Asymptotic behaviour of stochastic flows of diffeomorphisms: Two case studies

Abstract

The two stochastic flows studied are (i) the canonical stochastic flow on the orthonormal frame bundle of hyperbolic space (which gives stochastic parallel translation along Brownian paths in hyperbolic space) and (ii) a stochastic flow on the sphereS n-1 arising from its embedding as the unit sphere in ℝ n . Both flows are controlled by the same stochastic differential equation in a finite-dimensional Lie group. In each case the Lyapunov exponents are computed and a complete description is given of the local and global stability of the flow.