The present volume provides an introduction to Lévy processes in general Lie groups, and hopefully an accessible account on the limiting and dynamical properties of such processes in semi-simple Lie groups of non-compact type. Lévy processes in Euclidean spaces, including the famous Brownian motion, have always played a central role in probability theory. In recent times, there has been intense research activity in exploring the probabilistic connections of various algebraic and geometric structures, therefore, the study of stochastic processes in Lie groups has become increasingly important. This book is aimed at serving two purposes. First, it may provide a foundation to the theory of Lévy processes in Lie groups, as this is perhaps the first book written on the subject, and second it will present some important results in this area, revealing an interesting connection between probability and Lie groups.

Please note: when referring to a result in a referenced text, the chapter and enunciation numbering system in that text has been followed.

David Applebaum, Olav Kallenberg and Wang Longmin read portions of the manuscript and provided useful comments. Part of the book was written during the author's visit to Nankai University, Tianjin, China in the fall of 2002. I wish to take this opportunity to thank my hosts, Wu Rong and Zhou Xingwei, for their hospitality. It would be hard to imagine this work ever being completed without my wife's support and understanding.