DCDS
Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems
Yong Chen Hongjun Gao María J. Garrido–Atienza Björn Schmalfuss
Discrete & Continuous Dynamical Systems - A 2014, 34(1): 79-98 doi: 10.3934/dcds.2014.34.79
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 [17], 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.
keywords: Stochastic PDEs fractional Brownian motion pathwise solutions random dynamical systems.
CPAA
Inertial manifolds for stochastic pde with dynamical boundary conditions
Peter Brune Björn Schmalfuss
Communications on Pure & Applied Analysis 2011, 10(3): 831-846 doi: 10.3934/cpaa.2011.10.831
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.
keywords: dynamical boundary conditions. inertial manifolds stochastic partial differential equations Random dynamical systems
DCDS
Attractors for nonautonomous and random dynamical systems perturbed by impulses
Björn Schmalfuss
Discrete & Continuous Dynamical Systems - A 2003, 9(3): 727-744 doi: 10.3934/dcds.2003.9.727
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.
keywords: attractors. nonautonomous systems Impulse equations random dynamical systems
DCDS
Lyapunov functions and attractors under variable time-step discretization
Peter E. Kloeden Björn Schmalfuss
Discrete & Continuous Dynamical Systems - A 1996, 2(2): 163-172 doi: 10.3934/dcds.1996.2.163
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 family 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.
keywords: cocycle attractor. uniform asymptotic stability cocycle numerical scheme Lyapunov function
DCDS-B
Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay
Arne Ogrowsky Björn Schmalfuss
Discrete & Continuous Dynamical Systems - B 2013, 18(6): 1663-1681 doi: 10.3934/dcdsb.2013.18.1663
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.
keywords: nonautonomous delay cocycles. Unstable invariant manifolds nonautonomous differential equations
DCDS-B
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions
Tomás Caraballo María J. Garrido–Atienza Björn Schmalfuss José Valero
Discrete & Continuous Dynamical Systems - B 2010, 14(2): 439-455 doi: 10.3934/dcdsb.2010.14.439
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 conjugation properties.
keywords: Multivalued non-autonomous and random dynamical systems functional stochastic equations conjugacy method. pullback and random attractors
DCDS-B
Attractors for a random evolution equation with infinite memory: Theoretical results
Tomás Caraballo María J. Garrido-Atienza Björn Schmalfuss José Valero
Discrete & Continuous Dynamical Systems - B 2017, 22(5): 1779-1800 doi: 10.3934/dcdsb.2017106

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.

keywords: Pullback and random attractor random dynamical system random delay equation infinite delay
DCDS
Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions
Igor Chueshov Björn Schmalfuss
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 315-338 doi: 10.3934/dcds.2007.18.315
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 (pullback) attractor.
keywords: stochastic PDE monotonicity. random dynamical systems dynamical boundary conditions pullback attractor
DCDS-B
Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals
Markus Böhm Björn Schmalfuss
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-24 doi: 10.3934/dcdsb.2018303

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.

keywords: Multiplicative ergodic theorem random center manifolds Oseledets splitting
DCDS-B
Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion
María J. Garrido–Atienza Kening Lu Björn Schmalfuss
Discrete & Continuous Dynamical Systems - B 2010, 14(2): 473-493 doi: 10.3934/dcdsb.2010.14.473
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.
keywords: random dynamical systems. Stochastic PDEs fractional Brownian motion

Year of publication

Related Authors

Related Keywords

[Back to Top]