A selection of a weak KAM solution of the sub-riemannian Ma\~n\'e Lagrangian

H\'ector S\'anchez Morgado; Iker Mart\'inez Ju\'arez

A selection of a weak KAM solution of the sub-riemannian Ma\~n\'e Lagrangian

T0 review · 2 major / 2 minor · reviewed 2026-05-24 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read The zero-noise limit of the invariant measure selects a particular weak KAM solution of the sub-Riemannian Mañé Lagrangian on the torus.

desk verdict The paper shows that zero-noise limits of invariant measures select a weak KAM solution for sub-Riemannian Mañé Lagrangians on the torus, but only when the Aubry set has finitely many static classes. read the letter →

arxiv 2401.10335 v3 submitted 2024-01-18 math.DS

Iker Martínez Juárez , Héctor Sánchez Morgado This is my paper

classification math.DS

keywords weakKAMsolutionssub-RiemannianstructuresMañéLagrangianAubrysetstochasticperturbationsinvariantmeasurestorus

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper considers a sub-Riemannian structure on the torus satisfying the Hörmander condition together with the Mañé Lagrangian generated by a horizontal vector field. It assumes the Aubry set consists of finitely many static classes. Under this assumption the authors prove that the invariant measure of the horizontally perturbed stochastic flow converges, as the noise strength goes to zero, to a measure that singles out one specific weak KAM solution. A reader would care because the result supplies an explicit dynamical mechanism for choosing among weak KAM solutions in a constrained geometric setting.

What carries the argument

The zero-noise limit of the invariant measure of the horizontal stochastic perturbation of the vector-field flow, used as a selection device among weak KAM solutions.

What would settle it

An explicit sub-Riemannian structure on the torus with finitely many static classes in the Aubry set for which the zero-noise limit of the invariant measure fails to be supported on a weak KAM solution.

Watch

Extended reading notes

Core claim

For a sub-Riemannian structure on the torus satisfying the Hörmander condition, and assuming the Aubry set consists of a finite number of static classes, the invariant measure for the horizontal stochastic perturbation of the flow of the vector field determines a particular weak KAM solution of the associated Mañé Lagrangian as the perturbation intensity tends to zero.

Load-bearing premise

The Aubry set consists of a finite number of static classes.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core claim is that, given a sub-Riemannian structure on the torus satisfying Hörmander's condition and a horizontal vector field, the invariant measure of the stochastically perturbed flow converges to a measure supported on a single calibrated curve, thereby selecting one weak KAM solution as the noise parameter goes to zero. This holds provided the Aubry set consists of finitely many static classes. The paper states the result cleanly under those hypotheses and ties it to the Mañé Lagrangian setup. That combination for the sub-Riemannian case appears to be the concrete addition here, building on existing weak KAM and stochastic analysis tools rather than deriving new machinery from scratch. The argument looks like a standard limiting procedure once the finiteness assumption is in place, with no obvious circularity in the abstract framing. The main limitation is exactly that finiteness hypothesis. It is imposed rather than proved or shown to hold generically, and without it the limit measure could spread across multiple classes instead of isolating one solution. The paper treats the assumption as given, which narrows the result's reach. No other gaps stand out from the stated theorem, but the full proof details would need checking for any technical slips in the convergence step. This is specialized work aimed at people already working on weak KAM theory or sub-Riemannian dynamics. A reader looking for selection mechanisms via stochastic perturbations would find the statement useful as a reference point. It is worth sending to peer review because the claim is well-posed and the setting is precise, even though the assumption restricts its scope and some readers may want more discussion of when the finiteness condition applies.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that, for a sub-Riemannian structure on the torus satisfying the Hörmander condition, the Mañé Lagrangian associated to a horizontal vector field admits a selection of a particular weak KAM solution via the zero-noise limit of the invariant measure of its horizontal stochastic perturbation, provided the Aubry set consists of finitely many static classes.

Significance. If the limiting argument is rigorous, the result supplies a stochastic selection mechanism for weak KAM solutions in the sub-Riemannian setting, connecting stochastic analysis with Aubry-Mather theory and viscosity solutions on non-holonomic manifolds. This could be useful for identifying calibrated curves when multiple static classes are present.

major comments (2)

[Abstract] Abstract and §1: the finiteness of static classes in the Aubry set is stated as an enabling hypothesis but is not shown to hold for any concrete sub-Riemannian structure satisfying the Hörmander condition, nor is it shown to be generic; without this the limiting measure may converge to a convex combination rather than isolating a single weak KAM solution, making the assumption load-bearing for the central selection claim.
[§3] The limiting argument (presumably in §3 or §4): the passage from the invariant measure of the perturbed flow to a calibrated measure supported on a single static class relies on the finite-class assumption, yet the manuscript provides no explicit verification that the horizontal stochastic perturbation preserves the necessary tightness or convergence properties when the vector field is only horizontal.

minor comments (2)

The title phrasing 'A selection of a weak KAM solution' is slightly awkward; consider 'Selection of a weak KAM solution' or similar.
[Introduction] Notation for the stochastic perturbation (e.g., the horizontal noise term) should be introduced with a displayed equation in the introduction for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below, clarifying the role of the finite static classes assumption and the details of the limiting argument.

read point-by-point responses

Referee: [Abstract] Abstract and §1: the finiteness of static classes in the Aubry set is stated as an enabling hypothesis but is not shown to hold for any concrete sub-Riemannian structure satisfying the Hörmander condition, nor is it shown to be generic; without this the limiting measure may converge to a convex combination rather than isolating a single weak KAM solution, making the assumption load-bearing for the central selection claim.

Authors: The finiteness of static classes is explicitly introduced as a hypothesis under which the selection result holds, as stated in the abstract and §1. The manuscript does not assert that this condition holds for every Hörmander-satisfying sub-Riemannian structure on the torus, nor does it claim the condition is generic. The central contribution is the stochastic selection mechanism that isolates a single weak KAM solution precisely when the Aubry set has finitely many static classes; without the assumption the zero-noise limit may indeed be a convex combination, but the theorem remains valid as a conditional selection principle. Establishing genericity of the finite-class condition would require a separate analysis of the Aubry set for concrete examples and lies outside the scope of the present work. revision: no
Referee: [§3] The limiting argument (presumably in §3 or §4): the passage from the invariant measure of the perturbed flow to a calibrated measure supported on a single static class relies on the finite-class assumption, yet the manuscript provides no explicit verification that the horizontal stochastic perturbation preserves the necessary tightness or convergence properties when the vector field is only horizontal.

Authors: The limiting argument in §§3–4 establishes tightness and weak convergence of the invariant measures for the horizontal stochastic perturbation by exploiting the Hörmander condition and the sub-Riemannian metric to obtain uniform moment bounds and equicontinuity. The finite-class assumption is then used to deduce that any weak-* limit is supported on a single static class and is calibrated. While these estimates are written for the horizontal case, we acknowledge that an additional clarifying sentence emphasizing the preservation of tightness under horizontal noise would strengthen readability. We will insert such a remark in the revised version. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; result is a conditional theorem under explicit hypothesis

full rationale

The paper explicitly states the finiteness of static classes in the Aubry set as a hypothesis in the abstract and uses it to prove that the limiting invariant measure selects a particular weak KAM solution. No quoted equations or steps in the provided material reduce the claimed selection to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation rests on prior weak KAM and stochastic results treated as external, with the assumption serving as a standard enabling condition rather than an output smuggled in as a prediction. This is a normal non-circular mathematical argument.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The result rests on standard background assumptions of sub-Riemannian geometry and weak KAM theory; no free parameters or invented entities are visible in the abstract.

assumptions (2)

domain assumption The sub-Riemannian structure on the torus satisfies the Hörmander condition.
Invoked to guarantee that the horizontal distribution generates the full tangent space, enabling controllability.
domain assumption The Aubry set consists of a finite number of static classes.
Explicitly stated as the hypothesis under which the selection result holds.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A selection of a weak KAM solution of the sub-riemannian Ma\~n\'e Lagrangian." pith.science (2026). https://pith.science/paper/2401.10335

@misc{pith2026240110335,
  author       = {Pith},
  title        = {Pith review of: A selection of a weak KAM solution of the sub-riemannian Ma\~n\'e Lagrangian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2401.10335}},
  note         = {Machine review of arXiv:2401.10335}
}

read the original abstract

For a sub-riemannian structure on the torus, satisfying the H\"ormander condition, we consider the Ma\~n\'e Lagrangian associated to a horizontal vector field. Assuming that the Aubry set consists in a finite number of static classes, we show that the invariant measure, for the horizontal stochastic perturbation of the flow of the vector field, determines a particular weak KAM solution of the Lagrangian, as the perturbation tends to zero.

Discussion (0). Sign in to comment.

Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.