Björn Schmalfuss

Friedrich-Schiller-University Jena, Jena, Thuringia, Germany

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Publications (50)26.04 Total impact

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    Luu Hoang Duc, Björn Schmalfuss, Stefan Siegmund
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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.
    09/2013;
  • H. Gao, M. J. Garrido-Atienza, B. Schmalfuss
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    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 non-autonomous dynamical system generated by the pathwise mild solution of an nonlinear infinite-dimensional evolution equation with non--trivial 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.
    07/2013;
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    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 non-autonomous 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
    05/2013;
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    ABSTRACT: In this paper we study the longtime dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued 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 non-stochastic 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.
    02/2013;
  • María J. Garrido-Atienza, Kening Lu, Björn Schmalfuß
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    ABSTRACT: We are interested in developing a pathwise theory for mild solutions of stochastic evolution equations when the noise path is β-Hölder 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 area-elements 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 β-Hölder 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 semi-groupe. 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.
    Comptes Rendus Mathematique 12/2012; 350(s 23–24):1037–1042. · 0.48 Impact Factor
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    María J. Garrido-Atienza, Kening Lu, Björn Schmalfuss
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    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 well-known 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 $(1/3,1/2]$.
    05/2012;
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    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.
    International Journal of Bifurcation and Chaos 05/2012; 20(09). · 0.92 Impact Factor
  • A.ogrowsky, B.schmalfuss
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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 space-time 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 space-time discretizations to the random fixed point of the original partial differential equation.
    International Journal of Bifurcation and Chaos 05/2012; 20(09). · 0.92 Impact Factor
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    ABSTRACT: We investigate a random differential equation with random delay. First the non-autonomous 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.
    Stochastics and Dynamics 11/2011; 11(02n03). · 0.36 Impact Factor
  • María J. Garrido-Atienza, Björn Schmalfuß
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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 space-valued fractional Brownian motion
    Journal of Dynamics and Differential Equations 01/2011; 23(3):671-681. · 0.86 Impact Factor
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    Igor Chueshov, Björn Schmalfuß
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    ABSTRACT: We deal with abstract systems of two coupled nonlinear stochastic (infinite dimensional) equations subjected to additive white noise type process. This kind of systems may describe various interaction phe-nomena in a continuum random medium. Under suitable conditions we prove the existence of an exponentially attracting random invari-ant manifold for the coupled system and show that this system can be reduced to a single equation with modified nonlinearity. This result means that under some conditions we observe (nonlinear) synchro-nization phenomena in the coupled system. Our applications include stochastic systems consisting of (i) parabolic and hyperbolic equations, (ii) two hyperbolic equations, and (iii) Klein-Gordon and Schrödinger equations. We also show that the random manifold constructed con-verges to its deterministic counterpart when the intensity of noise tends to zero.
    Journal of Mathematical Physics 07/2010; 51(10). · 1.30 Impact Factor
  • BjÖrn Schmalfuss
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    ABSTRACT: The intention of the article is to show the existence of inertial manifolds for random dynamical systems generated by infinite dimensional random evolution equations. To find these manifolds we formulate a random graph transform. This transform allows us to introduce a random dynamical system on graphs. A random fixed point of this system defines the graph of the inertial manifold. In contrast to other publications dealing with these objects we also suppose that the linear part of such an evolution equation contains random operators. To deal with these objects we apply th e multiplicative ergodic theorem. The key assumption for the existence of an inertial manifold is an ω-wise gap condition. Key wordsInertial manifolds-random dynamical systems-stochastic partial de’s-multiplicative ergodic theorem
    06/2010: pages 213-236;
  • Tomás Caraballo, Björn Schmalfuss, José Valero
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    ABSTRACT: The asymptotic behaviour of a stochastic functional evolution equation in a separable Hilbert space is studied. The existence of the pullback and random attractors are investigated for random multivalued dynamical systems in the lack of the uniqueness of solutions.
    Discrete and Continuous Dynamical Systems-series B - DISCRETE CONTIN DYN SYS-SER B. 01/2010; 14(2):439-455.
  • María J. Garrido-Atienza, Björn Schmalfuß
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    ABSTRACT: In this paper we consider a class of nonlinear stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion with the Hurst parameter bigger than 1/2. We show that these SPDEs generate random dynamical systems.
    SeMA Journal. 01/2010; 51(1).
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    ABSTRACT: The long-time behavior of an integro-differential parabolic equation of diffusion type with memory terms, expressed by convolution integrals involving infinite delays and by a forcing term with bounded delay, is investigated in this paper. The assumptions imposed on the coefficients are weak in the sense that uniqueness of solutions of the corresponding initial value problems cannot be guaranteed. Then, it is proved that the model generates a multivalued non-autonomous dynamical system which possesses a pullback attractor. First, the analysis is carried out with an abstract parabolic equation. Then, the theory is applied to the particular integro-differential equation which is the objective of this paper. The general results obtained in the paper are also valid for other types of parabolic equations with memory.
    Nonlinear Analysis 01/2010; · 1.64 Impact Factor
  • Chengfeng Sun, Hongjun Gao, Jinqiao Duan, Björn Schmalfuß
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    ABSTRACT: The Boussinesq system models various phenomena in geophysical and climate dynamics. It is a coupled system of the Navier–Stokes equations and the salinity transport equation. Due to uncertainty in salinity flux on fluid boundary, this system is subject to random fluctuations on the boundary. This stochastic Boussinesq system can be transformed into a random dynamical system. Rare events, or small probability events, are investigated in the context of large deviations. A large deviations principle is established via a weak convergence approach based on a recently developed variational representation of functionals of infinite dimensional Brownian motion.
    Journal of Differential Equations 01/2010; 248(6):1269-1296. · 1.48 Impact Factor
  • Kening Lu, Björn Schmalfuss
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    ABSTRACT: The authors study nonlinear stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than 1/2. They prove that these SPDEs generate random dynamical systems (or stochastic flows) by using the stochastic calculus for an fBm where the stochastic integrals are defined by integrands given by fractional derivatives. In particular, they emphasize that the coefficients in front of the fractional noise are non-trivial. The obtained results can be applied to many concrete random dynamical systems generates from stochastic mathematical physics equations driven by a fractional Brownian motion.
    Discrete and Continuous Dynamical Systems-series B - DISCRETE CONTIN DYN SYS-SER B. 01/2010; 14(2):473-493.
  • María J. Garrido-Atienza, Kening Lu, Björn Schmalfuß
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    ABSTRACT: In this paper, we consider a class of stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than 1/2. The existence of local random unstable manifolds is shown if the linear parts of these SPDEs are hyperbolic. For this purpose we introduce a modified Lyapunov–Perron transform, which contains stochastic integrals. By the singularities inside these integrals we obtain a special Lyapunov–Perron's approach by treating a segment of the solution over time interval [0,1] as a starting point and setting up an infinite series equation involving these segments as time evolves. Using this approach, we establish the existence of local random unstable manifolds in a tempered neighborhood of an equilibrium.
    Journal of Differential Equations - J DIFFERENTIAL EQUATIONS. 01/2010; 248(7):1637-1667.
  • Peter Brune, Jinqiao Duan, Björn Schmalfuß
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    ABSTRACT: A coupled system of the two-dimensional Navier–Stokes equations and the salinity transport equation with spatially correlated white noise on the boundary as well as in fluid is investigated. The noise affects the system through a dynamical boundary condition. This system may be considered as a model for gravity currents in oceanic fluids. The noise is due to uncertainty in salinity flux on fluid boundary. After transforming this system into a random dynamical system, we first obtain asymptotic estimates on system evolution, and then show that the long time dynamics is captured by a random attractor.
    Stochastic Analysis and Applications 09/2009; 27(5):1096-1116. · 0.30 Impact Factor
  • Source
    David Cheban, Peter Kloeden, Bjoern Schmalfuss
    01/2009;

Publication Stats

621 Citations
26.04 Total Impact Points

Institutions

  • 2012
    • Friedrich-Schiller-University Jena
      • Department of Stochastics
      Jena, Thuringia, Germany
  • 2003–2012
    • Universität Paderborn
      • Department of Mathematics
      Paderborn, North Rhine-Westphalia, Germany
  • 2004
    • Fachhochschule der Wirtschaft
      Paderborn, North Rhine-Westphalia, Germany
    • Brigham Young University - Provo Main Campus
      • Department of Mathematics
      Provo, UT, United States
  • 2001
    • Moldova State University
      Kischinew, Chişinău, Moldova
    • Humboldt-Universität zu Berlin
      • Department of Mathematics
      Berlín, Berlin, Germany
  • 1999
    • Università di Pisa
      Pisa, Tuscany, Italy
  • 1998
    • Technische Hochschule Wildau
      Wildau, Brandenburg, Germany