1. General Facts about Random Dynamical Systems

Abstract

In this chapter we recall some basic definitions and facts about random dy-namical systems. For a more detailed discussion of the theory and applications of random dynamical systems we refer to the monograph Arnold [3]. We pay particular attention to dissipative systems and their random (pull back) attractors. These attractors were studied by many authors (see, e.g., Arnold [3], Crauel/Debussche/Flandoli [35], Crauel/Flandoli [36], Schenk-Hoppé [89], Schmalfuss [92, 93] and the references therein). The ideas that lead to the concept of a random attractor have their roots in the theory of deterministic dissipative systems which has been successfully developed in the last few decades (see, e.g., the monographs Babin/Vishik [13], Chueshov [20], Hale [50], Temam [104] and the literature quoted therein). The proof of the existence of random attractors given below follows almost step-by-step the corresponding deterministic argument (see, e.g., Chueshov [20], Temam [104]). Throughout this book we will be concerned with a probability space by which we mean a triple (Ω, F, P), where Ω is a space, F is a σ-algebra of sets in Ω, and P is a nonnegative σ-additive measure on F with P(Ω) = 1. We do not assume in general that the σ-algebra is complete. Below we will also use the symbol T for either R or Z and we will denote by T + all non-negative elements of T. We will denote by B(X) the Borel σ-algebra of sets in a topological space X. By definition B(X) is the σ-algebra generated by the collection of open subsets of X. If (X 1 , F 1) and (X 2 , F 2) are measurable spaces, we denote by F 1 × F 2 the product σ-algebra of subsets in X 1 × X 2 which is defined as the σ-algebra generated by the cylinder sets A = A 1 × A 2 , A i ∈ F i. We refer to Cohn [30] for basic definitions and facts from the measure theory.