Noncommutative ergodic theorems (Q2908718)

The authors present recent results about the asymptotic behavior of ergodic products of isometries of a metric space \(X.\) It is known that if the displacement is integrable, then either there is a sublinear diffusion or there is, for almost every trajectory in \(X,\) a preferred direction at the boundary. The authors' precise statement when \(X\) is a proper metric space [``On laws of large numbers for random walks'', Ann. Probab. 34, No. 5, 1693--1706 (2006; Zbl 1111.60005)] is discussed and compared with classical ergodic theorems. Applications are given to ergodic theorems for nonintegrable functions, random walks on groups, and Brownian motion on covering manifolds.NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].