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arxiv: 2605.05670 · v1 · submitted 2026-05-07 · 🧮 math.AP · math.DS

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On the inhomogeneous discounted Hamilton-Jacobi equations

Liang Jin , Jun Yan , Kai Zhao
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Pith reviewed 2026-05-08 07:35 UTC · model grok-4.3

classification 🧮 math.AP math.DS
keywords inhomogeneous discounted Hamilton-Jacobi equationviscosity solutionasymptotic stabilityMather measurecritical valuesolution semigroupergodic measureclosed manifold
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The pith

The inhomogeneous discounted Hamilton-Jacobi equation admits an asymptotically stable viscosity solution if and only if the constant c exceeds a critical value c0.

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

The paper studies the family of equations λ(x)u + h(x, d_x u) = c on a closed manifold, where λ is a continuous discount factor that does not vanish identically. It identifies a threshold c0 such that viscosity solutions exist precisely when c > c0. When this holds, the equation possesses an asymptotically stable solution whose basin can be described, and the long-time evolution of the associated solution semigroup is controlled by integrals of λ taken with respect to Mather measures. These integrals also govern the slowest convergence rate and imply a specific concentration behavior for the Mather measures themselves as c tends to infinity. For c at or above c0 the ergodic Mather measures are classified and located in phase space. The results link stability of solutions to ergodic theory on the cotangent bundle.

Core claim

The equation admits an asymptotically stable solution if and only if c > c0. In this case the lowest convergence rate equals the integral of λ over Mather measures, which yields an asymptotic behavior of Mather measures as c tends to infinity. Assume c ≥ c0 and the equation admits a solution; then ergodic Mather measures are classified and their distribution in the phase space is located.

What carries the argument

The critical threshold c0 together with the Mather measures of the Hamiltonian, whose integrals against the discount factor λ determine stability, convergence rates, and long-time asymptotics of solutions.

If this is right

  • Viscosity solutions exist exactly when c exceeds c0 and fail to exist below it.
  • The basin of attraction of the stable solution is fully determined by the dynamics of the Hamiltonian.
  • The solution semigroup converges at a rate given by the infimum of integrals of λ against Mather measures.
  • Mather measures concentrate asymptotically as c becomes large, according to the weighting by λ.
  • When solutions exist, all ergodic Mather measures are identified and placed in explicit regions of the phase space.

Where Pith is reading between the lines

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

  • The stability threshold and convergence formula may carry over to time-dependent or stochastic perturbations of the same equation.
  • The link between convergence rate and Mather-measure integrals suggests a way to approximate long-time behavior without solving the PDE directly.
  • The classification of ergodic measures could inform averaging or homogenization procedures for related Hamilton-Jacobi equations on manifolds.
  • Numerical schemes that track Mather measures might be used to locate the stable solution for large c.

Load-bearing premise

The discount factor λ is continuous and not identically zero on the closed manifold, while the Hamiltonian satisfies the standard conditions that guarantee existence of Mather measures.

What would settle it

An explicit example on the circle or torus where c > c0 yet no asymptotically stable solution exists, or a numerical computation showing the observed convergence rate differs from the integral of λ over the corresponding Mather measures.

read the original abstract

In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \begin{equation}\label{hjs1} \lambda(x)u+h(x,d_x u)=c \quad \tag{$\ast$} \end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $\lambda(x)$. There is a critical value $c_0\in[-\infty,\infty)$ such that \eqref{hjs1} admits a viscosity solution if $c>c_0$ and no solution if $c<c_0$. Inspired by the recent development [34] on the stability theory of viscosity solution, we show that the equation admits an asymptotically stable solution if and only if $c>c_0$. In this case, we determine the basin of the stable solution and investigate the long time behavior of the solution semigroup associated to \eqref{hjs1}. In particular, we relate the lowest convergence rate to the integral of $\lambda$ over Mather measures, which leads to an asymptotic behavior of Mather measures when $c$ goes to infinity. Assume $c\geqslant c_0$ and the equation admits a solution, we classify ergodic Mather measures and locate their distribution in the phase space.

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 studies the inhomogeneous discounted Hamilton-Jacobi equation λ(x)u + h(x, d_x u) = c on a closed manifold M with continuous, non-identically zero discount factor λ. It identifies a critical value c0 such that viscosity solutions exist precisely for c > c0. Building on stability theory from [34], it proves that an asymptotically stable viscosity solution exists if and only if c > c0, determines the basin of attraction, analyzes the long-time behavior of the associated solution semigroup, shows that the lowest convergence rate equals the infimum of ∫_M λ dμ over Mather measures (yielding asymptotics of those measures as c → ∞), and classifies ergodic Mather measures together with their phase-space distribution when c ≥ c0.

Significance. If the derivations hold, the work supplies a sharp if-and-only-if stability criterion and an explicit ergodic-theoretic formula for convergence rates in the variable-discount setting. The explicit link between the minimal rate and Mather integrals, together with the classification of ergodic measures, strengthens the connection between viscosity-solution theory and weak KAM/Mather theory; these are concrete, falsifiable statements that could be useful for further analysis of long-time behavior in discounted control problems.

minor comments (3)
  1. [§2] §2 (definition of c0): the critical value is introduced via the existence threshold, but an explicit variational characterization (e.g., via the Mather α-function or a minimax formula) would make the subsequent stability statements easier to follow without repeated cross-reference to [34].
  2. [Main stability theorem] The statement of the main stability theorem (presumably Theorem 1.1 or 3.1) invokes the semigroup analysis from [34] but does not list the precise hypotheses of the cited result that are being verified; a short paragraph confirming that λ ∈ C(M) and the standard convexity/ coercivity assumptions on h suffice would remove any ambiguity.
  3. [Throughout] Notation: the Hamiltonian is written h(x, d_x u) in the abstract and (∗), yet later passages occasionally use p as the momentum variable; a single consistent symbol throughout the text and in the statement of the Mather-measure integrals would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The provided summary accurately reflects the main results: the identification of the critical value c0 for existence of viscosity solutions to the inhomogeneous discounted Hamilton-Jacobi equation, the if-and-only-if criterion for asymptotic stability when c > c0, the determination of the basin of attraction, the analysis of the solution semigroup, the explicit formula for the minimal convergence rate in terms of integrals of λ over Mather measures, and the classification of ergodic Mather measures for c ≥ c0. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs the critical value c0 as the threshold for existence of viscosity solutions to the discounted HJ equation, a standard definition in the literature. It then applies prior stability results from reference [34] to prove asymptotic stability precisely when c > c0, and identifies the convergence rate as the infimum of integrals of λ against Mather measures using established weak KAM theory. No quoted step equates a claimed prediction or stability property to a fitted parameter or self-defined quantity by construction; the arguments rely on independently verifiable existence and stability theorems for Mather measures on closed manifolds rather than reducing the central claims to tautological inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence and properties of Mather measures for the underlying Hamiltonian flow and on the stability theory developed in the cited reference [34]. No new free parameters are introduced; c0 is defined as the infimum of admissible constants rather than fitted. No new entities are postulated.

axioms (2)
  • domain assumption The Hamiltonian satisfies the standard convexity and coercivity conditions that guarantee the existence of Mather measures (Aubry-Mather theory).
    Invoked to define the measures whose integrals give the convergence rate.
  • standard math Viscosity solution theory and comparison principles hold for the inhomogeneous discounted equation on a closed manifold.
    Required for the existence and uniqueness statements.

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