Semigroups and Moment Lyapunov Exponents

Let G be a noncompact semi-simple Lie group with finite center and a probability measure on G .We consider (i) the semigroup S generated by the support of (with the assumption that intS = ); (ii) The spectral radii r of the operators U () where U is a (nonunitary) representation of G induced by a real character and (iii) the moment Lyapunov exponents (, x) of the i.i.d.random product on G defined by .The equality r = (, x) holds in many cases.We give a necessary and sufficient condition to have S = G in terms of the analyticity of the map r .The condition is applied to measures obtained by solutions of invariant stochastic differential equations on G yielding a necessary and sufficient condition for the controllability of invariant control systems on G in terms of the largest eigenvalues of second order differential operators.