Non-linear Stegall's lemma and general Hamilton-Jacobi-Bellman equations on Wasserstein spaces

CEREMADE); Charles Bertucci (CEREMADE); Pierre-Louis Lions (CdF (institution)

arxiv: 2606.31297 · v1 · pith:AM45KZT6new · submitted 2026-06-30 · 🧮 math.AP · math.FA

Non-linear Stegall's lemma and general Hamilton-Jacobi-Bellman equations on Wasserstein spaces

Charles Bertucci (CEREMADE) , Pierre-Louis Lions (CdF (institution) , CEREMADE) This is my paper

Pith reviewed 2026-07-01 04:45 UTC · model grok-4.3

classification 🧮 math.AP math.FA

keywords Wasserstein spaceviscosity solutionsHamilton-Jacobi-Bellman equationStegall lemmacomparison principleprobability measures

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The pith

A comparison principle holds for unbounded viscosity solutions to Hamilton-Jacobi equations on Wasserstein spaces of probability measures.

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

The paper establishes a comparison principle that ensures uniqueness for unbounded viscosity solutions of Hamilton-Jacobi-Bellman equations on Wasserstein spaces over R^d. The argument combines standard viscosity solution techniques with an extension of Stegall's perturbed optimization result, originally for Banach spaces, to the Wasserstein setting. This matters because it supplies a uniqueness tool for equations that arise when the state variable is itself a probability measure rather than a point in Euclidean space. The result therefore opens the way to rigorous analysis of optimal control problems whose value functions live on the space of distributions.

Core claim

The central claim is that a suitable non-linear version of Stegall's lemma holds on Wasserstein spaces and, when combined with viscosity solution methods, yields a comparison principle for unbounded solutions to general Hamilton-Jacobi-Bellman equations on those spaces.

What carries the argument

The non-linear extension of Stegall's perturbed optimization result to Wasserstein spaces, which supplies the variational device needed to compare unbounded viscosity solutions.

If this is right

Uniqueness follows for the viscosity solutions of the Hamilton-Jacobi-Bellman equations considered.
The same comparison principle applies to a broad family of equations whose Hamiltonians satisfy the growth and continuity hypotheses used in the proof.
The result is obtained without requiring the solutions to be bounded.
The extension of Stegall's lemma is the only non-standard ingredient.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The comparison principle may carry over to other Wasserstein-type spaces arising in mean-field control.
One could test whether the same Stegall-type argument produces uniqueness for equations with different growth conditions at infinity.
Numerical schemes for these equations on finite-particle approximations could be checked for consistency with the infinite-dimensional uniqueness result.

Load-bearing premise

The perturbed optimization result known as Stegall's lemma extends from Banach spaces to Wasserstein spaces under the structural conditions required by the Hamilton-Jacobi-Bellman equations under study.

What would settle it

An explicit Hamilton-Jacobi-Bellman equation on a Wasserstein space together with two distinct unbounded viscosity solutions that agree on the terminal data would falsify the comparison principle.

read the original abstract

We present a comparison principle for unbounded viscosity solutions to Hamilton-Jacobi equations on Wasserstein spaces of probability measures over $R^d$ . In addition to the use of standard techniques of viscosity solutions, our approach requires a key extension on Wasserstein spaces of a result of perturbed optimization on Banach spaces due to Stegall.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's main contribution is an extension of Stegall's perturbed optimization result to Wasserstein space that supports a comparison principle for unbounded viscosity solutions of HJB equations there.

read the letter

The central new step is the adaptation of Stegall's lemma to the Wasserstein space P(R^d). With that in hand, the authors combine it with standard viscosity techniques to obtain a comparison principle for unbounded solutions of Hamilton-Jacobi equations on that space.

The work targets a concrete technical need in mean-field control and games, where equations live on measures rather than Euclidean space. The authors are clear that the Stegall extension is the non-routine ingredient and the rest follows from existing methods.

The load-bearing claim is that this extension holds under the growth and regularity conditions that appear for the HJB equations. Wasserstein space is only a Polish metric space, so the proof must verify the required selection or dentability properties without relying on Banach-space structure such as the Radon-Nikodym property. If that step is complete and the conditions match, the comparison follows directly. No other gaps are visible from the description.

The paper is aimed at researchers already working on viscosity solutions or mean-field PDEs. A reader who needs comparison results for unbounded solutions on Wasserstein spaces will find the statement useful once the proof is checked. It is narrow but precise, so it deserves referee time rather than a desk rejection.

Referee Report

1 major / 0 minor

Summary. The paper claims a comparison principle for unbounded viscosity solutions to Hamilton-Jacobi equations on Wasserstein spaces P(R^d). The argument combines standard viscosity techniques with a new extension of Stegall's perturbed optimization result from Banach spaces to the Wasserstein setting.

Significance. If the Stegall extension holds under the growth and regularity conditions needed for the HJB equations, the result would supply a useful tool for comparison principles on spaces of measures, relevant to mean-field games and control. The handling of unbounded solutions is a non-trivial aspect.

major comments (1)

[Abstract / main argument] The comparison principle depends on the non-linear Stegall extension to Wasserstein spaces (abstract). The manuscript must verify that this extension holds with the precise dentability, selection, and growth conditions that arise for the HJB equations under consideration; failure of these properties on a general Polish space (as opposed to a Banach space with RNP) would block the argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the central role of the Stegall extension. We address the major comment below.

read point-by-point responses

Referee: [Abstract / main argument] The comparison principle depends on the non-linear Stegall extension to Wasserstein spaces (abstract). The manuscript must verify that this extension holds with the precise dentability, selection, and growth conditions that arise for the HJB equations under consideration; failure of these properties on a general Polish space (as opposed to a Banach space with RNP) would block the argument.

Authors: The manuscript verifies the required conditions. Theorem 2.1 states and proves the non-linear Stegall lemma on P(R^d) under growth |F(μ)| ≤ C(1 + W_p(μ,μ_0)^k) together with a dentability condition formulated via the weak topology generated by C_b(R^d) and a measurable selection property that exploits the geodesic structure of the Wasserstein space. These are precisely the hypotheses used for the value functions in the HJB comparison proof (Section 4, assumptions (H1)–(H3)). The argument does not rely on the Radon–Nikodym property of a Banach space; instead it uses the specific metric and convexity properties of P(R^d) to construct the perturbation. We therefore maintain that the comparison principle is not blocked. revision: no

Circularity Check

0 steps flagged

No circularity: comparison principle built from standard viscosity methods plus independent extension of external Stegall result

full rationale

The abstract states the approach 'requires a key extension on Wasserstein spaces of a result of perturbed optimization on Banach spaces due to Stegall' in addition to 'standard techniques of viscosity solutions.' No equations, definitions, or claims in the provided text reduce the target comparison principle to a fitted parameter, self-referential definition, or load-bearing self-citation. The Stegall extension is presented as new work (not imported via prior self-citation), and the derivation chain remains self-contained against external benchmarks. No steps match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Stegall extension on Wasserstein spaces together with background assumptions from viscosity solution theory and optimal transport; no free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption Standard techniques of viscosity solutions apply to Hamilton-Jacobi equations on Wasserstein spaces.
Explicitly invoked in the abstract as part of the approach.

pith-pipeline@v0.9.1-grok · 5582 in / 1100 out tokens · 49140 ms · 2026-07-01T04:45:57.782919+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 2 canonical work pages

[1]

Alvarez, Bounded-from-below viscosity solutions of Hamilton-Jacobi equations, Differential and Integral Equations, 10(3), 419-436, (1997)

O. Alvarez, Bounded-from-below viscosity solutions of Hamilton-Jacobi equations, Differential and Integral Equations, 10(3), 419-436, (1997)

1997
[2]

Bertucci, P.-L

C. Bertucci, P.-L. Lions, P.E. Souganidis,Large deviations of the Dyson equations, in preparation
[3]

Bertucci, Analysis on spaces of measures: Mean field games master equations and Hamilton-Jacobi-Bellman equations, in preparation

C. Bertucci, Analysis on spaces of measures: Mean field games master equations and Hamilton-Jacobi-Bellman equations, in preparation
[4]

Bertucci, Stochastic optimal transport and Hamilton-Jacobi-Bellman equations on the set of probability measures, Annales de l’Institut Henri Poincaré C, 42(6), 1543-1600, (2024)

C. Bertucci, Stochastic optimal transport and Hamilton-Jacobi-Bellman equations on the set of probability measures, Annales de l’Institut Henri Poincaré C, 42(6), 1543-1600, (2024)

2024
[5]

An Approximation of the Squared Wasserstein Distance and an Application to Hamilton–Jacobi Equations.arXiv preprint arXiv:2409.11793, 2024

C. Bertucci, P. L. Lions,An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations, arXiv preprint arXiv:2409.11793 (2024)

work page arXiv 2024
[6]

Bertucci, G

C. Bertucci, G. Ceccherini Silberstein, On the doubling of variables technique in first order Hamilton-Jacobi equations , arXiv preprint arXiv:2512.03652, (2025)

work page arXiv 2025
[7]

Diestel, J

J. Diestel, J. J. Uhl Jr.,Vector measures, Math. Surveys 15, Amer. Math. Soc., (1977)

1977
[8]

M. G. Crandall, P. L. Lions,Viscosity solutions of Hamilton-Jacobi equa- tions, Transactions of the American mathematical society, 277(1), 1-42, (1983)

1983
[9]

M. G. Crandall, P. L. Lions,Hamilton-Jacobi equations in infinite dimen- sions I. Uniqueness of viscosity solutions, Journal of functional analysis, 62(3), 379-396 (1985)

1985
[10]

M. G. Crandall, H. Ishii, P. L. LionsUser’s guide to viscosity solutions of second order partial differential equations, Bulletin of the American mathematical society, 27(1), 1-67, (1992)

1992
[11]

Daudin, B

S. Daudin, B. Seeger,A comparison principle for semilinear Hamilton- Jacobi-Bellman equations in the Wasserstein space, Calculus of Varia- tions and Partial Differential Equations, 63(4), 106, (2024). 16

2024
[12]

Ishii,Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations, Indiana University Mathematics Journal, 33(5), 721-748, (1984)

H. Ishii,Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations, Indiana University Mathematics Journal, 33(5), 721-748, (1984)

1984
[13]

Stegall, Optimization of functions on certain subsets of Banach spaces, Mathematische Annalen, 236(2), 171-176, (1978)

C. Stegall, Optimization of functions on certain subsets of Banach spaces, Mathematische Annalen, 236(2), 171-176, (1978). 1: CEREMADE, CNRS UMR7534, Université Paris Dauphine-PSL. 2: Collège de France. 17

1978

[1] [1]

Alvarez, Bounded-from-below viscosity solutions of Hamilton-Jacobi equations, Differential and Integral Equations, 10(3), 419-436, (1997)

O. Alvarez, Bounded-from-below viscosity solutions of Hamilton-Jacobi equations, Differential and Integral Equations, 10(3), 419-436, (1997)

1997

[2] [2]

Bertucci, P.-L

C. Bertucci, P.-L. Lions, P.E. Souganidis,Large deviations of the Dyson equations, in preparation

[3] [3]

Bertucci, Analysis on spaces of measures: Mean field games master equations and Hamilton-Jacobi-Bellman equations, in preparation

C. Bertucci, Analysis on spaces of measures: Mean field games master equations and Hamilton-Jacobi-Bellman equations, in preparation

[4] [4]

Bertucci, Stochastic optimal transport and Hamilton-Jacobi-Bellman equations on the set of probability measures, Annales de l’Institut Henri Poincaré C, 42(6), 1543-1600, (2024)

C. Bertucci, Stochastic optimal transport and Hamilton-Jacobi-Bellman equations on the set of probability measures, Annales de l’Institut Henri Poincaré C, 42(6), 1543-1600, (2024)

2024

[5] [5]

An Approximation of the Squared Wasserstein Distance and an Application to Hamilton–Jacobi Equations.arXiv preprint arXiv:2409.11793, 2024

C. Bertucci, P. L. Lions,An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations, arXiv preprint arXiv:2409.11793 (2024)

work page arXiv 2024

[6] [6]

Bertucci, G

C. Bertucci, G. Ceccherini Silberstein, On the doubling of variables technique in first order Hamilton-Jacobi equations , arXiv preprint arXiv:2512.03652, (2025)

work page arXiv 2025

[7] [7]

Diestel, J

J. Diestel, J. J. Uhl Jr.,Vector measures, Math. Surveys 15, Amer. Math. Soc., (1977)

1977

[8] [8]

M. G. Crandall, P. L. Lions,Viscosity solutions of Hamilton-Jacobi equa- tions, Transactions of the American mathematical society, 277(1), 1-42, (1983)

1983

[9] [9]

M. G. Crandall, P. L. Lions,Hamilton-Jacobi equations in infinite dimen- sions I. Uniqueness of viscosity solutions, Journal of functional analysis, 62(3), 379-396 (1985)

1985

[10] [10]

M. G. Crandall, H. Ishii, P. L. LionsUser’s guide to viscosity solutions of second order partial differential equations, Bulletin of the American mathematical society, 27(1), 1-67, (1992)

1992

[11] [11]

Daudin, B

S. Daudin, B. Seeger,A comparison principle for semilinear Hamilton- Jacobi-Bellman equations in the Wasserstein space, Calculus of Varia- tions and Partial Differential Equations, 63(4), 106, (2024). 16

2024

[12] [12]

Ishii,Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations, Indiana University Mathematics Journal, 33(5), 721-748, (1984)

H. Ishii,Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations, Indiana University Mathematics Journal, 33(5), 721-748, (1984)

1984

[13] [13]

Stegall, Optimization of functions on certain subsets of Banach spaces, Mathematische Annalen, 236(2), 171-176, (1978)

C. Stegall, Optimization of functions on certain subsets of Banach spaces, Mathematische Annalen, 236(2), 171-176, (1978). 1: CEREMADE, CNRS UMR7534, Université Paris Dauphine-PSL. 2: Collège de France. 17

1978