DYNAMICAL PROPERTIES OF LÉVY PROCESSES IN LIE GROUPS

Let ϕ_t be a Lévy process in a semisimple Lie group G of noncompact type regarded as a stochastic flow on a homogeneous space of G, called a G-flow. We will determine the Lyapunov exponents and the stable manifolds of ϕ_t, and the stationary points of an associated vector field. As examples, SL(d,R)-flows and SO(1,d)-flows on SO(d) and S^{d - 1} are discussed in details.