Probability on Real Lie Algebras
Cambridge Tracts in Mathematics Series

This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.

Introduction; 1. Boson fock space; 2. Real Lie algebras; 3. Basic probability distributions on Lie algebras; 4. Noncommutative random variables; 5. Noncommutative stochastic integration; 6. Random variables on real Lie algebras; 7. Weyl calcuus on real Lie algebras; 8. Lévy processes on real Lie algebras; 9. A guide to the Malliavin calculus; 10. Noncommutative Girsanov theorem; 11. Noncommutative integration by parts; 12. Smoothness of densities on real Lie algebras; Appendix; Exercise solutions.

Uwe Franz is Professor at the Laboratoire de Mathématiques, UFR Sciences et Techniques, Université de Franche-Comté, Besançon, France.
Nicolas Privault is Associate Professor in the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, at Nanyang Technological University, Singapore.