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Onno van Gaans

Publications and source records attributed to Onno van Gaans.

14 recordsLinked to original sources
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 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.

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 It\^o processes with a negative drift.

math.PR↗
arXiv2025

A functional representation approach to vector lattice covers for spaces of compact operators

For ordered normed vector spaces $X, Y$, we consider the space $\mathcal{L}(X,Y)$ of bounded linear operators and characterize when its cone of positive operators has non-empty interior. When this is satisfied, we give a functional representation of the closure $\mathcal{C}(X,Y)$ of the finite rank operators in $\mathcal{L}(X,Y)$. This space is particularly interesting since it coincides in many cases with the space of compact operators from $X$ to $Y$. Our functional representation has very good order properties in the sense that it is a so-called vector lattice cover of $\mathcal{C}(X,Y)$. This can be used to characterize disjointness of operators in $\mathcal{C}(X,Y)$ and to determine which operators have a modulus in $\mathcal{C}(X,Y)$. We demonstrate how our results can be applied to a variety of concrete spaces.

math.FA↗
arXiv2024

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 L\'evy process.

math.DS↗
arXiv2023

Order theoretical structures in atomic JBW-algebras: disjointness, bands, and centres

Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors of type I. Extending Kadison's anti-lattice theorem, we show that each of these factors is a disjointness free anti-lattice. We characterise disjointness, bands, and disjointness preserving bijections with disjointness preserving inverses in direct sums of disjointness free anti-lattices and, therefore, in atomic JBW-algebras. We show that in unital JB-algebras the algebraic centre and the order theoretical centre are isomorphic. Moreover, the order theoretical centre is a Riesz space of multiplication operators. A survey of JBW-algebra factors of type I is included.

math.FA↗
arXiv2020

Dissipativity and positive off-diagonal property of operators on ordered Banach spaces

In this paper, we provide a sublinear function $p$ on ordered Banach spaces, which depends on the order structure of the space. With respect to this $p$, we study the relation between $p$-contractivity of positive semigroups and the $p$-dissipativity of its generators. The positive off-diagonal property of generators is also studied in ordered vector spaces.

math.FA↗
arXiv2019

On the linearity of order-isomorphisms

A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein-Avidan and Slomka to infinite dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.

math.FA↗
arXiv2018

Domination properties and extension of positive compact operators on pre-Riesz spaces

This paper concerns positive domination property of compact operators on pre-Riesz spaces. The method is embedding the pre-Riesz space to the Riesz completion. It extends the order continuous norms in pre-Riesz spaces to Riesz completions. The compactness of third power of a positive operator is obtained in a pre-Riesz space which has an order unit.

math.FA↗
arXiv2017

Monotone dynamical systems with dense periodic points

In this paper we prove a recent conjecture by M. Hirsch, which says that if $(f,Ω)$ is a discrete time monotone dynamical system, with $f\colon Ω\toΩ$ a homeomorphism on an open connected subset of a finite dimensional vector space, and the periodic points of $f$ are dense in $Ω$, then $f$ is periodic.

math.DS↗
arXiv2017

Disjointness preserving $\mathrm{C}_0$-semigroups and local operators on ordered Banach spaces

We generalize results concerning $\mathrm{C}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving $\mathrm{C}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local $\mathrm{C}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extensions of regular norms.

math.FA↗
arXiv2014

Bands in partially ordered vector spaces with order unit

In an Archimedean directed partially ordered vector space $X$ one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover $Y$ of $X$. If $X$ has an order unit, $Y$ can be represented as $C(Ω)$, where $Ω$ is a compact Hausdorff space. We characterize bands in $X$, and their disjoint complements, in terms of subsets of $Ω$. We also analyze two methods to extend bands in $X$ to $C(Ω)$ and show how the carriers of a band and its extensions are related. We use the results to show that in each $n$-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $\frac{1}{4}2^{2^n}$ for $n\geq 2$. We also construct examples of $(n+1)$-dimensional partially ordered vector spaces with ${2n\choose n}+2$ bands. This shows that there are $n$-dimensional partially ordered vector spaces that have more bands than an $n$-dimensional Archimedean vector lattice when $n\geq 4$.

math.FA↗
arXiv2005

Second derivatives of norms and contractive complementation in vector-valued spaces

We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $\ell_p(X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then an averaging operator. We apply our results to the space $\ell_p(\ell_q)$ with $p,q\in (1,2)\cup (2,\infty)$ and obtain a complete characterization of its 1-complemented subspaces.

math.FA↗