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arxiv: 2605.14134 · v1 · pith:RSVGOWY5new · submitted 2026-05-13 · 🧮 math.DS · math.PR

Stochastic Mackey-Glass Equations and Other Negative Feedback Systems: Existence of Invariant Measures

Mark van den Bosch , Onno van Gaans , Sjoerd Verduyn Lunel This is my paper

Pith reviewed 2026-05-15 02:02 UTC · model grok-4.3

classification 🧮 math.DS math.PR
keywords stochastic differential equationsinvariant measuresMackey-Glass equationnegative feedback systemsLévy processesKrylov-Bogoliubov methodpopulation dynamicsdelay equations
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The pith

Non-trivial invariant measures exist for stochastic negative feedback systems if and only if solutions from at least one initial condition remain bounded away from zero in probability.

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

The paper establishes conditions for the existence of non-trivial invariant measures in stochastic versions of equations like the Mackey-Glass and Nicholson's blowflies models, each with multiplicative noise. Under mild assumptions, solutions persist globally and stay bounded above in probability. A non-trivial invariant measure is shown to exist precisely when there is at least one initial condition such that the solution stays bounded away from zero in probability. This is proved using the Krylov-Bogoliubov method for square integrable Lévy processes with finite intensity. Understanding these measures helps describe the long-term statistical behavior of these systems, which model biological processes with feedback and random perturbations.

Core claim

For stochastic negative feedback systems like the perturbed Mackey-Glass equation, a non-trivial invariant measure exists if and only if there exists at least one initial condition for which the corresponding solution remains bounded away from zero in probability. This characterization holds when the driving noise is a square integrable Lévy process with finite intensity, and solutions are globally persistent and upper bounded in probability under the paper's mild assumptions. The result is obtained via the Krylov-Bogoliubov method, with numerical simulations illustrating the measures and their link to long-term dynamics.

What carries the argument

The equivalence between existence of a non-trivial invariant measure and the existence of an initial condition yielding solutions bounded away from zero in probability, established through the Krylov-Bogoliubov method.

If this is right

  • Solutions that avoid approaching zero in probability admit an invariant measure that captures their asymptotic distribution.
  • The result extends to other negative feedback systems such as Nicholson's blowflies equation under similar noise conditions.
  • Global persistence and upper boundedness in probability are sufficient to apply the Krylov-Bogoliubov theorem for existence.
  • Numerical simulations can identify these invariant measures and confirm their relation to the system's behavior over time.

Where Pith is reading between the lines

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

  • This condition may help determine when multiplicative noise preserves positive population densities in biological models.
  • Similar techniques could apply to other stochastic delay differential equations in population dynamics.
  • Testing the bounded-away-from-zero condition numerically could predict the existence of invariant measures without full proof.
  • If the noise intensity increases, it might push solutions toward zero, eliminating the invariant measure.

Load-bearing premise

The mild assumptions that ensure global persistence and upper boundedness in probability for the solutions hold, along with the noise being a square integrable Lévy process with finite intensity.

What would settle it

A counterexample where all solutions approach zero in probability yet a non-trivial invariant measure still exists, or conversely where some solutions stay bounded away from zero but no invariant measure is found.

Figures

Figures reproduced from arXiv: 2605.14134 by Mark van den Bosch, Onno van Gaans, Sjoerd Verduyn Lunel.

Figure 1
Figure 1. Figure 1: Simulation of the deterministic equation (1.6) on the time interval [0 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulation of 100 sample paths of the solution to equation (1.7) on the time interval [0 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The same as in Figure 2 for [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

We study equations like the Mackey-Glass equations and Nicholson's blowflies equation, each perturbed by a (small) multiplicative noise term. Solutions to these stochastic negative feedback systems persist globally and are bounded above in probability under mild assumptions. A non-trivial invariant measure is proved to exist if and only if there is at least one initial condition for which the solution remains bounded away from zero in probability. The noise driving the dynamical system is allowed to be a square integrable L\'evy process with finite intensity. Existence of invariant measures is obtained via the Krylov-Bogoliubov method. In addition to our theoretical results, we present numerical simulations identifying the invariant measures obtained via the Krylov-Bogoliubov method and illustrating their connection to the system's long-term behaviour.

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

1 major / 3 minor

Summary. The paper establishes an if-and-only-if criterion for the existence of a non-trivial invariant measure for stochastic negative-feedback delay equations (including stochastic Mackey-Glass and Nicholson's blowflies models) driven by square-integrable Lévy noise with finite intensity. Under mild assumptions guaranteeing global existence, persistence, and upper boundedness in probability of the scalar process, the authors apply the Krylov-Bogoliubov theorem to show that a non-trivial invariant measure exists precisely when there is at least one initial condition for which the solution remains bounded away from zero in probability. Numerical simulations are included to illustrate the long-term behavior and the obtained measures.

Significance. If the central result holds, the work supplies a concrete, checkable criterion for invariant-measure existence in a biologically relevant class of stochastic delay equations, extending classical deterministic persistence results to the Lévy-driven setting. The iff character of the theorem and the allowance for finite-intensity Lévy noise are practically useful; the numerical component further aids verification of the long-term dynamics.

major comments (1)
  1. [Proof of the main existence theorem (Krylov-Bogoliubov application)] In the proof that applies the Krylov-Bogoliubov theorem (the main existence result), tightness of the family of averaged occupation measures μ_T in the space of probability measures on D([-τ,0],ℝ) equipped with the Skorohod topology is asserted on the basis of global boundedness in probability of the scalar process X(t). However, no uniform-in-t Kolmogorov-type moment bounds of the form E[sup_{|θ|≤τ} |X(t+θ)-X(t)|^p] are derived to control the modulus of continuity of the history segments X_t under the jumps of the multiplicative Lévy noise. Without such bounds the passage from marginal boundedness to tightness in the function space is not justified and constitutes a load-bearing gap.
minor comments (3)
  1. [Theorem statement] The statement of the main theorem would benefit from an explicit list of the precise assumptions on the feedback function f and the Lévy measure that are used to obtain global existence and the upper bound in probability.
  2. [Numerical simulations] In the numerical section the discretization scheme for the stochastic delay equation and the method used to approximate the invariant measure from long trajectories should be stated more precisely, including step-size and burn-in choices.
  3. [Preliminaries] Notation for the history segment X_t(·) and the Skorohod metric is introduced without a dedicated preliminary subsection; a short paragraph recalling the topology on D([-τ,0],ℝ) would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and constructive criticism of our manuscript. We address the single major comment below and will revise the paper to incorporate the requested justification.

read point-by-point responses

  1. Referee: [Proof of the main existence theorem (Krylov-Bogoliubov application)] In the proof that applies the Krylov-Bogoliubov theorem (the main existence result), tightness of the family of averaged occupation measures μ_T in the space of probability measures on D([-τ,0],ℝ) equipped with the Skorohod topology is asserted on the basis of global boundedness in probability of the scalar process X(t). However, no uniform-in-t Kolmogorov-type moment bounds of the form E[sup_{|θ|≤τ} |X(t+θ)-X(t)|^p] are derived to control the modulus of continuity of the history segments X_t under the jumps of the multiplicative Lévy noise. Without such bounds the passage from marginal boundedness to tightness in the function space is not justified and constitutes a load-bearing gap.

    Authors: We agree with the referee that the tightness argument in the application of the Krylov-Bogoliubov theorem requires additional justification to be fully rigorous. The manuscript currently relies on marginal boundedness in probability of X(t) but does not explicitly derive the uniform Kolmogorov moment bounds needed for the history segments in the Skorohod topology on D([-τ,0],ℝ). In the revised version we will add a new lemma (placed before the main theorem) that establishes these bounds. Because the driving Lévy noise has finite intensity, it is a compound Poisson process plus a drift; on any finite time interval there are only finitely many jumps almost surely. Combined with the global existence, persistence, and upper boundedness in probability already assumed in the paper, and using the Lipschitz continuity of the negative-feedback drift term, we obtain E[sup_{|θ|≤τ} |X(t+θ)-X(t)|^p] ≤ C_p uniformly in t for a suitable p>0 (e.g., p=1). These estimates, together with the marginal tightness, verify the standard tightness criterion for cadlag processes (e.g., Billingsley, Convergence of Probability Measures, Theorem 15.6). The main existence result and its if-and-only-if character remain unchanged; only the proof is completed. revision: yes

Circularity Check

0 steps flagged

No circularity; standard Krylov-Bogoliubov application under external assumptions

full rationale

The derivation applies the classical Krylov-Bogoliubov theorem to averaged occupation measures once global existence, upper boundedness in probability, and the persistence-away-from-zero condition are established from the stochastic delay equation and Lévy noise assumptions. These inputs are independent of the target invariant measure; the iff statement is a characterization of when the theorem yields a non-trivial measure rather than a self-definition. No fitted parameters, self-citation chains, or ansatz smuggling appear in the load-bearing steps. The result is self-contained against the external theorem and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background results from stochastic analysis and delay equations without introducing new free parameters, axioms beyond domain standards, or invented entities.

axioms (2)
  • standard math Existence and properties of solutions to stochastic functional differential equations driven by square integrable Lévy processes with finite intensity
    Invoked to establish global persistence and boundedness under mild assumptions.
  • standard math Krylov-Bogoliubov theorem for existence of invariant measures from tightness conditions
    Used as the main tool to obtain the invariant measure from the boundedness condition.

pith-pipeline@v0.9.0 · 5440 in / 1230 out tokens · 70897 ms · 2026-05-15T02:02:29.222378+00:00 · methodology

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