Publications (70)47.35 Total impact

Article: Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise
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ABSTRACT: In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems.  [Show abstract] [Hide abstract]
ABSTRACT: We consider a stochastically perturbed coupled system consisting of linearized 3D NavierStokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for inplane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasistability estimates recently developed for deterministic systems.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the stochastic evolution equation $ du=Audt+G(u)d\omega,\quad u(0)=u_0 $ in a separable Hilbertspace $V$. Here $G$ is supposed to be three times Fr\'echetdifferentiable and $\omega$ is a trace class fractional Brownianmotion with Hurst parameter $H\in (1/3,1/2]$. We prove the existence of a global solution where exceptional sets are independent of the initial state $u_0\in V$. In addition, we show that the above equation generates a random dynamical system. 
Dataset: CKS 2001

Dataset: CKS 2001
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ABSTRACT: In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a H\"older continuous function with H\"older exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the wellknown Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting in a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a nonlinear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which is particular includes white noise.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we investigate the existence and some useful properties of the L\'evy areas of OrnsteinUhlenbeck processes associated to Hilbertspacevalued fractional Brownianmotions with Hurst parameter $H\in (1/3,1/2]$. We prove that this stochastic area has a H\"oldercontinuous version with sufficiently large H\"olderexponent and that can be approximated by smooth areas. In addition, we prove the stationarity of this area.  [Show abstract] [Hide abstract]
ABSTRACT: We prove existence and uniqueness of the solution of a stochastic shellmodel. The equation is driven by an infinite dimensional fractional Brownianmotion with Hurstparameter $H\in (1/2,1)$, and contains a nontrivial coefficient in front of the noise which satisfies special regularity conditions. The appearing stochastic integrals are defined in a fractional sense. First, we prove the existence and uniqueness of variational solutions to approximating equations driven by piecewise linear continuous noise, for which we are able to derive important uniform estimates in some functional spaces. Then, thanks to a compactness argument and these estimates, we prove that these variational solutions converge to a limit solution, which turns out to be the unique pathwise mild solution associated to the shellmodel with fractional noise as driving process. 
Article: Generation of random dynamical systems from fractional stochastic delay differential equations
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ABSTRACT: In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a H\"older space which is separable.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we deal with a nonautonomous differential equation with a nonautonomous delay. The aim is to establish the existence of an unstable invariant manifold to this differential equation for which we use the LyapunovPerron transformation. However, the delay is assumed to be unbounded which makes it necessary to use nonclassical methods.  [Show abstract] [Hide abstract]
ABSTRACT: The main goal of this article is to prove the existence of a random attractor for a stochastic evolution equation driven by a fractional Brownian motion with $H\in (1/2,1)$. We would like to emphasize that we do not use the usual cohomology method, consisting of transforming the stochastic equation into a random one, but we deal directly with the stochastic equation. In particular, in order to get adequate a priori estimates of the solution needed for the existence of an absorbing ball, we will introduce stopping times to control the size of the noise. In a first part of this article we shall obtain the existence of a pullback attractor for the nonautonomous dynamical system generated by the pathwise mild solution of an nonlinear infinitedimensional evolution equation with nontrivial H\"older continuous driving function. In a second part, we shall consider the random setup: stochastic equations having as driving process a fractional Brownian motion with $H\in (1/2,1)$. Under a smallness condition for that noise we will show the existence and uniqueness of a random attractor for the stochastic evolution equation.  [Show abstract] [Hide abstract]
ABSTRACT: This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a H\"older continuous function with H\"older exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart, we present here an existence and uniqueness result in the space of H\"older continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the H\"older continuous functions. That space of functions is appropriate to introduce a nonautonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion 
Article: Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients
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ABSTRACT: In this paper we study the longtime dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbertvalued Brownian motion. As a preparation for this purpose we have to show the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of nonstochastic integrals and the noisy path by using an integration by parts. This latter term causes that in a first moment only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the H\"older norm of the noisy path to be sufficiently small. Later, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the {\it Random Dynamical Systems theory}. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, have associated a random attractor.  [Show abstract] [Hide abstract]
ABSTRACT: We are interested in developing a pathwise theory for mild solutions of stochastic evolution equations when the noise path is βHölder continuous for β∈(1/3,1/2)β∈(1/3,1/2). From the point of view of the Rough Path Theory, stochastic integrals related to the solution of ordinary differential equations contain areaelements from a tensor space. Based on (compensated) fractional derivatives we are able to derive a second mild equation for these area components. We formulate sufficient conditions for the existence and uniqueness of a pathwise mild solution by using the Banach fixed point theorem provided that the coefficients of the system are sufficiently regular.RésuméDans cette Note, nous sommes intéressés à développer une théorie trajectorielle pour les solutions ‘mild’ dʼéquations dʼévolution stochastiques lorsque le bruit est βHölder continue pour β∈(1/3,1/2)β∈(1/3,1/2). Selon la théorie ‘Rough Path’, les intégrales stochastiques liés à la solution des équations différentielles ordinaires contiennent des éléments dʼun espace de tenseurs. Grâce aux dérivées fractionnaires (compensées), on peut formuler une deuxième équation pour ce tenseur, pour lequel nous construisons un autre tenseur en fonction non seulement sur le bruit, mais aussi sur le semigroupe. Nous formulons des conditions suffisantes pour lʼexistence et lʼunicité dʼune solution trajectorielle en utilisant le théoréme du point fixe de Banach lorsque des coefficients du système sont assez régulières.  [Show abstract] [Hide abstract]
ABSTRACT: Combining fractional calculus and the Rough Path Theory we study the existence and uniqueness of mild solutions to evolutions equations driven by a H\"older continuous function with H\"older exponent in $(1/3,1/2)$. Our stochastic integral is in some sense a generalization of the wellknown Young integral and can be defined independently of the initial condition. Similar to the Rough Path Theory we establish a second variable which is given, roughly speaking, by a tensor product. It is then necessary to formulate a second equation for this new variable, and we do in a mild sense. The crucial point in order to get this new equation is to construct a tensor depending on the noise path but also on the semigroup. We then prove the existence of a unique H\"older continuous solution of the system of equations, consisting of the path and the area components, if the nonlinear term and the initial condition are sufficiently smooth. Once the abstract theory is developed, we can present a pathwise nonlinear SPDE driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, the asymptotic behavior of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 is studied. In particular, it is shown that the corresponding solutions generate a random dynamical system for which the existence and uniqueness of a random attractor is proved. 
Article: DISCRETIZATION OF STATIONARY SOLUTIONS OF SPDE'S BY EXTERNAL APPROXIMATION IN SPACE AND TIME
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ABSTRACT: We consider a stochastic partial differential equation with additive noise satisfying a strong dissipativity condition for the nonlinear term such that this equation has a random fixed point. The goal of this article is to approximate this fixed point by space and spacetime discretizations of a stochastic differential equation or more precisely, a conjugate random partial differential equation. For these discretizations external schemes are used. We show the convergence of the random fixed points of the space and spacetime discretizations to the random fixed point of the original partial differential equation.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate a random differential equation with random delay. First the nonautonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case. 
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ABSTRACT: In this short note we prove that an infinite dimensional fractional Brownian motion B H of any Hurst parameter H Î (0, 1){H \in (0, 1)} forms an ergodic metric dynamical system. For the proof we mainly use the fundamental theorems of Kolmogorov. KeywordsErgodicity–Metric dynamical systems–Hilbert spacevalued fractional Brownian motion
Publication Stats
2k  Citations  
47.35  Total Impact Points  
Top Journals
Institutions

2015

ErnstAbbeHochschule Jena
Jena, Thuringia, Germany


20122014

Friedrich Schiller University Jena
 Department of Stochastics
Jena, Thuringia, Germany


20032013

Universität Paderborn
 Department of Mathematics
Paderborn, North RhineWestphalia, Germany


2005

Universidad de Sevilla
Hispalis, Andalusia, Spain


2004

Fachhochschule der Wirtschaft
Paderborn, North RhineWestphalia, Germany


1998

Technische Hochschule Wildau
Wildau, Brandenburg, Germany


19961997

Universität Bremen
Bremen, Bremen, Germany
