Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms

Abstract

The Lyapunov exponents λ1≧λ2≧...≧λd for a stochastic flow of diffeomorphisms of a d-dimensional manifold M (with a strongly recurrent one-point motion) describe the almost-sure limiting exponential growth rates of tangent vectors under the flow. This paper shows how the Lyapunov exponents are related to measure preserving properties of the stochastic flow on M and of the induced stochastic flow on the projective bundle PM. Relative entropy is used to quantify the extent to which a measure fails to be invariant under the flow. The results include the following. If M is compact and if the one-point motion on M is a non-degenerate diffusion with stationary probability measure ρ{variant} then λ1+...+λd≦0 with equality if and only if the flow preserves ρ{variant} almost surely; if in addition the induced one-point motion on PM satisfies a weak non-degeneracy condition then λ1=...=λd if and only if there is a smooth Riemannian structure on M with respect to which the flow is conformal almost surely. © 1989 Springer-Verlag.