Björn Schmalfuß

RANDOM ATTRACTORS FOR STOCHASTIC EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION

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.

Master-slave synchronization and invariant manifolds for coupled stochastic systems

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 phenomena in a continuum random medium. Under suitable conditions we prove the existence of an exponentially attracting random invariant 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) synchronization 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 Schrodinger equations. We also show that the random manifold constructed converges to its deterministic counterpart when the intensity of noise tends to zero.

INVARIANT FOLIATIONS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we study the existence of an invariant foliation for a class of stochastic partial differential equations with a multiplicative white noise. This invariant foliation is used to trace the long term behavior of all solutions of these equations.

Continuity Properties of Inertial Manifolds for Stochastic Retarded Semilinear Parabolic Equations

We study continuity properties of inertial manifolds for a class of random dynamical systems generated by retarded semilinear parabolic equations subjected to additive white noise. We focus on two cases: (i) the delay time tends to zero and (ii) the intensity of the noise becomes small.

Asymptotical Stability of Differential Equations Driven by Hölder Continuous Paths

In this manuscript, we establish local exponential stability of the trivial solution of differential equations driven by Hölder continuous paths with Hölder exponent greater than 1/2. This applies in particular to stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2. We motivate the study of local stability by giving a particular example of a scalar equation, where global stability of the trivial solution can be obtained.

A note on the generation of random dynamical systems from fractional stochastic delay differential equations

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 Holder space which is separable.

Lévy–Areas of Ornstein–Uhlenbeck Processes in Hilbert–Spaces

In this paper, we investigate the existence and some useful properties of the Levy-areas of Ornstein-Uhlenbeck processes associated to Hilbert space-valued fractional Brownian motions with Hurst parameter \(H\in (1/3,1/2]\). We prove that this stochastic area has a Holder-continuous version with sufficiently large Holder-exponent and that can be approximated by smooth areas. In addition, we prove the stationarity of this area.

Averaging of Attractors and Inertial Manifolds for Parabolic PDE With Random Coefficients

Abstract The averaging method has been used to study random or non-autonomous systems on a fast time scale. We apply this method to a random abstract evolution equation on a fast time scale whose long time behavior can be characterized by a random attractor or a random inertial manifold. The main purpose is to show that the long-time behavior of such a system can be described by a deterministic evolution equation with averaged coefficients. Our first result provides an averaging result on finite time intervals which we use to show that under a dissipativity assumption the attractors of the fast time scale systems are upper semicontinuous when the scaling parameter goes to zero. Our main result deals with a global averaging procedure. Under some spectral gap condition we show that inertial manifolds of the fast time scale system tend to an inertial manifold of the averaged system when the scaling parameter goes to zero. These general results can be applied to semilinear parabolic differential equations containing a scaled ergodic noise on a fast time scale which includes scaled almost periodic motions.

On 3D Navier-Stokes equations: regularization and uniqueness by delays

Abstract A modified version of the three dimensional Navier–Stokes equations is considered with periodic boundary conditions. A bounded constant delay is introduced into the convective term, that produces a regularizing effect on the solution. In fact, by assuming appropriate regularity on the initial data, the solutions of the delayed equations are proved to be regular and, as a consequence, existence and also uniqueness of a global weak solution are obtained. Moreover, the associated flow is constructed and the continuity of the semigroup is proved. Finally, we investigate the passage to the limit on the delay, obtaining that the limit is a weak solution of the Navier–Stokes equations.

A word from the editors

Two years have passed since the first issue of “Stochastics and Dynamics” appeared in March 2001. It was a good start for a new journal on a not so new subject in a new millenium. Due to personal reasons Ludwig Arnold (Bremen) has resigned from the Editorin-Chief position by the end of 2002, and Manfred Denker (Gottingen) took over in January 2003. This occasion is well suited to reflecting past and future of the journal, where we stand and in which direction we should move. There is no doubt that the combination of stochastics and dynamics is a field of research in its own right which has all the features such a field should have: its own examples and paradigms, its theorems, its specific methods and significant applications. It has developed into various branches, like SDE’s, stochasticity in dynamics, or random structures in dynamics, to name some. Other subjects with a natural intertwining of stochastics and dynamics are listed throughout the current classification scheme like, for example, in 28, 34, 35, 37, 47, 58, 60, 62, 70, 74, 76, 80. It is this description we will focus on in the future. An important aspect of the editorial work has been to include contributions to research in applied fields. We especially mention the September 2002 issue on stochastic climate models which grew out of Ludwig Arnold’s interest in the field and two conferences on the subject in Chorin (Germany). All this will be preserved. On the other hand, we believe it is necessary to broaden the scope of the journal to include more trend-setting articles and to incorporate the full spectrum of research as described above. Moreover, the journal should serve the needs of the community as a genuine board of information exchange in the area. In order to come closer to this goal, a new Advisory Board has been created to advise and inform the Editors about new trends and important directions in the field. Its three members will guarantee that the journal will be able to incorporate new scientific developments. The Editorial Board has been changed only slightly — which reflects the continuity of the editorial policy. Survey articles are important means for trend-setting and orientation. This needs to be done on an irregular basis. Also, the publication of important contributions to conferences which to a large part are devoted to stochastics and dynamics are very welcome.