This book is a self-contained introduction to the theory of Brownian motions and heat kernels on matrix Lie groups and manifolds, with an emphasis on the study of area type functionals. It offers graduate students a systematic account of the subject and serves as a convenient resource and reference for more experienced mathematicians. The book emphasizes methods rather than results and takes the reader to the frontiers of current research, starting with carefully motivated examples and constructions. These aspects are supported by the inclusion of several bibliographic notes at the end of each chapter and appendices at the end of the book. This book can be used as a self-study guide for readers interested in the interplay between geometry and probability or as a textbook for a special topics course.
