Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts
Jean-Pierre Conze Y. Guivarc'h
Discrete & Continuous Dynamical Systems - A 2013, 33(9): 4239-4269 doi: 10.3934/dcds.2013.33.4239
Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable group of $\nu$-preserving invertible maps of $X$ into itself. To a probability measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator $P$ given by $P\psi(x) = \sum_{a} \psi(ax) \, \mu(a)$. We consider various examples of ergodic $\Gamma$-actions and random walks and their extensions by a vector space: groups of automorphisms or affine transformations on compact nilmanifolds, random walks in random scenery on non amenable groups, translations on homogeneous spaces of simple Lie groups, random walks on motion groups. A powerful tool in this study is the spectral gap property for the operator $P$ when it holds. We use it to obtain limit theorems, recurrence/transience property and ergodicity for random walks on non compact extensions of the corresponding dynamical systems.
keywords: spectral gap recurrence local limit theorem non compact extension of dynamical system random walk random scenery. Nilmanifold

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