Björn Schmalfuss

Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise

The random attractor to the stochastic 3D Navier-Stokes equation will be studied. In the first part we formulate an existence theorem for attractors of non-autonomous dynamical systems on a bundle of metric spaces. Using this theorem we can prove the existence of an attractor for the 3D Navier-Stokes equation with multiplicative white noise. In addition we prove that this attractor is a random multi-function

RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS

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.

Random Attractors for Stochastic Evolution Equations Driven by Fractional Brownian Motion

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 Hurst parameter

Stochastic Lattice Dynamical Systems with Fractional Noise

H\in (1/2,1)

Local Stability of Differential Equations Driven by Hölder-Continuous Paths with Hölder Index in (1/3,1/2)

. 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 the 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 a nontrivial Holder continuous driving function. In the second part, we shall consider the random setup: stochastic equations having as a driving process a fractional Brownian motion with

Non--autonomous and random attractors for delay random semilinear equations without uniqueness

H\in (1/2,1)

Random dynamical systems for stochastic partial differentialequations driven by a fractional Brownian motion

. Under a smal...

Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions

This article is devoted to study stochastic lattice dynamical systems driven by a fractional Brownian motion with Hurst parameter

Qualitative properties for the stochastic Navier-Stokes equation

H\in(1/2,1)

Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems

. First of all, we investigate the existence and uniqueness of pathwise mild solutions to such systems by the Young integration setting and prove that the solution generates a random dynamical system. Further, we analyze the exponential stability of the trivial solution.