Random walks on groups and random transformations. (English) Zbl 1053.60045

Hasselblatt, B. (ed.) et al., Handbook of dynamical systems. Volume 1A. Amsterdam: North-Holland (ISBN 0-444-82669-6/hbk). 931-1014 (2002).
This paper, which is a chapter in a contributed book, deals with three topics: random walks on matrix groups (first section), random walks on general (locally or discrete) groups (second section), and random walks on groups of transformation of measure spaces and manifolds (third section). Section 1 is mainly concerned with laws of large numbers with random matrices and the techniques so involved are, among others, Markov processes, algebraic group and unitary representation. Section 2 focuses on the connections between the properties of the group and the behaviour of the random walks defined on it. Section 2 goes from the random ergodic theorem to the random-walk-based notion of entropy, in the framework of diffeomrphisms of manifolds.
For the entire collection see [Zbl 1013.00016].


60G50 Sums of independent random variables; random walks
60J05 Discrete-time Markov processes on general state spaces
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
37A50 Dynamical systems and their relations with probability theory and stochastic processes