This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder 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 Hölder 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 Hölder 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.
In this article we investigate the dynamics of stochastic partial differential equations with dynamical boundary conditions. We prove that such a problem with Lipschitz continuous non--linearity generates a random dynamical system. The main result is to show that this random dynamical system has an inertial manifold.
Under additional assumptions on the non--linearity this manifold is differentiable.
Nonautonomous and random dynamical systems perturbed by impulses are considered.
The impulses form a flow. Over this flow the perturbed system also has
the structure of a new nonautonomous/random dynamical system. The long time behavior of
this system is considered. In particular the existence of an attractor is proven.
The result can be applied to a large class of dissipative systems given by
partial or ordinary differential equations. As an example of this class of problems the
Lorenz system is studied. For another problem given by a one-dimensional affine differential
equation and perturbed by affine impulses, the attractor can be calculated explicitly.
A one-step numerical scheme with variable
time--steps is applied to an autonomous differential equation with a uniformly
asymptotically stable set, which is compact but otherwise of arbitrary
geometric shape. A Lyapunov function characterizing this set is used to
show that the
resulting nonautonomous difference equation generated by the numerical scheme
has an absorbing set. The existence of a cocycle attractor consisting of a
of equivariant sets for the associated discrete time cocycle is then
established and shown to be close in the Hausdorff separation to the original
stable set for sufficiently small maximal time-steps.
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 Lyapunov-Perron transformation. However, the delay is assumed to be unbounded which makes it necessary to use nonclassical methods.
In this work we present the existence and uniqueness of pullback
and random attractors for stochastic evolution equations with
infinite delays when the uniqueness of solutions for these
equations is not required. Our results are obtained by means of
the theory of set-valued random dynamical systems and their
The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.
We consider non-linear parabolic stochastic partial differential
equations with dynamical boundary conditions and with a noise
which acts in the domain but also on the boundary and is
presented by the temporal generalized derivative of an infinite
dimensional Wiener process.
We prove that solutions to this stochastic partial differential equation
generate a random dynamical system. Under additional conditions we
show that this system is monotone.
Our main result states the existence of a compact global
We consider a stochastic nonlinear evolution equation where the domain is given by a fractal set. The linear part of the equation is given by a Laplacian defined on the fractal. This equation generates a random dynamical system. The long time behavior is given by an attractor which has a finite Hausdorff dimension. We would like to reveal the connections between upper and lower estimates of this Hausdorff dimension and the geometry of the fractal. In particular, the parameter which determines these bounds is the spectral exponent of the fractal. Especially for the lower estimate we construct a local unstable random Lipschitz manifold.
In this paper we study nonlinear stochastic partial differential
equations (SPDEs) driven by a fractional Brownian motion (fBm)
with the Hurst parameter bigger than $1/2$. We show 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, we emphasize that the coefficients in
front of the fractional noise are non-trivial.