Abstract. With an intrinsic approach on semi-simple Lie groups we find a Furstenberg--Khasminskii type formula for the limit of the diagonal component in the Iwasawa decomposition. It is an integral formula with respect to the invariant measure in the maximal flag manifold of the group (i.e. the Furstenberg boundary $B=G/MAN$). Its integrand involves the Borel type Riemannian metric in the flag manifolds. When applied to linear stochastic systems which generate a semi-simple group the formula provides a diagonal matrix whose entries are the Lyapunov spectrum. Some Brownian motions on homogeneous spaces are discussed.
AMSclassification. 60H10 (58G32, 22E46)
Keywords. Lyapunov exponents, stochastic differential equations, semi-simple Lie groups, flag manifolds.