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Existence of Invariant Measures for Delay Equations with Stochastic Negative Feedback

Mark van den Bosch, Onno W. van Gaans, Sjoerd M. Verduyn Lunel

TL;DR

The paper develops a rigorous framework for SDDEs with finite delay under stochastic perturbations, including Brownian and integrable Lévy noise, to establish the existence of invariant measures and stationary solutions. A central strategy combines transforming the system to enforce positivity, proving global existence and boundedness in probability, and applying Krylov–Bogoliubov tightness arguments for segment processes. The results are applied to delay models with negative feedback (notably Mackey–Glass and Nicholson’s blowflies), showing that nontrivial invariant measures exist under suitable conditions (e.g., $f(0)>0$) and that the Dirac measure at zero is not the only stationary state in certain regimes. The work provides a general, extensible probabilistic approach to ergodic behavior in stochastic delay systems and offers tools for analyzing both continuous and regulated Lévy perturbations, with implications for understanding long-run dynamics beyond deterministic chaos.

Abstract

We provide sufficient conditions for the existence of invariant probability measures for generic stochastic differential equations with finite time delay. Applications include 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 all solutions are bounded above in probability. It turns out that the occurrence of finite time blowups and boundedness in probability of solutions and solution segments are closely related. A non-trivial invariant measure is shown 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 an integrable Lévy process.

Existence of Invariant Measures for Delay Equations with Stochastic Negative Feedback

TL;DR

The paper develops a rigorous framework for SDDEs with finite delay under stochastic perturbations, including Brownian and integrable Lévy noise, to establish the existence of invariant measures and stationary solutions. A central strategy combines transforming the system to enforce positivity, proving global existence and boundedness in probability, and applying Krylov–Bogoliubov tightness arguments for segment processes. The results are applied to delay models with negative feedback (notably Mackey–Glass and Nicholson’s blowflies), showing that nontrivial invariant measures exist under suitable conditions (e.g., ) and that the Dirac measure at zero is not the only stationary state in certain regimes. The work provides a general, extensible probabilistic approach to ergodic behavior in stochastic delay systems and offers tools for analyzing both continuous and regulated Lévy perturbations, with implications for understanding long-run dynamics beyond deterministic chaos.

Abstract

We provide sufficient conditions for the existence of invariant probability measures for generic stochastic differential equations with finite time delay. Applications include 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 all solutions are bounded above in probability. It turns out that the occurrence of finite time blowups and boundedness in probability of solutions and solution segments are closely related. A non-trivial invariant measure is shown 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 an integrable Lévy process.
Paper Structure (38 sections, 51 theorems, 323 equations, 1 figure)

This paper contains 38 sections, 51 theorems, 323 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Key objectives
  3. Noisy systems
  4. Main results
  5. Bounded in probability
  6. Global existence and the permanence property
  7. Towards invariant measures
  8. Scope and outlook
  9. Organisation
  10. Existence and uniqueness of SDEs with finite time delay
  11. Autonomous equations
  12. About the functionals
  13. About the integrals
  14. Non-autonomous equations
  15. Boundedness in probability of solutions as well as their segments
  16. ...and 23 more sections

Key Result

Theorem 1.1

Suppose $f:\mathbb R\to \mathbb R$ is locally Lipschitz continuous, non-negative on $(0,\infty)$, and bounded from above. Assume $\inf_{t\geq 0}\gamma(t)>0$, $\sup_{t\geq 0}r(t)<\infty$ and $\sigma\geq 0$. Then the solution to where $c:C[-\tau,0]\to\mathbb R$ is bounded and locally Lipschitz with respect to the supremum norm $\|\cdot\|_\infty$, is unique, persists globally, and is bounded in prob

Figures (1)

  • Figure 1: On the left, we see an illustration of some change of time $\lambda\in\Lambda[-1,0]$. In the centre, we see two càdlàg functions $\psi$ and $\varphi$ in $D[-1,0]$ with $\|\varphi-\psi\|_\infty=7.75.$ The right illustration shows the graph of the two càdlàg functions $\psi$ and $\varphi\circ \lambda$ whose difference in supremum equals 0.6.

Theorems & Definitions (119)

  • Theorem 1.1: see §\ref{['Sec6']}
  • Corollary 1.2
  • Proposition 1.3
  • proof
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • ...and 109 more