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arxiv: 2305.18981 · v3 · pith:XMNG4TRWnew · submitted 2023-05-30 · 🧮 math.AP · math.OC· math.PR

Convergence of infinitesimal generators and stability of convex monotone semigroups

Jonas Blessing , Michael Kupper , Max Nendel This is my paper

Pith reviewed 2026-05-24 08:30 UTC · model grok-4.3

classification 🧮 math.AP math.OCmath.PR
keywords convex monotone semigroupsinfinitesimal generatorsmixed topologystabilitycomparison principleΓ-generatorLipschitz functionsHamilton-Jacobi-Bellman equations
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The pith

Convergence of infinitesimal generators in the mixed topology implies stability of convex monotone semigroups.

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

The paper establishes that convergence of the infinitesimal generators of convex monotone semigroups in the mixed topology yields convergence of the semigroups themselves. This holds for strongly continuous semigroups acting on spaces of continuous functions. The proof relies on a comparison principle that uses the Γ-generator restricted to the Lipschitz set to uniquely determine each semigroup, avoiding viscosity solution techniques. The same framework covers both time and space discretizations and directly yields stability statements for several classes of approximations. A reader would care because the result transfers generator-level information to the full semigroup dynamics in settings that include optimal control and nonlinear PDEs.

Core claim

Based on the convergence of their infinitesimal generators in the mixed topology, the paper provides a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions. The result relies on a comparison principle that uniquely determines the semigroup via its Γ-generator on the Lipschitz set, resembling the classical linear analogue.

What carries the argument

The Γ-generator defined on the Lipschitz set, which with the comparison principle uniquely identifies the semigroup and transfers generator convergence to semigroup convergence.

If this is right

  • Stability holds for Euler schemes and Yosida-type approximations of upper envelopes of linear semigroups.
  • Finite-difference schemes for convex HJB equations inherit stability from generator convergence.
  • Freidlin-Wentzell-type large-deviation results and Markov chain approximations are stable for a class of stochastic optimal control problems.
  • Continuous-time Markov processes with uncertain transition probabilities admit stability statements under the same generator condition.

Where Pith is reading between the lines

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

  • The generator-convergence approach could extend to other classes of nonlinear semigroups once an analogous comparison principle is available.
  • In applications, verifying mixed-topology convergence of generators for concrete discretizations would directly certify the corresponding semigroup approximations.
  • The method may connect to monotone-operator theory by treating the Γ-generator as a nonlinear analogue of the classical generator.

Load-bearing premise

The comparison principle must hold so that the Γ-generator on the Lipschitz set uniquely determines the semigroup.

What would settle it

A pair of distinct convex monotone semigroups whose generators converge in the mixed topology but whose semigroups themselves fail to converge would falsify the stability claim.

read the original abstract

Based on the convergence of their infinitesimal generators in the mixed topology, we provide a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions. In contrast to previous results, we do not rely on the theory of viscosity solutions but use a recent comparison principle which uniquely determines the semigroup via its $\Gamma$-generator defined on the Lipschitz set and therefore resembles the classical analogue from the linear case. The framework also allows for discretizations both in time and space and covers a variety of applications. This includes Euler schemes and Yosida-type approximations for upper envelopes of families of linear semigroups, stability results and finite-difference schemes for convex HJB equations, Freidlin-Wentzell-type results and Markov chain approximations for a class of stochastic optimal control problems and continuous-time Markov processes with uncertain transition probabilities.

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 establishes a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions, obtained from convergence of their infinitesimal generators (specifically Γ-generators) in the mixed topology. It replaces viscosity solution theory with a recent comparison principle that uniquely determines the semigroup from its Γ-generator on the Lipschitz set, thereby mirroring the classical linear case more directly. The framework is shown to accommodate time and space discretizations and is applied to Euler schemes, Yosida approximations for upper envelopes of linear semigroups, finite-difference schemes for convex HJB equations, Freidlin-Wentzell-type results, and Markov chain approximations for stochastic optimal control and processes with uncertain transitions.

Significance. If the central claims hold, the result supplies a direct, comparison-principle-based route to semigroup stability that parallels the linear theory and avoids the technical overhead of viscosity solutions. The explicit coverage of both continuous and discrete approximations, together with concrete applications to HJB equations and stochastic control, indicates that the framework can support numerical analysis and approximation theory in nonlinear settings. The parameter-free character of the comparison principle (as described) and the absence of ad-hoc fitting strengthen the result's generality.

minor comments (3)
  1. [Abstract / Introduction] The abstract and introduction should explicitly recall the definition of the mixed topology and the precise domain of the Γ-generator (Lipschitz set) to make the comparison with the linear case immediate for readers outside the immediate subfield.
  2. [Applications] In the applications section, each discretization (Euler, Yosida, finite-difference, Markov chain) should contain a one-sentence pointer back to the precise hypothesis of the main stability theorem that is being verified.
  3. [Preliminaries] Notation for the space of continuous functions (e.g., whether bounded or unbounded, topology) should be fixed at first use and used consistently thereafter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no point-by-point responses to provide. We remain available to incorporate any minor editorial changes the editor may request.

Circularity Check

0 steps flagged

No significant circularity; stability follows from external comparison principle

full rationale

The derivation proceeds from convergence of infinitesimal generators in the mixed topology to semigroup stability. The key uniqueness device is an external recent comparison principle that pins down the semigroup via its Γ-generator on the Lipschitz set; this is invoked as an independent input rather than derived or fitted inside the paper. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citation chains appear. The abstract and structure treat the comparison principle as a black-box external fact analogous to the linear case, leaving the generator-convergence step as the novel contribution with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates in the standard setting of strongly continuous semigroups on continuous functions and invokes a recent comparison principle; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Semigroups under consideration are strongly continuous, convex, and monotone on spaces of continuous functions.
    This defines the class of objects for which the stability result is claimed.
  • domain assumption A recent comparison principle uniquely determines the semigroup via its Γ-generator on the Lipschitz set.
    This is the key tool replacing viscosity solutions and is invoked to establish uniqueness and stability.

pith-pipeline@v0.9.0 · 5668 in / 1246 out tokens · 22416 ms · 2026-05-24T08:30:18.140570+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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