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arxiv: 2501.00141 · v2 · pith:27N2DN2Vnew · submitted 2024-12-30 · 🧮 math.DS · math.PR

Existence of Invariant Probability Measures for Stochastic Differential Equations with Finite Time Delay

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

Pith reviewed 2026-05-23 06:46 UTC · model grok-4.3

classification 🧮 math.DS math.PR
keywords invariant probability measuresstochastic delay differential equationsKrylov-Bogoliubov methodone-sided boundLévy noiseMackey-Glass equationWright equation
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The pith

Sufficient conditions on the deterministic coefficient guarantee invariant probability measures for stochastic differential equations with finite time delay.

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

The paper gives conditions under which stochastic differential equations that include a finite time delay possess an invariant probability measure. It uses the Krylov-Bogoliubov method and shows that when the deterministic term obeys a one-sided bound, any solution bounded in probability automatically makes its entire history segment bounded in probability as well. This reduces the existence question to the simpler task of locating at least one bounded solution. The result covers a broad class of systems, including the stochastic Mackey-Glass and Wright equations, and holds when the driving noise is any integrable Lévy process. A reader would care because the reduction turns an infinite-dimensional problem into a check that can often be performed directly on sample paths.

Core claim

For generic stochastic differential equations with finite time delay, the Krylov-Bogoliubov method yields sufficient conditions for the existence of invariant probability measures. When the deterministic coefficient satisfies a one-sided bound, boundedness in probability of the solution X(t) transfers to boundedness in probability of the solution segment X_t. Therefore an invariant measure exists whenever at least one solution remains bounded in probability. The noise may be any integrable Lévy process, and the conclusion applies in particular to the stochastic Mackey-Glass equation and the stochastic Wright equation.

What carries the argument

The Krylov-Bogoliubov method applied to the solution-segment process, together with the one-sided bound on the deterministic coefficient that transfers boundedness in probability from X(t) to X_t.

If this is right

  • Existence of an invariant measure reduces to the existence of one bounded-in-probability solution for every system whose deterministic part obeys the one-sided bound.
  • The same conclusion holds when the driving noise is any integrable Lévy process.
  • The stochastic Mackey-Glass equation and the stochastic Wright equation therefore possess invariant measures once a single bounded solution is identified.
  • Boundedness in probability of the pointwise solution automatically produces boundedness in probability of the history segment.

Where Pith is reading between the lines

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

  • The same one-sided bound technique could be tested on other biological or physical models that naturally satisfy comparable growth restrictions.
  • Adding a contraction or dissipativity condition might allow the same argument to prove uniqueness of the invariant measure.
  • Direct numerical sampling of paths could serve as a practical test for the existence of a bounded solution and hence of the invariant measure.

Load-bearing premise

The deterministic coefficient must satisfy a one-sided bound that transfers boundedness in probability from the value X(t) to the full history segment X_t.

What would settle it

An explicit stochastic delay equation whose deterministic coefficient obeys a one-sided bound, that possesses at least one solution bounded in probability, yet admits no invariant probability measure, would falsify the main claim.

Figures

Figures reproduced from arXiv: 2501.00141 by Mark van den Bosch, Onno van Gaans, Sjoerd Verduyn Lunel.

Figure 1
Figure 1. Figure 1: On the left, we see an illustration of some change of time [PITH_FULL_IMAGE:figures/full_fig_p051_1.png] view at source ↗
read the original abstract

We provide sufficient conditions for the existence of invariant probability measures for generic stochastic differential equations with finite time delay. This is achieved by means of the Krylov-Bogoliubov method. Furthermore, we focus on stochastic delay equations whose deterministic coefficient satisfies a one-sided bound, which enables us to show that boundedness in probability of a solution $X(t)$ entails boundedness in probability of its solution segment $X_t$. This implies that for a large set of systems, we can infer that an invariant measure exists if only there is at least one solution that is bounded in probability. Applications include, but are not limited to, the stochastic Mackey-Glass equations and the stochastic Wright's equation. The noise driving the dynamical system is allowed to be an integrable L\'evy process.

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 / 0 minor

Summary. The manuscript establishes sufficient conditions for the existence of invariant probability measures for stochastic differential equations with finite time delay, driven by integrable Lévy processes. It applies the Krylov-Bogoliubov theorem and focuses on the case where the deterministic coefficient satisfies a one-sided bound; under this condition, boundedness in probability of the solution process X(t) is shown to imply boundedness in probability of the solution segment X_t. This reduces the existence question to the existence of at least one bounded solution. Applications to the stochastic Mackey-Glass equation and stochastic Wright's equation are indicated.

Significance. If the transfer of boundedness in probability holds rigorously, the result would supply a concrete, checkable criterion for invariant measures in a broad family of delay SDEs under one-sided conditions on the drift, extending the classical Krylov-Bogoliubov approach to the delay setting with general Lévy noise. The reduction to a single bounded solution is potentially useful for applications, provided the one-sided bound suffices for the segment.

major comments (1)
  1. [Abstract and the main boundedness-transfer argument] Abstract (statement on the one-sided bound and the implication for X_t): the claim that boundedness in probability of X(t) entails boundedness in probability of the segment X_t under the one-sided bound alone requires explicit justification for càdlàg solutions. A jump of the driving Lévy process at time t can make the sup-norm of X_t over [-τ,0] arbitrarily large while leaving the post-jump value X(t) controlled; the one-sided inner-product inequality does not automatically supply uniform integrability or moment control on the Lévy measure to prevent this. This step is load-bearing for the central reduction that existence follows from a single bounded solution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the valuable comment regarding the boundedness-transfer argument. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract and the main boundedness-transfer argument] Abstract (statement on the one-sided bound and the implication for X_t): the claim that boundedness in probability of X(t) entails boundedness in probability of the segment X_t under the one-sided bound alone requires explicit justification for càdlàg solutions. A jump of the driving Lévy process at time t can make the sup-norm of X_t over [-τ,0] arbitrarily large while leaving the post-jump value X(t) controlled; the one-sided inner-product inequality does not automatically supply uniform integrability or moment control on the Lévy measure to prevent this. This step is load-bearing for the central reduction that existence follows from a single bounded solution.

    Authors: We appreciate the referee pointing out the need for explicit justification in the càdlàg case. The manuscript establishes the implication in the proof of the main theorem by leveraging the one-sided bound to obtain uniform estimates on the solution segments. Regarding the jump concern, note that any jump at time t directly impacts X(t), so the boundedness in probability of X(t) already constrains the size of possible jumps at t. For jumps in the delay interval, the one-sided condition and the integrability of the Lévy process ensure that the probability of large historical values is controlled uniformly, as large past jumps would lead to growth that violates the bound at the current time unless the measure is controlled. We will, however, revise the paper to include a more detailed remark or appendix explaining this step explicitly for càdlàg paths to make the argument fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external Krylov-Bogoliubov theorem under stated one-sided bound assumption

full rationale

The paper's central claim rests on applying the standard Krylov-Bogoliubov method to obtain existence of invariant measures once a solution is shown to be bounded in probability. The one-sided bound is introduced as an explicit hypothesis on the deterministic coefficient, not derived from the target result. Boundedness transfer from X(t) to the segment X_t is presented as a consequence of this hypothesis plus the equation structure, without any self-referential fitting, renaming, or load-bearing self-citation chain. No equations or steps reduce the existence statement to a tautology or to a parameter fitted from the same data. The argument is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from stochastic analysis and the theory of delay equations; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • standard math Standard existence and uniqueness assumptions for solutions of stochastic delay differential equations driven by Lévy processes
    Implicit in the setup of the equations considered.
  • domain assumption The noise is an integrable Lévy process
    Explicitly stated as allowed in the abstract.

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