The Annals of Probability

Random Lie group actions on compact manifolds: A perturbative analysis

Christian Sadel and Hermann Schulz-Baldes

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A random Lie group action on a compact manifold generates a discrete time Markov process. The main object of this paper is the evaluation of associated Birkhoff sums in a regime of weak, but sufficiently effective coupling of the randomness. This effectiveness is expressed in terms of random Lie algebra elements and replaces the transience or Furstenberg’s irreducibility hypothesis in related problems. The Birkhoff sum of any given smooth function then turns out to be equal to its integral w.r.t. a unique smooth measure on the manifold up to errors of the order of the coupling constant. Applications to the theory of products of random matrices and a model of a disordered quantum wire are presented.

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Ann. Probab. Volume 38, Number 6 (2010), 2224-2257.

First available: 24 September 2010

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Primary: 60J05: Discrete-time Markov processes on general state spaces 37H05: Foundations, general theory of cocycles, algebraic ergodic theory [See also 37Axx] 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx]

Group action invariant measure Birkhoff sum


Sadel, Christian; Schulz-Baldes, Hermann. Random Lie group actions on compact manifolds: A perturbative analysis. The Annals of Probability 38 (2010), no. 6, 2224--2257. doi:10.1214/10-AOP544.

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