Rotation numbers for linear stochastic differential equations

Let dx = Sigma(i=0)(m) A(i)x circle dW(i) be a linear SDE in R-d, generatin g the flow Phi(t), of Linear isomorphisms. The multiplicative ergodic theor em asserts that every vector v is an element of R-d \ {0} possesses a Lyapu nov exponent (exponential gran th rate) lambda(v) under Phi(t), which is a random variable taking its values from a finite list of canonical exponents lambda(i) realized in the invariant Oseledets spaces E-i. We prove that, i n the case of simple Lyapunov spectrum, every 2-plane p in R-d possesses a rotation number rho(p) under Phi(t) which is defined as the linear growth r ate of the cumulative infinitesimal rotations of a vector u(t) inside Phi(t )(p). Again, rho(p) is a random variable taking its values from a finite li st of canonical rotation numbers rho(ij) realized in span (E-i, E-j) We giv e rather explicit Furstenberg-Khasminskii-type formulas for the rho(ij). Th is carries over results of Arnold and San Martin from random to stochastic differential equations, which is made possible by utilizing anticipative ca lculus.