arXiv subjects

Sjoerd Verduyn Lunel

Publications and source records attributed to Sjoerd Verduyn Lunel.

5 recordsLinked to original sources

arXiv2026

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

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 Lévy process.

math.DS↗

arXiv2026

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

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 Lévy 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.

math.DS↗

arXiv2026

Stochastic Wright's Equation: Existence of Invariant Measures

Wright's delay differential equation is one of the prime examples of a fully nonlinear equation without an explicit solution and whose dynamics can be understood by analytic means. In this paper, we introduce stochastic perturbations by adding Brownian noise with a bounded Lipschitz noise coefficient to a transformed version of Wright's equation. The transformation considered plays an important role in the deterministic theory as well. We demonstrate that this stochastically perturbed equation has (at least) two invariant measures: a trivial measure concentrated at $-1$ and a nontrivial measure on $(-1,\infty)$. The crucial and most challenging step of the proof is showing that every solution is bounded away from $-1$ in probability. In addition, a major part of our analysis is devoted to deriving detailed estimates for Itô processes with a negative drift.

math.PR↗

arXiv2021

Twin semigroups and delay equations

In the standard theory of delay equations, the fundamental solution does not 'live' in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with "retarded" equations, and in Part III we show how the Stieltjes integral enables incorporation of unbounded perturbations corresponding to neutral delay equations.

math.DS↗

arXiv2020

Pseudospectral approximation of Hopf bifurcation for delay differential equations

Pseudospectral approximation reduces DDE (delay differential equations) to ODE (ordinary differential equations). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyse certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity.

math.DS↗