.2 from [dlPnG99, Lat97], summarized in Corollary 8.3.
    Provides the optimal L^p moment scaling in the abstract averaging Lemma 8.4; not proved in this paper.
  • standard math Aizenman avoidance argument and Ambrosio superposition principle, used in Section 11.
    Needed to pass from a.e. ODE uniqueness to uniqueness for the continuity equation and renormalized solutions.
  • domain assumption In Corollary 1.4: ∇·U = 0 and the zero-level set satisfies lim_{epsilon→0} epsilon^{-(1-alpha+delta)} |Z_epsilon| = 0.
    These are additional structural hypotheses, not consequences of Assumption 1.1.

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Pith. "Pith review of Effective Lagrangian regularity and the uniqueness threshold for random H\"older velocity fields." pith.science (2026). https://pith.science/paper/SNZZM7Z6

@misc{pith2026260805931,
  author       = {Pith},
  title        = {Pith review of: Effective Lagrangian regularity and the uniqueness threshold for random H\"older velocity fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SNZZM7Z6}},
  note         = {Machine review of arXiv:2608.05931}
}
abstract

We study the behavior of the ordinary differential equations, flow maps, and continuity equations associated to autonomous random velocity fields that admit a natural multiscale finite range decomposition. The velocity fields we consider are only H\"older regular in space---$C^{\alpha-}(\mathbb{T}^d)$ for some $\alpha \in (0,1)$---thus the associated ODE and continuity equation are not a priori well-posed. However, above the critical threshold of $\alpha = 1/2$, due to multiscale stochastic cancellations, we prove well-posedness is almost surely restored away from the zero level set of the velocity field. This threshold marks a genuine transition, as demonstrated by examples lying below the threshold that exhibit robust ill-posedness. We additionally provide effective regularity estimates below the critical threshold and prove analogous results in the related "refreshing" regime.

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Works this paper leans on

118 extracted references · 41 canonical work pages

  1. [1]

    Structure of level sets and S ard-type properties of L ipschitz maps

    Giovanni Alberti, Stefano Bianchini, and Gianluca Crippa. Structure of level sets and S ard-type properties of L ipschitz maps. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 12(4):863--902, 2013

    2013

  2. [2]

    A uniqueness result for the continuity equation in two dimensions

    Giovanni Alberti, Stefano Bianchini, and Gianluca Crippa. A uniqueness result for the continuity equation in two dimensions. J. Eur. Math. Soc. (JEMS) , 16(2):201--234, 2014

    2014

  3. [3]

    Strong existence and uniqueness for stable stochastic differential equations with distributional drift

    Siva Athreya, Oleg Butkovsky, and Leonid Mytnik. Strong existence and uniqueness for stable stochastic differential equations with distributional drift. Ann. Probab. , 48(1):178--210, 2020

    2020

  4. [4]

    Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift, September 2024

    Scott Armstrong, Ahmed Bou-Rabee, and Tuomo Kuusi. Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift, September 2024. arXiv:2404.01115 [math]

    arXiv 2024

  5. [5]

    Superdiffusion and anomalous regularization in self-similar random incompressible flows, January 2026

    Scott Armstrong, Ahmed Bou-Rabee, and Tuomo Kuusi. Superdiffusion and anomalous regularization in self-similar random incompressible flows, January 2026. arXiv:2601.22142 [math]

    arXiv 2026

  6. [6]

    Continuity equations and ODE flows with non-smooth velocity

    Luigi Ambrosio and Gianluca Crippa. Continuity equations and ODE flows with non-smooth velocity. Proc. Roy. Soc. Edinburgh Sect. A , 144(6):1191--1244, 2014

    2014

  7. [7]

    A sufficient condition for the avoidance of sets by measure preserving flows in R n

    Michael Aizenman. A sufficient condition for the avoidance of sets by measure preserving flows in R n . Duke Math. J. , 45(4):809--813, 1978

    1978

  8. [8]

    Renormalization group and elliptic homogenization in high contrast

    Scott Armstrong and Tuomo Kuusi. Renormalization group and elliptic homogenization in high contrast. Invent. Math. , 242(3):895--1086, 2025

    2025

Show all 118 references
  1. [9]

    Quantitative stochastic homogenization and large-scale regularity , volume 352 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    Scott Armstrong, Tuomo Kuusi, and Jean-Christophe Mourrat. Quantitative stochastic homogenization and large-scale regularity , volume 352 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, Cham, 2019

    2019

  2. [10]

    Path-by-path uniqueness for stochastic differential equations under Krylov - Röckner condition, July 2025

    Lukas Anzeletti, Khoa Lê, and Chengcheng Ling. Path-by-path uniqueness for stochastic differential equations under Krylov - Röckner condition, July 2025. arXiv:2304.06802 [math]

    2025 arXiv

  3. [11]

    Transport equation and C auchy problem for BV vector fields

    Luigi Ambrosio. Transport equation and C auchy problem for BV vector fields. Invent. Math. , 158(2):227--260, 2004

    2004

  4. [12]

    Transport equation and C auchy problem for non-smooth vector fields

    Luigi Ambrosio. Transport equation and C auchy problem for non-smooth vector fields. In Calculus of variations and nonlinear partial differential equations , volume 1927 of Lecture Notes in Math. , pages 1--41. Springer, Berlin, 2008

    1927

  5. [13]

    Anomalous diffusion by fractal homogenization

    Scott Armstrong and Vlad Vicol. Anomalous diffusion by fractal homogenization. Ann. PDE , 11(1):Paper No. 2, 145, 2025

    2025

  6. [14]

    A simple method for finite range decomposition of quadratic forms and G aussian fields

    Roland Bauerschmidt. A simple method for finite range decomposition of quadratic forms and G aussian fields. Probab. Theory Related Fields , 157(3-4):817--845, 2013

    2013

  7. [15]

    Boutros, Camillo De Lellis, and Svitlana Mayboroda

    Marco Bagnara, Daniel W. Boutros, Camillo De Lellis, and Svitlana Mayboroda. Regularity thresholds for anomalous dissipation and related phenomena in passive scalars, April 2026. arXiv:2603.11466 [math.AP]

    2026 arXiv

  8. [16]

    Lagrangian chaos and scalar advection in stochastic fluid mechanics

    Jacob Bedrossian, Alex Blumenthal, and Sam Punshon-Smith. Lagrangian chaos and scalar advection in stochastic fluid mechanics. J. Eur. Math. Soc. (JEMS) , 24(6):1893--1990, 2022

    1990

  9. [17]

    Almost-sure exponential mixing of passive scalars by the stochastic N avier- S tokes equations

    Jacob Bedrossian, Alex Blumenthal, and Samuel Punshon-Smith. Almost-sure exponential mixing of passive scalars by the stochastic N avier- S tokes equations. Ann. Probab. , 50(1):241--303, 2022

    2022

  10. [18]

    Brydges, and Gordon Slade

    Roland Bauerschmidt, David C. Brydges, and Gordon Slade. Introduction to a renormalisation group method , volume 2242 of Lecture Notes in Mathematics . Springer, Singapore, 2019

    2019

  11. [19]

    Positive solutions of transport equations and classical nonuniqueness of characteristic curves

    Elia Bru\'e, Maria Colombo, and Camillo De Lellis. Positive solutions of transport equations and classical nonuniqueness of characteristic curves. Arch. Ration. Mech. Anal. , 240(2):1055--1090, 2021

    2021

  12. [20]

    Sharp nonuniqueness in the transport equation with sobolev velocity field

    Elia Bru \`e , Maria Colombo, and Anuj Kumar. Sharp nonuniqueness in the transport equation with sobolev velocity field. Journal of the European Mathematical Society , 2026. Published online

    2026

  13. [21]

    Alex Blumenthal, Michele Coti Zelati, and Rishabh S. Gvalani. Exponential mixing for random dynamical systems and an example of P ierrehumbert. Ann. Probab. , 51(4):1559--1601, 2023

    2023

  14. [22]

    Bouchut and L

    F. Bouchut and L. Desvillettes. On two-dimensional H amiltonian transport equations with continuous coefficients. Differential Integral Equations , 14(8):1015--1024, 2001

    2001

  15. [23]

    Stochastic ODE s and stochastic linear PDE s with critical drift: regularity, duality and uniqueness

    Lisa Beck, Franco Flandoli, Massimiliano Gubinelli, and Mario Maurelli. Stochastic ODE s and stochastic linear PDE s with critical drift: regularity, duality and uniqueness. Electron. J. Probab. , 24:Paper No. 136, 72, 2019

    2019

  16. [24]

    Negative regularity mixing for random volume preserving diffeomorphisms, October 2024

    Jacob Bedrossian, Patrick Flynn, and Sam Punshon-Smith. Negative regularity mixing for random volume preserving diffeomorphisms, October 2024. arXiv:2410.19251 [math]

    2024 arXiv

  17. [25]

    Slow modes in passive advection

    Denis Bernard, Krzysztof Gawedzki, and Antti Kupiainen. Slow modes in passive advection. J. Statist. Phys. , 90(3-4):519--569, 1998

    1998

  18. [26]

    Brydges, G

    David C. Brydges, G. Guadagni, and P. K. Mitter. Finite range decomposition of G aussian processes. J. Statist. Phys. , 115(1-2):415--449, 2004

    2004

  19. [27]

    Stochastic equations with singular drift driven by fractional B rownian motion

    Oleg Butkovsky, Khoa L\^e, and Leonid Mytnik. Stochastic equations with singular drift driven by fractional B rownian motion. Probab. Math. Phys. , 6(3):857--912, 2025

    2025

  20. [28]

    Weak uniqueness for singular stochastic equations, March 2025

    Oleg Butkovsky and Leonid Mytnik. Weak uniqueness for singular stochastic equations, March 2025. arXiv:2405.13780 [math]

    2025 arXiv

  21. [29]

    On the probabilistic C auchy theory of the cubic nonlinear S chr\"odinger equation on R ^d , d 3

    \'Arp\'ad B\'enyi, Tadahiro Oh, and Oana Pocovnicu. On the probabilistic C auchy theory of the cubic nonlinear S chr\"odinger equation on R ^d , d 3 . Trans. Amer. Math. Soc. Ser. B , 2:1--50, 2015

    2015

  22. [30]

    Anomalous dissipation and Euler flows, October 2023

    Jan Burczak, László Székelyhidi, and Bian Wu. Anomalous dissipation and Euler flows, October 2023. arXiv:2310.02934 [math]

    2023 arXiv

  23. [31]

    Scalar anomalous dissipation and optimal regularity via iterated homogenization, April 2026

    Jan Burczak, László Székelyhidi, and Bian Wu. Scalar anomalous dissipation and optimal regularity via iterated homogenization, April 2026. arXiv:2604.13912 [math]

    2026 arXiv

  24. [32]

    Random data C auchy theory for supercritical wave equations

    Nicolas Burq and Nikolay Tzvetkov. Random data C auchy theory for supercritical wave equations. I . L ocal theory. Invent. Math. , 173(3):449--475, 2008

    2008

  25. [33]

    Lyapunov exponents for random perturbations of some area-preserving maps including the standard map

    Alex Blumenthal, Jinxin Xue, and Lai-Sang Young. Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. Ann. of Math. (2) , 185(1):285--310, 2017

    2017

  26. [34]

    Anomalous dissipation and lack of selection in the O bukhov- C orrsin theory of scalar turbulence

    Maria Colombo, Gianluca Crippa, and Massimo Sorella. Anomalous dissipation and lack of selection in the O bukhov- C orrsin theory of scalar turbulence. Ann. PDE , 9(2):Paper No. 21, 48, 2023

    2023

  27. [35]

    Ill-posedness for nonlinear Schrodinger and wave equations, November 2003

    Michael Christ, James Colliander, and Terence Tao. Ill-posedness for nonlinear Schrodinger and wave equations, November 2003. arXiv:math/0311048

    2003 arXiv

  28. [36]

    Catellier and M

    R. Catellier and M. Gubinelli. Averaging along irregular curves and regularisation of ODE s. Stochastic Process. Appl. , 126(8):2323--2366, 2016

    2016

  29. [37]

    Lagrangian dispersion in G aussian self-similar velocity ensembles

    Marta Chaves, Krzysztof Gawedzki, Peter Horvai, Antti Kupiainen, and Massimo Vergassola. Lagrangian dispersion in G aussian self-similar velocity ensembles. J. Statist. Phys. , 113(5-6):643--692, 2003. Progress in statistical hydrodynamics (Santa Fe, NM, 2002)

    2003

  30. [38]

    Sharpness of the Osgood Criterion for the Continuity Equation with Divergence -free Vector Fields , January 2026

    Roberto Colombo and Anuj Kumar. Sharpness of the Osgood Criterion for the Continuity Equation with Divergence -free Vector Fields , January 2026. arXiv:2601.12096 [math.AP]

    2026

  31. [39]

    The G aussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off

    Georgiana Chatzigeorgiou, Peter Morfe, Felix Otto, and Lihan Wang. The G aussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off. Ann. Probab. , 53(4):1510--1536, 2025

    2025

  32. [40]

    Exponential scalar mixing for the 2 D N avier- S tokes equations with degenerate stochastic forcing

    William Cooperman and Keefer Rowan. Exponential scalar mixing for the 2 D N avier- S tokes equations with degenerate stochastic forcing. Invent. Math. , 243(3):863--959, 2026

    2026

  33. [41]

    A. M. Davie. Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. , 2007

    2007

  34. [42]

    Equality of critical parameters for percolation of G aussian free field level sets

    Hugo Duminil-Copin, Subhajit Goswami, Pierre-François Rodriguez, and Franco Severo. Equality of critical parameters for percolation of G aussian free field level sets. Duke Math. J. , 172(5):839--913, 2023

    2023

  35. [43]

    Drivas and Gregory L

    Theodore D. Drivas and Gregory L. Eyink. A L agrangian fluctuation-dissipation relation for scalar turbulence. P art I . F lows with no bounding walls. J. Fluid Mech. , 829:153--189, 2017

    2017

  36. [44]

    Drivas, Tarek M

    Theodore D. Drivas, Tarek M. Elgindi, Gautam Iyer, and In-Jee Jeong. Anomalous dissipation in passive scalar transport. Arch. Ration. Mech. Anal. , 243(3):1151--1180, 2022

    2022

  37. [45]

    Drivas, Lucio Galeati, and Umberto Pappalettera

    Theodore D. Drivas, Lucio Galeati, and Umberto Pappalettera. Anomalous dissipation and regularization in isotropic Gaussian turbulence, September 2025. arXiv:2509.10211 [math]

    2025 arXiv

  38. [46]

    Sample path properties of the stochastic flows

    Dmitry Dolgopyat, Vadim Kaloshin, and Leonid Koralov. Sample path properties of the stochastic flows. Ann. Probab. , 32(1A):1--27, 2004

    2004

  39. [47]

    R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and S obolev spaces. Invent. Math. , 98(3):511--547, 1989

    1989

  40. [48]

    de la Pe\ na and Evarist Gin\'e

    V\'ictor H. de la Pe\ na and Evarist Gin\'e. Decoupling . Probability and its Applications (New York). Springer-Verlag, New York, 1999. From dependence to independence, Randomly stopped processes. U -statistics and processes. Martingales and beyond

    1999

  41. [49]

    Elgindi and Kyle Liss

    Tarek M. Elgindi and Kyle Liss. Norm growth, non-uniqueness, and anomalous dissipation in passive scalars. Arch. Ration. Mech. Anal. , 248(6):Paper No. 120, 28, 2024

    2024

  42. [50]

    Curvilinear integrals along enriched paths

    Denis Feyel and Arnaud de La Pradelle. Curvilinear integrals along enriched paths. Electron. J. Probab. , 11:no. 34, 860--892, 2006

    2006

  43. [51]

    Delayed blow-up by transport noise

    Franco Flandoli, Lucio Galeati, and Dejun Luo. Delayed blow-up by transport noise. Comm. Partial Differential Equations , 46(9):1757--1788, 2021

    2021

  44. [52]

    Flandoli, M

    F. Flandoli, M. Gubinelli, and E. Priola. Well-posedness of the transport equation by stochastic perturbation. Invent. Math. , 180(1):1--53, 2010

    2010

  45. [53]

    Particles and fields in fluid turbulence

    Gregory Falkovich, Krzysztof Gawedzki, and Massimo Vergassola. Particles and fields in fluid turbulence. Reviews of Modern Physics , 73:913--975, November 2001. Publisher: American Physical Society

    2001

  46. [54]

    Friz and Martin Hairer

    Peter K. Friz and Martin Hairer. A course on rough paths . Universitext. Springer, Cham, second edition, 2020. With an introduction to regularity structures

    2020

  47. [55]

    Multidimensional stochastic differential equations with distributional drift

    Franco Flandoli, Elena Issoglio, and Francesco Russo. Multidimensional stochastic differential equations with distributional drift. Trans. Amer. Math. Soc. , 369(3):1665--1688, 2017

    2017

  48. [56]

    Turbulence

    Uriel Frisch. Turbulence . Cambridge University Press, Cambridge, 1995. The legacy of A. N. Kolmogorov

    1995

  49. [57]

    The 2D Euler equations are well-posed for generic initial data in L^2 , April 2026

    Lucio Galeati. The 2D Euler equations are well-posed for generic initial data in L^2 , April 2026. arXiv:2604.14100 [math.AP]

    2026 arXiv

  50. [58]

    Soluble models of turbulent transport

    Krzysztof Gawedzki. Soluble models of turbulent transport. In Non-equilibrium statistical mechanics and turbulence , volume 355 of London Math. Soc. Lecture Note Ser. , pages 44--107. Cambridge Univ. Press, Cambridge, 2008

    2008

  51. [59]

    Noiseless regularisation by noise

    Lucio Galeati and Massimiliano Gubinelli. Noiseless regularisation by noise. Rev. Mat. Iberoam. , 38(2):433--502, 2022

    2022

  52. [60]

    Solution theory of fractional SDE s in complete subcritical regimes

    Lucio Galeati and M\'at\'e Gerencs\'er. Solution theory of fractional SDE s in complete subcritical regimes. Forum Math. Sigma , 13:Paper No. e12, 66, 2025

    2025

  53. [61]

    Anomalous Regularization in Kraichnan 's Passive Scalar Model , 2024

    Lucio Galeati, Francesco Grotto, and Mario Maurelli. Anomalous Regularization in Kraichnan 's Passive Scalar Model , 2024. arXiv:2407.16668

    2024 arXiv

  54. [62]

    Anomalous Scaling of the Passive Scalar

    Krzysztof Gawedzki and Antti Kupiainen. Anomalous Scaling of the Passive Scalar . Physical Review Letters , 75(21):3834--3837, November 1995

    1995

  55. [63]

    Quantitative results on the corrector equation in stochastic homogenization

    Antoine Gloria and Felix Otto. Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc. (JEMS) , 19(11):3489--3548, 2017

    2017

  56. [64]

    Gubinelli

    M. Gubinelli. Controlling rough paths. J. Funct. Anal. , 216(1):86--140, 2004

    2004

  57. [65]

    Stabilization by transport noise and enhanced dissipation in the Kraichnan model, 2021

    Benjamin Gess and Ivan Yaroslavtsev. Stabilization by transport noise and enhanced dissipation in the Kraichnan model, 2021. arXiv:2104.03949

    2021 arXiv

  58. [66]

    G\"otze and A

    F. G\"otze and A. Yu. Zaitsev. Bounds for the rate of strong approximation in the multidimensional invariance principle. Teor. Veroyatn. Primen. , 53(1):100--123, 2008

    2008

  59. [67]

    Turbulent and intermittent phenomena in a universal total anomalous dissipator, July 2025

    Elias Hess-Childs and Keefer Rowan. Turbulent and intermittent phenomena in a universal total anomalous dissipator, July 2025. arXiv:2508.00115 [math]

    2025 arXiv

  60. [68]

    A universal total anomalous dissipator, January 2025

    Elias Hess-Childs and Keefer Rowan. A universal total anomalous dissipator, January 2025. arXiv:2501.18526 [math]

    2025 arXiv

  61. [69]

    Sharp pathwise nonuniqueness for additive SDEs , April 2026

    Elias Hess-Childs and Keefer Rowan. Sharp pathwise nonuniqueness for additive SDEs , April 2026. arXiv:2604.23883 [math]

    2026 arXiv

  62. [70]

    On a domination of sums of random variables by sums of conditionally independent ones

    Paweł Hitczenko. On a domination of sums of random variables by sums of conditionally independent ones. Ann. Probab. , 22(1):453--468, 1994

    1994

  63. [71]

    Nonlinear Y oung integrals and differential systems in H \"older media

    Yaozhong Hu and Khoa L\^e. Nonlinear Y oung integrals and differential systems in H \"older media. Trans. Amer. Math. Soc. , 369(3):1935--2002, 2017

    1935

  64. [72]

    Anomalous and total dissipation due to advection by solutions of randomly forced N avier-- S tokes equations

    Martina Hofmanov\'a, Umberto Pappalettera, Rongchan Zhu, and Xiangchan Zhu. Anomalous and total dissipation due to advection by solutions of randomly forced N avier-- S tokes equations. Ann. Appl. Probab. , 35(5):3119--3149, 2025

    2025

  65. [73]

    Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field, 2024

    Carl Johan Peter Johansson and Massimo Sorella. Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field, 2024. arXiv:2409.03599

    2024

  66. [74]

    Rough weak solutions for singular L \'evy SDE s

    Helena Kremp and Nicolas Perkowski. Rough weak solutions for singular L \'evy SDE s. Probab. Theory Related Fields , 193(1-2):483--537, 2025

    2025

  67. [75]

    N. V. Krylov and M. R\"ockner. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields , 131(2):154--196, 2005

    2005

  68. [76]

    Kraichnan

    Robert H. Kraichnan. Small‐ Scale Structure of a Scalar Field Convected by Turbulence . The Physics of Fluids , 11(5):945--953, May 1968

    1968

  69. [77]

    Nonuniqueness of trajectories on a set of full measure for S obolev vector fields

    Anuj Kumar. Nonuniqueness of trajectories on a set of full measure for S obolev vector fields. Arch. Ration. Mech. Anal. , 248(6):Paper No. 114, 26, 2024

    2024

  70. [78]

    Woyczy\'nski

    Stanisław Kwapie\'n and Wojbor A. Woyczy\'nski. Semimartingale integrals via decoupling inequalities and tangent processes. Probab. Math. Statist. , 12(2):165--200, 1991

    1991

  71. [79]

    Estimation of moments of sums of independent real random variables

    Rafał Latała. Estimation of moments of sums of independent real random variables. Ann. Probab. , 25(3):1502--1513, 1997

    1997

  72. [80]

    A stochastic sewing lemma and applications

    Khoa Le. A stochastic sewing lemma and applications. Electron. J. Probab. , 25:Paper No. 38, 55, 2020

    2020

  73. [81]

    Perte de r\'egularit\'e pour les \'equations d'ondes sur-critiques

    Gilles Lebeau. Perte de r\'egularit\'e pour les \'equations d'ondes sur-critiques. Bull. Soc. Math. France , 133(1):145--157, 2005

    2005

  74. [82]

    Sur les \'equations diff\'erentielles ordinaires et les \'equations de transport

    Pierre-Louis Lions. Sur les \'equations diff\'erentielles ordinaires et les \'equations de transport. C. R. Acad. Sci. Paris S\'er. I Math. , 326(7):833--838, 1998

    1998

  75. [83]

    Integration of B rownian vector fields

    Yves Le Jan and Olivier Raimond. Integration of B rownian vector fields. Ann. Probab. , 30(2):826--873, 2002

    2002

  76. [84]

    Flows, coalescence and noise

    Yves Le Jan and Olivier Raimond. Flows, coalescence and noise. Ann. Probab. , 32(2):1247--1315, 2004

    2004

  77. [85]

    Lawler and Vlada Limic

    Gregory F. Lawler and Vlada Limic. Random walk: a modern introduction , volume 123 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2010

    2010

  78. [86]

    Lectures on the P oisson process , volume 7 of Institute of Mathematical Statistics Textbooks

    G\"unter Last and Mathew Penrose. Lectures on the P oisson process , volume 7 of Institute of Mathematical Statistics Textbooks . Cambridge University Press, Cambridge, 2018

    2018

  79. [87]

    S. V. Lototskii and B. L. Rozovskii. The passive scalar equation in a turbulent incompressible G aussian velocity field. Uspekhi Mat. Nauk , 59(2(356)):105--120, 2004

    2004

  80. [88]

    S. V. Lototsky and B. L. Rozovskii. Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab. , 34(2):638--662, 2006

    2006

  81. [89]

    An elementary approach to mixing and dissipation enhancement by transport noise, 2024

    Dejun Luo, Bin Tang, and Guohuan Zhao. An elementary approach to mixing and dissipation enhancement by transport noise, 2024. arXiv:2402.07484

    2024 arXiv

  82. [90]

    Terry J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana , 14(2):215--310, 1998

    1998

  83. [91]

    Morfe, Felix Otto, and Christian Wagner

    Peter S. Morfe, Felix Otto, and Christian Wagner. A Critical Drift - Diffusion Equation : Intermittent Behavior via Geometric Brownian Motion on sl(n), November 2025

    2025

  84. [92]

    Modena and L

    S. Modena and L. Sz\' e kelyhidi, Jr. Non-uniqueness for the transport equation with S obolev vector fields. Ann. PDE , 4(2):Art. 18, 38, 2018

    2018

  85. [93]

    Modena and L

    S. Modena and L. Sz\' e kelyhidi, Jr. Non-renormalized solutions to the continuity equation. Calc. Var. Partial Differential Equations , 58(6):Art. 208, 30, 2019

    2019

  86. [94]

    Modena and G

    S. Modena and G. Sattig. Convex integration solutions to the transport equation with full dimensional concentration. Ann. Inst. H. Poincar\' e Anal. Non Lin\' e aire , 37(5):1075--1108, 2020

    2020

  87. [95]

    Percolation of strongly correlated G aussian fields, I : D ecay of subcritical connection probabilities

    Stephen Muirhead and Franco Severo. Percolation of strongly correlated G aussian fields, I : D ecay of subcritical connection probabilities. Probab. Math. Phys. , 5(2):357--412, 2024

    2024

  88. [96]

    Percolation of strongly correlated G aussian fields II

    Stephen Muirhead. Percolation of strongly correlated G aussian fields II . S harpness of the phase transition. Ann. Probab. , 52(3):838--881, 2024

    2024

  89. [97]

    Meerschaert, Wensheng Wang, and Yimin Xiao

    Mark M. Meerschaert, Wensheng Wang, and Yimin Xiao. Fernique-type inequalities and moduli of continuity for anisotropic G aussian random fields. Trans. Amer. Math. Soc. , 365(2):1081--1107, 2013

    2013

  90. [98]

    J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. , 80:931--954, 1958

    1958

  91. [99]

    Exponential mixing by random cellular flows

    V\'ictor Navarro-Fern\'andez and Christian Seis. Exponential mixing by random cellular flows. J. Funct. Anal. , 290(2):Paper No. 111227, 54, 2026

    2026

  92. [100]

    Regularization of differential equations by fractional noise

    David Nualart and Youssef Ouknine. Regularization of differential equations by fractional noise. Stochastic Process. Appl. , 102(1):103--116, 2002

    2002

  93. [101]

    Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on R ^3

    Tadahiro Oh and Oana Pocovnicu. Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on R ^3 . J. Math. Pures Appl. (9) , 105(3):342--366, 2016

    2016

  94. [102]

    Zur theorie der differentialgleichung y'=f(x,y)

    Władysław Orlicz. Zur theorie der differentialgleichung y'=f(x,y) . Bulletin International de l'Acad \'e mie Polonaise des Sciences et des Lettres, S \'e rie A: Sciences Math \'e matiques , pages 221--228, 1932

    1932

  95. [103]

    Pathwise uniqueness for singular SDE s driven by stable processes

    Enrico Priola. Pathwise uniqueness for singular SDE s driven by stable processes. Osaka J. Math. , 49(2):421--447, 2012

    2012

  96. [104]

    Davie's type uniqueness for a class of SDE s with jumps

    Enrico Priola. Davie's type uniqueness for a class of SDE s with jumps. Ann. Inst. Henri Poincar\'e Probab. Stat. , 54(2):694--725, 2018

    2018

  97. [105]

    G. C. Papanicolaou and S. R. S. Varadhan. Boundary value problems with rapidly oscillating random coefficients. In Random fields, V ol. I , II ( E sztergom, 1979) , volume 27 of Colloq. Math. Soc. J\'anos Bolyai , pages 835--873. North-Holland, Amsterdam-New York, 1981

    1979

  98. [106]

    Atmospheric diffusion shown on a distance-neighbour graph

    Lewis Fry Richardson. Atmospheric diffusion shown on a distance-neighbour graph. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character , 110(756):709--737, 1926

    1926

  99. [107]

    On anomalous diffusion in the K raichnan model and correlated-in-time variants

    Keefer Rowan. On anomalous diffusion in the K raichnan model and correlated-in-time variants. Arch. Ration. Mech. Anal. , 248(5):Paper No. 93, 47, 2024

    2024

  100. [108]

    The Obukhov -- Corrsin spectrum of passive scalar turbulence through anomalous regularization, December 2025

    Keefer Rowan. The Obukhov -- Corrsin spectrum of passive scalar turbulence through anomalous regularization, December 2025. arXiv:2512.02853 [math.AP]

    2025

  101. [109]

    S DE s with critical time dependent drifts: weak solutions

    Michael R\"ockner and Guohuan Zhao. S DE s with critical time dependent drifts: weak solutions. Bernoulli , 29(1):757--784, 2023

    2023

  102. [110]

    S DE s with critical time dependent drifts: strong solutions

    Michael R\"ockner and Guohuan Zhao. S DE s with critical time dependent drifts: strong solutions. Probab. Theory Related Fields , 192(3-4):1071--1111, 2025

    2025

  103. [111]

    Finite range decompositions of G aussian fields with applications to level-set percolation

    Florian Schweiger. Finite range decompositions of G aussian fields with applications to level-set percolation. J. Funct. Anal. , 286(5):Paper No. 110285, 34, 2024

    2024

  104. [112]

    On the loss of continuity for super-critical drift-diffusion equations

    Luis Silvestre, Vlad Vicol, and Andrej Zlatos. On the loss of continuity for super-critical drift-diffusion equations. Arch. Ration. Mech. Anal. , 207(3):845--877, 2013

    2013

  105. [113]

    Theory of function spaces , volume 78 of Monographs in Mathematics

    Hans Triebel. Theory of function spaces , volume 78 of Monographs in Mathematics . Birkh\"auser Verlag, Basel, 1983

    1983

  106. [114]

    A. Ju. Veretennikov. Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.) , 111(153)(3):434--452, 480, 1980

    1980

  107. [115]

    Wilson and J

    Kenneth G. Wilson and J. Kogut. The renormalization group and the expansion. Physics Reports , 12(2):75--199, August 1974

    1974

  108. [116]

    L. C. Young. An inequality of the H \"older type, connected with S tieltjes integration. Acta Math. , 67(1):251--282, 1936

    1936

  109. [117]

    A. K. Zvonkin. A transformation of the phase space of a diffusion process that will remove the drift. Mat. Sb. (N.S.) , 93(135):129--149, 152, 1974

    1974

  110. [118]

    Heat kernel and ergodicity of SDEs with distributional drifts, April 2018

    Xicheng Zhang and Guohuan Zhao. Heat kernel and ergodicity of SDEs with distributional drifts, April 2018. arXiv:1710.10537 [math]

    2018 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.