Decomposition of stochastic flows and Lyapunov exponents (Q1584542)

Consider a stochastic differential equation on a compact Riemannian manifold, and let \(\phi_t\) be the flow generated by this equation. For a fixed \(m\) in \(M\) and a fixed orthonormal frame \(u\) at \(m\), the author proves that this flow has a unique decomposition \(\phi_t=\xi_t\psi_t\), where \(\xi_t\) is an isometry, \(\psi_t(m)=m\), and the differential of \(\psi_t\) satisfies \(D\psi_t(u)=us_t\) for an upper triangular matrix \(s_t\). This decomposition can be applied to the study of the Lyapunov exponents \((\lambda_i)\) of the flow; the component \(\psi_t\) preserves the directions of tangent vectors at \(m\) corresponding to the rates \(\lambda_i\).