The Annals of Probability

Random Lie group actions on compact manifolds: A perturbative analysis

Christian Sadel and Hermann Schulz-Baldes

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 24 September 2010

Permanent link to this document
http://projecteuclid.org/euclid.aop/1285334205

Digital Object Identifier
doi:10.1214/10-AOP544

Zentralblatt MATH identifier
05817047

Mathematical Reviews number (MathSciNet)
MR2683629

Subjects
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]

Keywords
Group action invariant measure Birkhoff sum

Citation

Sadel, Christian; Schulz-Baldes, Hermann. Random Lie group actions on compact manifolds: A perturbative analysis. Ann. Probab. 38 (2010), no. 6, 2224--2257. doi:10.1214/10-AOP544. http://projecteuclid.org/euclid.aop/1285334205.


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