Many aspects of the classical probability theory based on vector spaces were generalised in the second half of the twentieth century to measures on groups, especially Lie groups. The subject of Probability measures on groups, that emerged out this research has continued to grow and many interesting new developments have occurred in the area in recent years. A School was organised jointly with CIMPA, France and the Tata Institute of Fundamental Research entitled ``Probability Measures on Groups: Recent Directions and Trends'' during 9--22 September 2002 in Mumbai. Lecture courses were given at the School by M. Babillot (Orlean, France), D. Bakry (Toulouse, France), S.G. Dani (Tata Institute, Mumbai), J. Faraut (Paris), Y. Guivarc'h (Rennes, France) and M. McCrudden (Manchester, U.K.), aimed at introducing various advanced topics on the theme to students as well as teachers and practicing mathematicians desirous of getting acquainted with the area. The prerequisites for the courses were limited to only with basic background in measure theory, harmonic analysis and elementary Lie group theory. The courses were well-received. Notes were prepared and distributed to the participants during the courses. The present volume represents improved, edited and refereed versions of the notes, brought out for dissemination of the topics to the wider community.

Preface / An Introduction to Poisson Boundaries of Lie Groups / Functional Inequalities for Markov Semigroups / Asymptotic Behaviour of Measures Under Automorphisms / Infinite Dimensional Harmonic Analysis and Probability / Limit Theorems for Random Walks and Products of Random Matrices / The Embedding Problem for Probabilities on Locally Compact Groups.

Postgraduate Students, Teachers and Researchers