# Random Dynamical Systems.

equilibrium points, bifurcations, existence of invariant manifolds, attractors, etc. The theory of random

dynamical systems studies the qualitative behavior of systems under the influence of noise. A noise is an

ergodic stochastic process. The main topic of these three lectures is to introduce the foundation of this theory.

In particular we give a mathematical description of several types of noise. In addition, we present the definition

of a random dynamical system as a measurable cocycle. Then we give several examples of random (partial)

differential equations and stochastic (partial) differential equations generating a random dynamical system.

**Lecture 1: Basics on random dynamical systems.**

We introduce a *metric dynamical system* as a model of a noise. As an example for such a metric dynamical

system we introduce the (fractional) Brownian motion. We then are able to introduce the ergodic theorem and

tempered random variables. Based on these foundations we can define a random dynamical system. We then

describe how to generate these systems by simple random differential equations. For these systems random

fixed points and describe their bifurcation are studied, [3], [1].

**Lecture 2: Partial differential equations and random dynamical systems.**

We consider several partial differential equations like 2D Navier-Stokes equations, reaction {diffusions equations, wave equations driven by a white or multiplicative noise. To generate a random dynamical system we

transform these equations by a

*random cohomology*into partial differential equations with random coefficients.

These transformed equation then allows us to generate a random dynamical system. We also present some

ideas how to define random dynamical system for stochastic differential equations driven by a fractional Brownian motion where modern techniques of stochastic integration are used. These techniques allow to generate

random dynamical systems for stochastic partial differential equations with general coefficients in front of the

noise, [7], [5], [4].

**Lecture 3: Random attractors.**

We present the theory of

*pullback attractors*. The existence of pullback attractors is based on two properties:

the existence of a pullback absorbing set or (asymptotical) compactness of the random dynamical systems. We

then explain methods how these properties can be found for particular equations from mathematical physics.

We will also discuss several questions related to the existence of random attractors, [2], [6].

References

[1] L. Arnold. Random dynamical systems. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.

[2] L. Arnold and B. Schmalfuss. Fixed points and attractors for random dynamical systems. In Advances in nonlinear stochastic

mechanics (Trondheim, 1995), volume 47 of Solid Mech. Appl., pages 19{28. Kluwer Acad. Publ., Dordrecht, 1996.

[3] H. Bauer. Probability theory, volume 23 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1996.

Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author.

[4] M. Garrido-Atienza, K. Lu, and B. Schmalfuss. Unstable manifolds for a stochastic partial differential equation driven by a

ractional Brownian. J. Differential Equations, 248(7):1637{1667, 2010.

[5] B. Malowski and Nualart. D. Evolution equations driven by a fractional brownian motion. Journal of Functional Analysis,

202:277305, 2003.

[6] B. Schmalfuss. The random attractor of the stochastic Lorenz system. Z. Angew. Math. Phys., 48(6):951{975, 1997.

[7] B. Schmalfuss. Attractors for the non-autonomous dynamical systems. In International Conference on Differential Equations,

Vol. 1, 2 (Berlin, 1999), pages 684{689. World Sci. Publishing, River Edge, NJ, 2000.