Furstenberg's theorem for nonlinear stochastic systems

✍ Scribed by Andrew Carverhill

• Book ID: 104740535
• Publisher: Springer
• Year: 1987
• Tongue: English
• Weight: 301 KB
• Volume: 74
• Category: Article
• ISSN: 1432-2064
• DOI: 10.1007/bf00363514

✦ Synopsis

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~ N on IR m with determinant 1, and says that the highest Lyapunov exponent /3=--lim I ilx, ~ .... XIH n~oo ~/ 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 /?>0 is ensured by a Lie algebra condition on the induced system on the principal bundle over M.