arXiv subjects

Mark van den Bosch

Publications and source records attributed to Mark van den Bosch.

7 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↗

arXiv2025

Local Phase Tracking and Metastability of Planar Waves in Stochastic Reaction-Diffusion Systems

Planar travelling waves on $\mathbb R^d,$ with $ d\geq 2,$ are shown to persist in systems of reaction-diffusion equations with multiplicative noise on significantly long timescales with high probability, provided that the wave is orbitally stable in dimension one ($d=1$). While a global phase tracking mechanism is required to determine the location of the stochastically perturbed wave in one dimension, or on a cylindrical domain, we show that the travelling wave on the full unbounded space can be controlled by keeping track of local deviations only. In particular, the energy infinitesimally added to or withdrawn from the system by noise dissipates almost fully into the transverse direction, leaving behind small localised phase shifts. The noise process considered is white in time and coloured in space, possibly weighted, and either translation invariant or trace class.

math.AP↗

arXiv2025

Conditional Speed and Shape Corrections for Travelling Wave Solutions to Stochastically Perturbed Reaction-Diffusion Systems

In this work we perform rigorous small noise expansions to study the impact of stochastic forcing on the behaviour of planar travelling wave solutions to reaction-diffusion equations on cylindrical domains. In particular, we use a stochastic freezing approach that allows effective limiting information to be extracted concerning the behaviour of the stochastic perturbations from the deterministic wave. As an application, this allows us to provide a rigorous definition for the stochastic corrections to the wave speed. In addition, our approach allows their size to be computed to any desired order in the noise strength, provided that sufficient smoothness is available.

math.AP↗

arXiv2024

Notes on stochastic integration theory with respect to càdlàg semimartingales and a brief introduction to Lévy processes

The purpose of these notes is to distribute, mostly without proofs, fundamental definitions and results concerning the theory of semimartingales and stochastic integration. The material serves as a foundational guide for those interested in applying these concepts, particularly in the study of stochastic (functional) differential equations driven by Lévy processes. These notes are adapted from the preliminary chapter of the author's master's thesis (with only minor changes) and are intended to introduce newcomers to the essentials of càdlàg semimartingale theory while also discussing the advantages, limitations, and subtleties as compared to stochastic integration in the continuous setting.

math.PR↗

arXiv2024

Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations. In the deterministic setting, multidimensional stability of planar waves on the whole space $\mathbb R^d$ has been studied by many. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. This is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.

math.AP↗