arXiv subjects

Peter E. Kloeden

Publications and source records attributed to Peter E. Kloeden.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of the dynamics of Caputo fractional differential equations

It is known that a finite-dimensional Caputo fractional differential equation, though itself need not generate a semiflow, can be represented as a Volterra integral equation which generates an infinite-dimensional semiflow on the space $\mathfrak{C}=C([0,\infty); \mathbb{R}^d)$ under the standard compact-open topology. In this paper we construct a compact absorbing set and an attractor for this semiflow on $\mathfrak{C}$, and then prove that the attractor consists of equi globally Hölder continuous functions. This strengthens the previous work of Doan \& Kloeden \cite{DK21} where a bounded (with respect to a weighted norm) attractor was constructed.

math.DS↗

arXiv2025

Strong order-one convergence of the Euler method for random ordinary differential equations driven by semi-martingale noises

It is well known that the Euler method for a random ordinary differential equation $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$ driven by a stochastic process $\{Y_t\}_t$ with $θ$-Hölder sample paths is estimated to be of strong order $θ$ with respect to the time step, provided $f=f(t, x, y)$ is sufficiently regular and with suitable bounds. This order is known to increase to $1$ in some special cases. Here, it is proved that, in many more typical cases, further structures on the noise can be exploited so that the strong convergence is of order 1. In fact, we prove so for any semi-martingale noise. This includes Itô diffusion processes, point-process noises, transport-type processes with sample paths of bounded variation, and time-changed Brownian motion. The result follows from estimating the global error as an iterated integral over both large and small mesh scales, and by switching the order of integration to move the critical regularity to the large scale. The work is complemented with numerical simulations showing the optimality of the strong order 1 convergence in those cases, and with an example with fractional Brownian motion noise with Hurst parameter $0 < H < 1/2,$ which is not a semi-martingale and for which the order of convergence is $H + 1/2$, hence lower than the attained order 1 in the semi-martingale case, but still higher than the order $H$ of convergence expected from previous works.

math.PR↗

arXiv2025

Fokker-Planck equation for stochastic heat equations

This work is devoted to the study of the Fokker--Planck equation for a stochastic heat equation with an additive $Q$-Wiener noise and non-homogeneous boundary conditions. We explicitly construct the probability density function and establish the associated Fokker--Planck equation by applying the eigenfunction expansion technique. Moreover, the Feynman--Kac formula is used to obtain the probabilistic representation of the solution. The analysis is further extended to cases with multiplicative noise involving nonlocal diffusion operators under homogeneous boundary conditions, as well as the corresponding Kardar--Parisi--Zhang (KPZ) equation. Notably, the evolution of the probability density function for the stochastic heat equation depends critically on the spatial location.

math.PR↗

arXiv2024

Sigmoidal approximations of a nonautonomous neural network with infinite delay and Heaviside function

In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attractors in both cases, and the convergence of the attractors of the sigmoidal models to those of the Heaviside inclusion.

math.DS↗

arXiv2023

Random attractors of a stochastic Hopfield neural network model with delays

The global asymptotic behavior of a stochastic Hopfield neural network model (HNNM) with delays is explored by studying the existence and structure of random attractors. It is first proved that the trajectory field of the stochastic delayed HNNM admits an almost sure continuous version, which is compact for $t>τ$ (where $τ$ is the delay) by a delicate construction based on the random semiflow generated by the diffusion term. Then, this version is shown to generate a random dynamical system (RDS) by piece-wise linear approximation, after which the existence of a random absorbing set is obtained by a careful uniform apriori estimate of the solutions. Subsequently, the pullback asymptotic compactness of the RDS generated by the stochastic delayed HNNM is proved and hence the existence of random attractors is obtained. Moreover, sufficient conditions under which the attractors turn out to be an exponential attracting stationary solution are given. Numerical simulations are also conducted at last to illustrate the effectiveness of the established results.

math.DS↗

arXiv2022

Numerical Dynamics of Integrodifference Equations: Forward Dynamics and Pullback Attractors

In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior need to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pullback attractors, as well as forward limit sets persist and that the latter two notions even converge under perturbation. As concrete application, we study integrodifference equation under spatial discretization of collocation type.

math.DS↗

arXiv2022

Forward and Pullback Dynamics of Nonautonomous Integrodifference Equations: Basic Constructions

In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture their long-term behaviour comprehensively, both pullback and forward attractors, as well as forward limit sets are constructed for general infinite-dimensional nonautonomous dynamical systems in discrete time. While the theory of pullback attractors, but not their application to integrodifference equations, is meanwhile well-established, the present novel approach is needed in order to understand their future behaviour.

math.DS↗

arXiv2021

Attractors of Caputo fractional differential equations with triangular vector fields

It is shown that the attractor of an autonomous Caputo fractional differential equation of order $α\in(0,1)$ in $\mathbb{R}^d$ whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of "N. D. Cong and H.T. Tuan, Generation of nonlocal fractional dynamical systems by fractional differential equations. Journal of Integral Equations and Applications, 29 (2017), 1-24" which shows that no two solutions of such a Caputo FDE can intersect in finite time

math.CA↗

arXiv2020

Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations

Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence rates for full-discrete approximations of stochastic Allen-Cahn equations with space-time white noise which have recently been obtained in [Becker, S., Gess, B., Jentzen, A., and Kloeden, P. E., Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. arXiv:1711.02423 (2017)].

math.PR↗

arXiv2019

Semi-dynamical systems generated by autonomous Caputo fractional differential equations

An autonomous Caputo fractional differential equation of order $α\in(0,1)$ in $\mathbb{R}^d$ whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space $\mathfrak{C}$ of continuous functions $f:\R^+\rightarrow \R^d$ with the topology uniform convergence on compact subsets. This contrasts with a recent result of Cong \& Tuan \cite{cong}, which showed that such equations do not, in general, generate a dynamical system on the space $\mathbb{R}^d$.

math.CA↗

arXiv2019

Strong $(L^2,L^γ\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension

In this paper, we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary $p>2$ order nonlinearity and in any space dimension $N\geq 1$. It is proved that the weak solutions can be $(L^2, L^γ\cap H_0^1)$-continuous in initial data for any $γ\geq 2$ (independent of the physical parameters of the system), i.e., can converge in the norm of any $L^γ\cap H_0^1$ as the corresponding initial values converge in $L^2$. Applying this to the global attractor we find that, with external forcing only in $ L^2$, the attractor $\mathscr{A}$ attracts bounded subsets of $L^2$ in the norm of any $L^γ\cap H_0^1$, and that every translation set $\mathscr{A}-z_0$ of $\mathscr{A}$ for any $z_0 \in \mathscr{A}$ is a finite dimensional compact subset of $L^γ\cap H_0^1$. The main technique we employ is a combination of the mathematical induction and a decomposition of the nonlinearity by which the continuity result is strengthened to $(L^2, L^γ\cap H_0^1)$-continuity and, since interpolation inequalities are avoided, the restriction on space dimension is removed.

math.DS↗

arXiv2017

Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations

The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities. In particular, in the scientific literature there exists neither a result which proves strong convergence rates nor a result which proves weak convergence rates for full-discrete numerical approximations of stochastic Allen-Cahn equations. In this article we bridge this gap and establish strong convergence rates for full-discrete numerical approximations of space-time white noise driven SPDEs with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. Moreover, we also establish lower bounds for strong temporal and spatial approximation errors which demonstrate that our strong convergence rates are essentially sharp and can, in general, not be improved.

math.PR↗

arXiv2014

The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor

The dichotomy spectrum is introduced for linear mean-square random dynamical systems, and it is shown that for finite-dimensional mean-field stochastic differential equations, the dichotomy spectrum consists of finitely many compact intervals. It is then demonstrated that a change in the sign of the dichotomy spectrum is associated with a bifurcation from a trivial to a non-trivial mean-square random attractor.

math.DS↗

arXiv2013

An exponential Wagner-Platen type scheme for SPDEs

The strong numerical approximation of semilinear stochastic partial differential equations (SPDEs) driven by infinite dimensional Wiener processes is investigated. There are a number of results in the literature that show that Euler-type approximation methods converge strongly, under suitable assumptions, to the exact solutions of such SPDEs with strong order 1/2 or at least with strong order 1/2 - epsilon where epsilon > 0 is arbitrarily small. Recent results extend these results and show that Milstein-type approximation methods converge, under suitable assumptions, to the exact solutions of such SPDEs with strong order 1 - epsilon. It has also been shown that splitting-up approximation methods converge, under suitable assumptions, with strong order 1 to the exact solutions of such SPDEs. In this article an exponential Wagner-Platen type numerical approximation method for such SPDEs is proposed and shown to converge, under suitable assumptions, with strong order 3/2 - epsilon to the exact solutions of such SPDEs.

math.NA↗

arXiv2013

Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations

The Euler-Maruyama scheme is known to diverge strongly and numerically weakly when applied to nonlinear stochastic differential equations (SDEs) with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. Classical Monte Carlo simulations do, however, not suffer from this divergence behavior of Euler's method because this divergence behavior happens on rare events. Indeed, for such nonlinear SDEs the classical Monte Carlo Euler method has been shown to converge by exploiting that the Euler approximations diverge only on events whose probabilities decay to zero very rapidly. Significantly more efficient than the classical Monte Carlo Euler method is the recently introduced multilevel Monte Carlo Euler method. The main observation of this article is that this multilevel Monte Carlo Euler method does - in contrast to classical Monte Carlo methods - not converge in general in the case of such nonlinear SDEs. More precisely, we establish divergence of the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. In particular, the multilevel Monte Carlo Euler method diverges for these nonlinear SDEs on an event that is not at all rare but has probability one. As a consequence for applications, we recommend not to use the multilevel Monte Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead we propose to combine the multilevel Monte Carlo method with a slightly modified Euler method. More precisely, we show that the multilevel Monte Carlo method combined with a tamed Euler method converges for nonlinear SDEs with globally one-sided Lipschitz continuous drift coefficients and preserves its strikingly higher order convergence rate from the Lipschitz case.

math.PR↗

arXiv2012

Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

math.NA↗

arXiv2012

Asymptotic behaviour of random tridiagonal Markov chains in biological applications

Discrete-time discrete-state random Markov chains with a tridiagonal generator are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses the Hilbert projection metric and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself. The proof does not involve probabilistic properties of the sample path and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transitions probabilities, in which case the attractor is a periodic path.

math.DS↗

arXiv2011

Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz continuous drift and diffusion coefficient. Recent results extend this convergence to coefficients which grow at most linearly. For superlinearly growing coefficients finite-time convergence in the strong mean square sense remained an open question according to [Higham, Mao & Stuart (2002); Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40, no. 3, 1041-1063]. In this article we answer this question to the negative and prove for a large class of stochastic differential equations with non-globally Lipschitz continuous coefficients that Euler's approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense.

math.NA↗