Summary
In this first chapter we introduce the concept of a random dynamical system and study its invariant measures — objects which characterize the possible long term behavior of the system.
The definition of a random dynamical system or cocycle is given in Sect. 1.1 and basic properties are derived (Theorem 1.1.6). The local concept (allowing for explosion) is introduced in Sect. 1.2.
Sect. 1.3 deals with the key but tricky technical problem of “perfecting” a “crude” cocycle (as generated by a stochastic differential equation). We give a satisfactory answer in Theorem 1.3.2 and its two corollaries. This section can be omitted at first reading.
The basic Sects. 1.4 to 1.7 are devoted to the study of invariant measures of random dynamical systems. We describe invariance in terms of the factorization of the measure (Theorem 1.4.5). For a Polish state space we introduce a topology of weak convergence of measures which permits us to carry over the Krylov-Bogolyubov procedure (Theorem 1.6.4) and to prove that each continuous random dynamical system on a compact state space has at least one invariant measure (Theorem 1.5.10; for a useful generalization to a random compact set see Theorem 1.6.13).
In Sect. 1.7 we relate our general definition of an invariant measure to the classical one for Markov processes (Corollary 1.7.6).
Theorem 1.8.4 in Sect. 1.8 stating that all invariant measures for continuous random dynamical systems with state space ℝ are random Dirac measures will be frequently quoted later as we will explicitly work out several one-dimensional examples.
Finally, in Sect. 1.9 we introduce the bundle version of a random dynamical system and provide several notions of isomorphism which are used later to identify different systems of similar structure.
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© 1998 Springer-Verlag Berlin Heidelberg
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Arnold, L. (1998). Basic Definitions. Invariant Measures. In: Random Dynamical Systems. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12878-7_1
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DOI: https://doi.org/10.1007/978-3-662-12878-7_1
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