Noncommutative Ergodic Theorems

2011/10/31 by Anders Karlsson, Karlsson, Anders, François Ledrappier +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.DS #math.PR #msc:22D40 #msc:37Axx #msc:37H15

paper · pdf · doi:10.48550/arxiv.1110.6847

25 pages

arxiv created 2011/10/31 · arxiv updated 2011/11/01

Abstract

We present recent results about the asymptotic behavior of ergodic products of isometries of a metric space X. If we assume that 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. We discuss the precise statement when X is a proper metric space and compare it with classical ergodic theorems. Applications are given to ergodic theorems for nonintegrable functions, random walks on groups and Brownian motion on covering manifolds.

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• Brownian motion on stationary random manifolds

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