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Furstenberg's theorem for nonlinear stochastic systems
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  • Published: April 1987

Furstenberg's theorem for nonlinear stochastic systems

  • Andrew Carverhill1 

Probability Theory and Related Fields volume 74, pages 529–534 (1987)Cite this article

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Summary

We extend Furstenberg's theorem to the case of an i.i.d. random composition of incompressible diffeomorphisms of a compact manifold M. The original theorem applies to linear maps {X i} i ∈N on ℝm with determinant 1, and says that the highest Lyapunov exponent

$$\beta \equiv \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}||X_n \circ ... \circ X_1 ||$$

is strictly positive unless there is a probability measure on the projective (m-1)-space which is a.s. invariant under the action of X i . Our extension refers to a probability measure on the projective bundle over M.

We show that when our diffeomorphism is the flow of a stochastic differential equation, the criterion for β>0 is ensured by a Lie algebra condition on the induced system on the principal bundle over M.

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, 27514, Chapel Hill, NC, USA

    Andrew Carverhill

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  1. Andrew Carverhill
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Carverhill, A. Furstenberg's theorem for nonlinear stochastic systems. Probab. Th. Rel. Fields 74, 529–534 (1987). https://doi.org/10.1007/BF00363514

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  • Received: 17 April 1985

  • Revised: 15 October 1986

  • Issue Date: April 1987

  • DOI: https://doi.org/10.1007/BF00363514

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Keywords

  • Differential Equation
  • Manifold
  • Stochastic Process
  • Probability Measure
  • Probability Theory
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