On geometric properties of stochastic flows related to the Lyapunov spectrum (Q1590676)

Consider the stochastic differential equations \[ d\xi_t(x)= \sum^n_{i=1} X_i\bigl(\xi_t(x)\bigr) \circ dW^i_t,\;\xi_0(x)= x\text{ and }d \eta_t(x)= \sum^n_{i,j=1} X_{ij}\bigl(\eta_t (x)\bigr)\circ dW_t^{ij},\;\eta_0 (x)=x, \] where \(\circ\;dW\) denotes the Stratonovich differential. Let \(\gamma\) be a smooth curve on the sphere \(S^{n-1}\), with \(\gamma(0)=x\). If \(\kappa_i (i=1, 2)\) are the curvature of \(\xi_t(\gamma)\) at \(\xi_t (\gamma(0))\) and \(\eta_t (\gamma)\) at \(\eta_t (\gamma(0))\) respectively, then \(\kappa^2_1(t)\) is a transient diffusion with generator \[ Lf(\rho)=2 (\rho^2+ \rho)f'' (\rho)+ (n+1) \rho+ (n-2)f' (\rho) \] and \(\kappa_2 (t)=\exp (\sqrt 6B_t)\), where \(B_t\) is a one-dimensional Brownian motion. Moreover for \(n=2\) or 3 \[ P(\lim_{t\to \infty} \kappa^2_t= 0)+P(\lim_{t \to\infty} \kappa^2_t= \infty)=1 \] and for \(n\geq 4\) \[ P(\lim_{t \to\infty} \kappa^2_t= \infty)=1. \]