# The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors

@article{Imkeller2001TheCO, title={The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors}, author={Peter Imkeller and Bj{\"o}rn Schmalfu{\ss}}, journal={Journal of Dynamics and Differential Equations}, year={2001}, volume={13}, pages={215-249} }

We consider stochastic differential equations in d-dimensional Euclidean space driven by an m-dimensional Wiener process, determined by the drift vector field f0 and the diffusion vector fields f1,...,fm, and investigate the existence of global random attractors for the associated flows φ. For this purpose φ is decomposed into a stationary diffeomorphism Φ given by the stochastic differential equation on the space of smooth flows on Rd driven by m independent stationary Ornstein Uhlenbeck… Expand

#### 86 Citations

On the cohomology of flows of stochastic and random differential equations

- Mathematics
- 2001

Abstract. We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional… Expand

Gevrey regularity of random attractors for stochastic reaction-diffusion equations

- Mathematics
- 2000

We prove the Gevrey regularity of the global attractor of the random dynamical system generated by a semilinear parabolic equation with periodic boundary conditions which is subjected to a spatially… Expand

THE COHOMOLOGY OF STOCHASTIC AND RANDOM DIFFERENTIAL EQUATIONS, AND LOCAL LINEARIZATION OF STOCHASTIC FLOWS

- Mathematics
- 2002

Random dynamical systems can be generated by stochastic differential equations (sde) on the one hand, and by random differential equations (rde), i.e. randomly parametrized ordinary differential… Expand

On attractors, spectra and bifurcations of random dynamical systems

- Computer Science
- 2014

It is shown that one may still observe qualitative changes in the dynamics at the underlying deterministic bifurcation point, in terms of a loss of hyperbolicity of the dichotomy spectrum; a lossOf uniform attractivity; a qualitative change in the 5 distribution of finite-time Lyapunov exponents; and that whilst for small parameter values the systems are topologically equivalent, there is a lossof uniform topological equivalence. Expand

Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients

- Mathematics
- 2013

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… Expand

Numerical simulation of nonlinear dynamical systems driven by commutative noise

- Mathematics, Computer Science
- J. Comput. Phys.
- 2007

The local linearization (LL) approach has become an effective technique for the numerical integration of ordinary, random and stochastic differential equations. One of the reasons for this success is… Expand

Random exponential attractor for stochastic reaction–diffusion equation with multiplicative noise in R3

- Mathematics
- 2017

Abstract In this paper, we first improve the existing conditions for the existence of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we consider the… Expand

Random attractors for partly dissipative stochastic lattice dynamical systems1

- Mathematics
- 2008

In this paper, we consider the long term behavior for the stochastic lattice dynamical systems with some partly dissipative nonlinear term in l 2 × l 2. The main purpose of this paper is to establish… Expand

Random attractors for dissipative systems with rough noises

- 2020

We provide an analytic approach to study the asymptotic dynamics of rough differential equations. The driving noises in consideration are ν Hölder continuous with ν ∈ ( 13 , 1) for simplicity of… Expand

Weak Random Attractors

- 1999

We deene point attractors and set attractors for random dynamical systems via convergence in probability forward in time. This deenition of a set attractor is weaker than the usual one via almost… Expand

#### References

SHOWING 1-10 OF 29 REFERENCES

Random attractors--general properties, existence and applications to stochastic bifurcation theory

- Mathematics
- 1997

This paper is concerned with attractors of randomly perturbed
dynamical systems, called random attractors. The framework used is
provided by the theory of random dynamical systems.
We first… Expand

The random attractor of the stochastic Lorenz system

- Mathematics
- 1997

We study the existence of attractors for dynamical systems under the influence of random parameters. Such a random attractor is a measurable multi-function with compact images fulfilling particular… Expand

Di erential equations driven by rough signals

- Mathematics
- 1998

This paper aims to provide a systematic approach to the treatment of differential equations of the type
dyt = Si fi(yt) dxti
where the driving signal xt is a rough path. Such equations are very… Expand

Markov measures for random dynamical systems

- Mathematics
- 1991

Random dynamical systems with Markovian coefficients allow two distinct ways of description. The description via the Markov semigroup is the classical one. More recently, the “dynamical systems ” or… Expand

Numerical Approximation of Random Attractors

- Mathematics
- 1999

In this article an algorithm for the numerical approximation of random attractors based on the subdivision algorithm of Dellnitz and Hohmann is presented. It is applied to the stochastic Duffing-van… Expand

Attractors for random dynamical systems

- Mathematics
- 1994

SummaryA criterion for existence of global random attractors for RDS is established. Existence of invariant Markov measures supported by the random attractor is proved. For SPDE this yields invariant… Expand

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

- Mathematics
- 1996

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… Expand

Perfect cocycles through stochastic differential equations

- Mathematics
- 1995

SummaryWe prove that if ϕ is a random dynamical system (cocycle) for whicht→ϕ(t, ω)x is a semimartingale, then it is generated by a stochastic differential equation driven by a vector field valued… Expand

A comprehensive introduction to differential geometry

- Physics
- 1975

Spivak's Comprehensive introduction takes as its theme the classical roots of contemporary differential geometry. Spivak explains his Main Premise (my term) as follows: "in order for an introduction… Expand

Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors

- Mathematics
- 1992

This paper surveys results of the authors and others conceming estimates for the Hausdorff dimension of strange attractors, particularly in the case of (generalized) Lorenz systems and Rössler… Expand