Probability on Real Lie Algebras

This monograph is a progressive introduction to non-commutativity in probability theory.

Uwe Franz (Author), Nicolas Privault (Author)

9781107128651, Cambridge University Press

Hardback, published 25 January 2016

302 pages, 2 b/w illus. 27 exercises
22.9 x 15.2 x 2.1 cm, 0.6 kg

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.

Subject Areas: Quantum physics [quantum mechanics & quantum field theory PHQ], Stochastics [PBWL], Probability & statistics [PBT], Calculus & mathematical analysis [PBK], Algebra [PBF]