Chapter 12 Random walks on groups and random transformations

Abstract

This chapter discusses three aspects of random walks. It focuses on random walks on matrix groups G = SL k (R). This theory is mainly concerned with Laws of Large Numbers and other limit theorems. The chapter discusses random walks on general groups. It focuses on the connections between the properties of the group and the behavior of the random walks on it. The chapter also discusses the recurrence properties of random walks and bounded μ-harmonic functions, the concept of the Poisson boundary, and the related notions of boundary entropy and random walk entropy. The chapter discusses random walks on the groups of the transformations of measure spaces and manifolds. The chapter presents Random Ergodic Theorem and related results. It presents the random-walk-based notion of entropy. This notion is discussed in the chapter in the context of diffeomorphism of manifolds and in the general measurable setting. The chapter discusses the connection between the entropy of random volume-preserving diffeomorphism.