Levy processes in Euclidian spaces are essential in probability theory. Here, Liao (mathematics, Auburn U.) describes the limiting and dynamical properties in semi-simple Lie groups of the non-compact type. He includes not only a new description of these processes but also significant results that reveal connections between probability and Lie groups. Liao describes Levy processes in Lie groups, induced processes, generators and stochastic integral equation of a Levy process, the Levy process in compact Lie groups and Fourier analysis, semi-simple groups of the noncompact type, limiting properties of Levy processes, the rate of convergence, and Levy processes as stochastic flows. He includes appendices with background information on Lie groups and stochastic analysis