arXiv subjects

Julian Hölz

Publications and source records attributed to Julian Hölz.

5 recordsLinked to original sources

arXiv2025

Entropy based lower dimension bounds for finite-time prediction of Dynamic Mode Decomposition algorithms

Motivated by Dynamic Mode Decomposition algorithms, we provide lower bounds on the dimension of a finite-dimensional subspace $F \subseteq \mathrm{L}^2(\mathrm{X})$ required for predicting the behavior of dynamical systems over long time horizons. We distinguish between two cases: (i) If $F$ is determined by a finite partition of $X$ we derive a lower bound that depends on the dynamical measure-theoretic entropy of the partition. (ii) We consider general finite-dimensional subspaces $F$ and establish a lower bound for the dimension of $F$ that is contingent on the spectral structure of the Koopman operator of the system, via the approximation entropy of $F$ as studied by Voiculescu. Furthermore, we motivate the use of delay observables to improve the predictive qualities of Dynamic Mode Decomposition algorithms.

math.DS↗

arXiv2024

Ultra Feller operators from a functional analytic perspective

It is a widely acknowledged fact that the product of two positive strong Feller operators on a Polish space $E$ enjoys the ultra Feller property. We present a functional analytic proof of this fact that allows us to drop the assumption that the operators are positive and also extends the applicability of this result to more general state spaces. As it turns out, this result can be considered a variant of the theorem that on a Banach space with the Dunford--Pettis property, the product of two weakly compact operators is compact.

math.FA↗

arXiv2024

Uniform convergence of solutions to stochastic hybrid models of gene regulatory networks

In a recent paper by Kurasov, Lück, Mugnolo and Wolf, a hybrid gene regulatory network was proposed to model gene expression dynamics by using a stochastic system of coupled partial differential equations. This approach approximates protein counts in cells by modelling them as distributions. In a follow-up paper, the existence and strong convergence of the solutions to equilibrium was proven. In this paper, we make use of a recent convergence theorem for stochastic, irreducible semigroups that contain partial integral operators. In particular, we improve upon their results by showing that the convergence rate is independent of the initial distribution of proteins, therefore proving that the solutions converge not only strongly but even uniformly to equilibrium.

math.FA↗

arXiv2024

A Note on the Uniform Ergodicity of Dynamical Systems

We study the uniform ergodicity property for non-invertible topological and measure-preserving dynamical systems. It is shown that for topological dynamical systems uniform ergodicity is equivalent to eventually periodicity and that for measure preserving systems it is equivalent to periodicity. To obtain our results, we prove a result on the long-term behavior of lattice homomorphisms that have $1$ isolated in its spectrum.

math.DS↗

arXiv2023

Eventual cone invariance revisited

We consider finite-dimensional real vector spaces $X$ ordered by a closed cone $X_+$ with non-empty interior and study eventual nonnegativity of matrix semigroups $(e^{tA})_{t \ge 0}$ with respect to this cone. Our first contribution is the observation that, for general cones, one needs to distinguish between different notions of eventual nonnegativity: (i) uniform eventual nonnegativity means that $e^{tA}$ maps $X_+$ into $X_+$ for all sufficiently large times $t$; (ii) individual eventual nonnegativity means that for each $x \in X_+$ the vector $e^{tA}x$ is in $X_+$ for all $t$ larger than an $x$-dependent time $t_0$; and (iii) weak eventual nonnegativity means that for each $x \in X_+$ and each functional $x'$ in the dual cone $X'_+$ the value $\langle x', e^{tA} x \rangle$ is in $[0,\infty)$ for all $t$ larger than an $x$- and $x'$-dependent time $t_0$. Until now, only the first of these notions has been studied in the literature. We demonstrate by examples that, somewhat surprisingly for finite-dimensional spaces, all three notions are different. Our second contribution is to show that typical Perron-Frobenius like properties remain valid under the weakest of the above notions. Third, we study a strengthened form of the above mentioned concepts, namely eventual positivity. We prove that uniform, individual and weak versions of this property are - in contrast to the nonnegative case - equivalent, and that they can be characterized by spectral properties.

math.RA↗