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arxiv: 2604.25305 · v1 · submitted 2026-04-28 · 🧮 math.AP · math.OC

Path-dependent Hamilton--Jacobi equations: Uniqueness results for viscosity solutions defined via families of compact sets

Mikhail I. Gomoyunov This is my paper

Pith reviewed 2026-05-07 15:35 UTC · model grok-4.3

classification 🧮 math.AP math.OC
keywords continuousuniquenesssolutionsviscosityclassconditionestablishedfunctionals
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The pith

Uniqueness holds for continuous viscosity solutions of path-dependent HJ equations when the Hamiltonian is continuous and locally Lipschitz in the functional variable, either with sublinear growth in the gradient or with an added local Lipschitz condition on the solution itself.

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

Path-dependent Hamilton-Jacobi equations model optimal control or differential games where the payoff or dynamics depend on the entire history of a path rather than only its current value. Classical solutions often fail to exist, so researchers use viscosity solutions, which are defined indirectly through test functions that touch the solution from above or below. Here the test functions must be smooth in a coinvariant sense with respect to a family of compact sets. The authors establish uniqueness in two settings. When the Hamiltonian's Lipschitz constant grows at most linearly in the gradient variable, uniqueness holds among all continuous viscosity solutions. Without that growth bound, uniqueness still holds if the solutions themselves are locally Lipschitz. The proofs adapt the classical doubling-of-variables technique but introduce a new penalty functional that produces the required coinvariantly smooth test functions. This approach generalizes earlier uniqueness theorems by weakening conditions on the Hamiltonian and allowing a larger solution class.

Core claim

We prove two uniqueness results for viscosity (generalized) solutions defined in terms of coinvariantly smooth test functionals and a dense family of compact subsets of the space of continuous functions.

Load-bearing premise

The Hamiltonian is continuous and satisfies a local Lipschitz condition in the functional variable with respect to the supremum norm; uniqueness then holds either under an additional sublinear growth condition on the Lipschitz constant in the gradient variable or under an extra local Lipschitz condition on the viscosity solution itself.

read the original abstract

We consider a path-dependent Hamilton--Jacobi equation with coinvariant derivatives over the space of continuous functions. We prove two uniqueness results for viscosity (generalized) solutions defined in terms of coinvariantly smooth test functionals and a dense family of compact subsets of the space of continuous functions. It is assumed that the Hamiltonian is continuous and satisfies a local Lipschitz condition in the functional variable with respect to the supremum norm. When the Lipschitz constant satisfies a sublinear growth condition in the gradient (impulse) variable, uniqueness is established in the class of continuous viscosity solutions. In the general case, without any such growth conditions, uniqueness is established in the class of continuous viscosity solutions that satisfy an additional local Lipschitz condition. The proofs are based on the standard method of doubling variables, but use a novel penalty functional for constructing coinvariantly smooth test functionals. The obtained results generalize previously known ones by relaxing the assumptions on the Hamiltonian and/or enlarging the class of functionals in which uniqueness is established.

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.

Referee Report

0 major / 3 minor

Summary. The paper proves two uniqueness theorems for viscosity solutions to path-dependent Hamilton-Jacobi equations with coinvariant derivatives on the space of continuous functions. Viscosity solutions are defined using coinvariantly smooth test functionals over a dense family of compact subsets. Under the assumptions that the Hamiltonian is continuous and locally Lipschitz in the functional variable with respect to the supremum norm, uniqueness holds for all continuous solutions when the Lipschitz constant satisfies a sublinear growth condition in the gradient variable; alternatively, uniqueness holds without the growth condition but only for solutions that are themselves locally Lipschitz. The proofs rely on a doubling-of-variables argument employing a novel penalty functional to construct suitable test functionals. The results relax prior assumptions on the Hamiltonian and/or enlarge the class of functionals considered.

Significance. If the central claims hold, the work advances the theory of path-dependent PDEs by providing uniqueness under weaker conditions than in the existing literature on coinvariant viscosity solutions. The novel penalty functional for generating coinvariantly smooth tests over compact families is a technical contribution that may apply to related problems in stochastic control and rough-path theory. The paper is a pure existence-uniqueness proof with clearly stated assumptions and no free parameters or ad-hoc normalizations.

minor comments (3)
  1. The introduction should explicitly compare the new penalty functional to those used in prior works on path-dependent HJ equations (e.g., the references cited for the doubling-variables method) to clarify the precise relaxation achieved.
  2. Notation for the coinvariant derivative and the family of compact sets should be introduced with a short example in §2 to aid readability for readers unfamiliar with the framework.
  3. In the statement of the main theorems, the precise dependence of the local Lipschitz constant on the compact set should be made explicit to avoid ambiguity when applying the results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for recommending minor revision. No specific major comments were provided in the report, so we have no points requiring detailed rebuttal or revision at this stage. We appreciate the recognition of our technical contributions, including the novel penalty functional and the relaxed assumptions.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is a pure mathematical proof paper establishing uniqueness theorems for viscosity solutions of path-dependent Hamilton-Jacobi equations. The derivation proceeds from explicit assumptions on the Hamiltonian (continuity and local Lipschitz conditions) and the definition of viscosity solutions via coinvariantly smooth test functionals over dense compact sets, using a doubling-variables argument with a novel penalty. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the results generalize prior work under relaxed hypotheses without importing uniqueness from the authors' own prior results as an unverified axiom. The proof chain is self-contained against the stated functional-analytic framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates entirely within the established framework of viscosity solutions for path-dependent equations; no free parameters are fitted, no new entities are postulated, and the axioms invoked are standard background results from functional analysis and PDE theory.

axioms (2)
  • standard math The space of continuous functions equipped with the supremum norm and the notion of coinvariant derivatives form a suitable setting for defining viscosity solutions.
    Invoked throughout the statement of the problem and the definition of solutions.
  • domain assumption Viscosity solutions are defined via test functionals that are coinvariantly smooth with respect to a dense family of compact sets.
    This is the specific notion of solution adopted in the paper.

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18 extracted references · 18 canonical work pages

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