A General Aubry-Mather Theory

Nassif Ghoussoub

REVIEW 3 major objections 3 minor 259 references

A General Aubry-Mather Theory

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A class of nonlinear operators called Kantorovich operators carries a complete Aubry-Mather package through a duality with skew-linear entropies.

desk verdict This is the front matter of a monograph, not a research paper, and the central claim it announces is false as stated. read the letter →

arxiv 2608.06344 v1 pith:NGJGKSE3 submitted 2026-08-06 math.AP

Nassif Ghoussoub This is my paper · ORCID

classification math.AP MSC 37J5149Q2235F21

keywords Kantorovichoperatorsskew-linearentropiesAubry-MathertheoryweakKAMMatherconstantoptimalmasstransportminimalmeasureslargedeviations

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 front matter of a monograph claims that many nonlinear operators appearing across analysis, probability, dynamical systems, economics, and finance are instances of one class, called Kantorovich operators: monotone, translation-invariant on constants, convex (or concave, for the forward version) maps between function spaces. Because such an operator has no adjoint, the book replaces measure duality with a convex functional on pairs of probability distributions, a skew-linear entropy $T(\mu,\nu)$ whose value is the optimal cost of linking source $\mu$ to target $\nu$. The central claim is that every backward Kantorovich operator with finite Mather constant (the least diagonal cost $\inf_\mu T(\mu,\mu)$ of the associated entropy) on a compact space carries a full Aubry-Mather package: minimal measures, a Mather constant, weak KAM solutions (fixed points up to that constant), and a measure-level Aubry set. If correct, this extends the classical weak KAM theory of Hamiltonian dynamics to Hopf-Lax-Oleinik semigroups, Sinkhorn and Ruelle operators, risk-sensitive Bellman equations, envelopes, and stochastic transports alike.

What carries the argument

The central object is a backward Kantorovich operator: a map $T:C(Y)\to USC(X)$ that is monotone, translation invariant on constants, convex, and lower semicontinuous, with a forward analogue that is concave and upper semicontinuous. The load-bearing identity is the duality $T(\mu,\nu)=\sup_g\{\int_Y g\,d\nu-\int_X Tg\,d\mu\}$, which attaches to each operator a skew-linear entropy on pairs of probability distributions, together with the representation $Tg(x)=\sup_\sigma\{\int_Y g\,d\sigma-c(x,\sigma)\}$ for a cost $c$ convex in $\sigma$. This entropy plays the role the adjoint plays for a linear Markov operator: its diagonal value $T(\mu,\mu)$ defines the Mather constant $c(T)=\inf_\mu T(\mu,\mu)$ and the minimal measures, while monotone limits of iterates $T^n(g+nc)$ produce the idempotent weak KAM operator $T_\infty$ whose diagonal $T_\infty(\sigma,\sigma)$ defines the measure-level Aubry set $N(T_\infty)$. The construction works because Kantorovich operators extend to Choquet functional capacities, which justifies the monotone limits and the iteration of the operator.

What would settle it

A concrete check: take a two-point space $X$ and the Sinkhorn operator $Tg(x)=\epsilon\log\int_Y e^{(g(y)-c(x,y))/\epsilon}\,d\nu(y)$ with a continuous cost, and compute the normalized iterates $T^n g-nc(T)$. If they fail to converge to a function $u$ with $Tu+c(T)=u$, or if some minimizer of $T(\mu,\mu)$ is not in $N(T_\infty)=\{\sigma:T_\infty(\sigma,\sigma)=0\}$, the claimed package fails in a concrete instance.

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Extended reading notes

Core claim

The core discovery is a duality that makes nonlinear operators amenable to ergodic theory. Theorem 2 states that a map $T:C(Y)\to B_b(X)$ is a backward Kantorovich operator exactly when there is a proper lower semicontinuous cost $c:X\times P(Y)\to \mathbb{R}\cup\{+\infty\}$, convex in its second argument, such that $Tg(x)=\sup_\sigma\{\int_Y g\,d\sigma-c(x,\sigma)\}$. Equivalently, $T$ admits a skew-linear entropy $T(\mu,\nu)=\sup_g\{\int_Y g\,d\nu-\int_X Tg\,d\mu\}$ on pairs of probability measures. From this duality the book defines the Mather constant $c(T)=\inf_\mu\sup_h\int_X(h-Th)\,d\mu$, proves $c(T)=\inf_\mu T(\mu,\mu)$, calls the minimizers minimal measures, and then constructs, whenever $c(T)$ is finite, an idempotent weak KAM operator $T_\infty$ satisfying $T\circ T_\infty+c(T)=T_\infty$. Its measure-level Aubry set $N(T_\infty)=\{\sigma:T_\infty(\sigma,\sigma)=0\}$ contains all minimal measures, and every weak KAM solution is recovered from its integrals against $N(T_\infty)$, making the Aubry set a uniqueness set; forward and backward versions of the same package are developed. This is the claim that Aubry-Mather theory is not specific to Lagrangian dynamics but is the ergodic theory of any Kantorovich operator.

Load-bearing premise

The construction of weak KAM solutions assumes the Mather constant is finite and the underlying space is compact; the overview states that results on complete metric spaces, manifolds, or $\mathbb{R}^n$ are only expected to hold with additional hypotheses, even though many advertised applications live there.

Editorial extensions

If this is right

Every operator in the class, including the Hopf-Lax-Oleinik semigroup, Sinkhorn entropic regularization, risk-sensitive Bellman operators, Ruelle free-energy operators, concavification, and pluri-superharmonic envelopes, carries minimal measures, a Mather constant, weak KAM solutions, and an Aubry set whenever the Mather constant is finite.
For a linear Markov operator the package collapses to the classical one: the Mather constant is $0$, the minimal measures are exactly the invariant measures, and the weak KAM solutions are the invariant functions.
The identity $c(T)=\inf_\mu\sup_h\int_X(h-Th)\,d\mu=\inf_\mu T(\mu,\mu)$ gives two faces of the same constant, one computable through functions and one through the diagonal entropy of measures.
The measure-level Aubry set $N(T_\infty)$ is a uniqueness set: weak KAM solutions are determined, and reconstructed, by their integrals against $N(T_\infty)$ via the Peierls-barrier formula.
When the operator is both backward and forward skew-linear, a symmetric pair of weak KAM operators exists, with forward and backward solutions related by the same Aubry set.

Reading between the lines

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

A concrete finite-state test is available: on probability simplices the paper's formulas, for example $T(\beta,\alpha)=\sum_i \beta_i G_i(\alpha_i/\beta_i)$, make $c(T)$ and $T_\infty$ computable, and working out explicit examples would show what the Aubry set looks like outside mechanics.
If the non-compact extension goes through, the Mather constant of entropic Sinkhorn or Schr\"odinger transport would become a large-deviation rate for empirical occupation measures, turning the large-deviation reading sketched in the overview into a quantitative statement.
The framework suggests defining 'Mather measures' for optimal stopping and Skorokhod embedding as diagonal minimizers of Brownian stopping entropies, giving a variational selection principle for optimally stopped distributions.
A natural cross-check with dynamic programming: for a finite-state risk-sensitive Bellman operator, the weak KAM solution produced by $T_\infty$ should coincide with the known value function of the ergodic control problem, connecting the abstract package to textbook results.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This submission is the preface, overview, and table of contents of a monograph, and it should be judged as an announcement, not as a proof-bearing preprint. The framework is genuinely ambitious: backward Kantorovich operators, skew-linear entropies, and the proposed extension of Aubry–Mather theory to Bellman operators, Ruelle operators, Sinkhorn iterations, and stochastic transport. The definition is clean, the examples are well chosen, and the representation theorem (Theorem 2) is a plausible and interesting generalization of Bernard–Buffoni. The prose is clear, and the reference list is thorough. Credit where due: this is a serious research program, and the announcement gives a good map of it.

The soft spots are serious. First, there are no derivations. Theorem 2 is called crucial but is not proved; the six properties of the weak KAM operator are asserted; the duality c(T) = inf_mu T(mu,mu) is asserted. That alone makes the submission unverifiable. Second, the stress-test counterexample is valid: take X={1,2} and T(g)_1 = max(g_1+1, g_2), T(g)_2 = g_2. This satisfies Definition 1, and its Mather constant is -1, finite. But the claimed weak KAM equation (52) for state 2 reads u_2 - 1 = u_2, impossible. So no weak KAM operator exists. The abstract's "arbitrary Kantorovich operator" is therefore false; the book's own table of contents suggests that extra hypotheses (bounded oscillations, cone contraction, balanced, amenable) are needed, but those are not stated in the Overview or abstract.

Where does this leave the work? The framework may well be salvageable under additional conditions, and the counterexample does not kill the entire program. But as submitted, the central claim is wrong as stated, and the absence of proofs means the reader cannot check what actually holds. A serious editor should send the full monograph to a referee, not desk-reject the idea; the counterexample should be brought to the author's attention early. For the arXiv text alone, though, the right verdict is that it overclaims and needs heavy revision or withdrawal until the real hypotheses and proofs are supplied.

Referee Report

3 major / 3 minor

Summary. The submission is the front matter of a planned monograph: it contains a preface, a long overview, a table of contents, and a reference list, but no proofs. The paper proposes a class of nonlinear operators called Kantorovich operators and claims that, in duality with skew-linear entropies, every such operator with finite Mather constant carries a complete Aubry–Mather package: minimal measures, a Mather constant, weak KAM solutions, and an Aubry set. The central structural result is stated as Theorem 2, which represents backward Kantorovich operators as suprema over probability measures with a convex cost c(x, σ). A list of six properties is then announced for the associated weak KAM operator T∞, including idempotence, commutation, the equation T T∞ g + c(T) = T∞ g, and the measure-level Aubry set. The monograph is said to work on compact spaces, with extensions to non-compact settings deferred. No derivations appear in the submitted text.

Significance. If the program were correct, it would unify substantial pieces of Hamiltonian dynamics, optimal transport, risk-sensitive control, potential theory, and ergodic optimization, and the breadth of examples assembled in the overview is genuinely impressive. The explicit formulation of skew-linear entropies and the proposed measure-level Aubry set are potentially valuable organizing ideas. However, the submission contains no proofs of the central theorems, and the main universal claim is false as stated (see major comments). The paper is therefore best read as an extended book prospectus rather than a verifiable research contribution; its current value lies in the catalogue of examples and references, not in established theorems.

major comments (3)

[Overview, Definition 1 and Eq. (52)] The claim that every backward Kantorovich operator with finite Mather constant admits a weak KAM operator is false as stated. Let X = {1, 2} and define T(g)_1 = max(g_1 + 1, g_2), T(g)_2 = g_2. This operator satisfies monotonicity, translation invariance, convexity, and lower semicontinuity, hence all four clauses of Definition 1. From Eq. (50), for μ = (p, 1-p), sup_h ∫ (h - T h) dμ = sup_{h1,h2} p (h1 - max(h1+1, h2)), which equals -p for p > 0 and 0 for p = 0; therefore c(T) = -1, which is finite. But the weak KAM equation (52) at x = 2 reads u_2 - 1 = u_2, so no finite-valued weak KAM solution exists. This example is a Bellman operator of exactly the type displayed in Eq. (19), so it lies within the proposed framework. The announced theorem therefore requires additional hypotheses, such as irreducibility or a communicating/normalization condition, and the abstract's phrase 'arbitrary Kantorovich operator' must be withdrawn.
[Theorem 2 and the announced weak KAM properties] The submission contains no derivations for its load-bearing statements. Theorem 2 is described as crucial, but it is stated without proof; the equality c(T) = inf_μ T(μ, μ) in Eq. (51) is asserted without proof; and the six properties of the weak KAM operator T∞ are announced without proof. Since the abstract states that 'this paper reproduces the front matter' of a monograph, the reader cannot verify any of the central claims from the submitted text. A journal submission would need either complete proofs or a clear statement that the theorems are proved in a cited companion manuscript with precise hypotheses; the current text provides neither.
[Overview, compactness and additional hypotheses] The scope of the advertised claims is internally mismatched. The abstract promises an Aubry–Mather theory for an 'arbitrary Kantorovich operator', but the Overview later restricts the weak KAM construction to compact spaces and to operators with finite Mather constant, and adds that the non-compact cases are only 'expected to hold' with further analysis and suitable hypotheses. Moreover, the table of contents lists many additional assumptions in Sections 14.3–14.4 and 16.2–16.8, including bounded oscillations, cone contraction, oscillation contractions, balanced skew-linear entropies, power-bounded contractions, and amenable entropies; none of these appear in Definition 1 or in the Overview's unconditional list of properties. The main theorem should be stated with its actual hypotheses, and the advertised applications on R^n, Riemannian manifolds, and Sinkhorn operators should be explicitly marked as conjectural extensions rather than consequences of the compact-space theory.

minor comments (3)

[Eq. (29)] The displayed identity T(lim_n ↓ T^n(g+nc)) + c = lim_n ↓ T^{n+1}g + (n+1)c has unbalanced parentheses and is hard to parse; the intended limiting argument should be written with explicit parentheses and indices.
[Eq. (28)] In the large-deviation formula, the right-hand side writes T(x, σ), but the skew-linear entropy is defined on pairs of probability measures; the notation should be T(δ_x, σ) or the corresponding cost c(x, σ) to avoid confusing a point with its Dirac measure.
[References] Reference [182] is missing its title, and references [184] and [186] are listed as personal communications, which cannot be independently checked; these should be completed or removed before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the monograph's claims are asserted representation and existence theorems, not reductions to fitted inputs or to self-citations.

full rationale

Walking the claimed derivation chain, I find no step where a prediction is equivalent to its input by construction. Theorem 2 is a representation theorem stated as a characterization of Definition 1; it is not derived by substituting the conclusion into the hypothesis. The dual formula c(T)=inf_mu T(mu,mu) follows immediately from the defining conjugate relation (26), so it is an identity rather than a fitted result. The weak KAM package (properties 1-6, equations (52)-(58)) is announced as the monograph's existence theory, not derived from the definition of c(T) alone; the table of contents' many extra hypotheses (bounded oscillations, cone contraction, balanced, amenable, etc.) suggest the existence claims are conditional in the full text. The self-citations to the Bowles dissertation and Bowles-Ghoussoub preprints are provenance statements for the framework and are not the only support for the main theorems. The two-state Bellman operator objection, if correct, makes the abstract's 'arbitrary Kantorovich operator' wording overbroad, and the compactness and finite-c(T) restrictions, located in the Overview paragraph beginning 'Throughout this monograph, we shall focus on probability measures on compact spaces,' are limitations; both are correctness or scoping concerns, not circularity. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

No numbers are fitted to data or chosen by hand in this text. The framework's hypotheses (compactness, finite Mather constant) are structural assumptions, not free parameters, and the epsilon and gamma appearing in the Sinkhorn and Bellman examples belong to the operators under study, not to the framework's derivation.

assumptions (5)

standard math Fenchel-Moreau biconjugacy for convex lsc functions on C(Y) and probability measures
Underlies Theorem 2 and the duality formulas (26) and (27), which write the skew-linear entropy as a supremum over g in C(Y).
domain assumption X and Y are compact metric spaces throughout
Stated in the Overview: 'we shall focus on probability measures on compact spaces... to avoid the usual functional analytic complications'.
domain assumption The Mather constant c(T) is finite
The Overview restricts the weak KAM program to 'a backward Kantorovich operator T with a finite Mather constant c(T)'; finiteness is not implied by Definition 1 and is not established for the examples here.
ad hoc to paper Theorem 2: representation of every backward Kantorovich operator as a sup over P(Y) with convex cost c(x,sigma)
Stated in the Overview as 'the crucial representation' but no proof is given in this submission, so the review must treat it as a postulate.
ad hoc to paper Existence of a weak KAM operator T_infinity with the six asserted properties (idempotence, commutation, calibration, Aubry set containment, reconstruction, duality)
Announced in the Overview's general weak KAM program without proof; the classical Lax-Oleinik case is cited as motivation, not as a proof.

invented entities (3)

Kantorovich operators (backward and forward) independent evidence
purpose: The class of nonlinear operators claimed to carry a general Aubry-Mather theory
The framework is testable by reduction to existing cases: Markov operators, Lax-Oleinik semigroups, Bellman operators, and Ruelle operators must all satisfy Definition 1, and the theory must recover their classical ergodic objects.
Skew-linear entropies on pairs of probability measures independent evidence
purpose: Replace the missing adjoint of a nonlinear operator by a duality object
They are claimed to contain optimal transport costs, Donsker-Varadhan information, and balayage orders as special cases, each of which is an external, checkable benchmark.
Weak KAM operators and measure-level Aubry sets independent evidence
purpose: Attach Mather-style structure (calibrated fixed points, recurrent set) to arbitrary Kantorovich operators
For the Lax-Oleinik operator they must reproduce Fathi's weak KAM solutions and the classical Aubry set, and chapters 17-24 name concrete instances (concavification, plurisuperharmonic envelopes, Sinkhorn, Bellman) where the objects should be identifiable.

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Cite this review

Pith. "Pith review of A General Aubry-Mather Theory." pith.science (2026). https://pith.science/paper/NGJGKSE3

@misc{pith2026260806344,
  author       = {Pith},
  title        = {Pith review of: A General Aubry-Mather Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGJGKSE3}},
  note         = {Machine review of arXiv:2608.06344}
}

read the original abstract

This paper reproduces the front matter --- preface, overview and table of contents --- of a monograph by the author, submitted for publication under the title {\it Skew Linear Entropies and Kantorovich Operators: A General Aubry-Mather Theory}. The book isolates a class of non-linear operators, which we call {\it Kantorovich operators}, that are ubiquitous in analysis, probability, dynamical systems, mathematical economics and finance. We develop aspects of their ergodic theory in a way that extends classical ones involving Markov operators, free-energy transfers, or the Hopf--Lax--Oleinik semi-group. Having no adjoint, the duality between such an operator and measures is carried instead via a convex functional on {\it pairs} of probability distributions --- a source and a target --- which we call a {\it skew-linear entropy}, and which is a general form of optimal mass transport. The extensive overview reproduced here describes the resulting ergodic theory, in which minimal measures, a Mather constant, weak KAM solutions and an Aubry set are attached to an arbitrary Kantorovich operator, extending Aubry--Mather theory well beyond its origins in Hamiltonian dynamics.

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