Furstenberg's theorem for nonlinear stochastic systems

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)529-534
Journal / PublicationProbability Theory and Related Fields
Issue number4
Publication statusPublished - Apr 1987
Externally publishedYes


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 {Xi}i∈N on ℝm with determinant 1, and says that the highest Lyapunov exponent {Mathematical expression} is strictly positive unless there is a probability measure on the projective (m-1)-space which is a.s. invariant under the action of Xi. 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. © 1987 Springer-Verlag.