This monograph develops a pedagogical approach to the role of noncommutativity in probability theory, starting in the first chapter at a level suitable for graduate and advanced undergraduate students. The contents also aim at being relevant to the physics student and to the algebraist interested in connections with probability and statistics.

Our presentation of noncommutativity in probability revolves around concrete examples of relations between algebraic structures and probability distribution, especially via recursive relations among moments and their generating functions. In this way, basic Lie algebras such as the Heisenberg–Weyl algebra hw, the oscillator algebra osc, the special linear algebra sl(2,R), and other Lie algebras such as so(2) and so(3), can be connected with classical probability distributions, notably the Gaussian, Poisson, and gamma distributions, as well as some other infinitely divisible distributions.

Based on this framework, the Chapters 1–3 allow the reader to directly manipulate examples and as such they remain accessible to advanced undergraduates seeking an introduction to noncommutative probability. This setting also allows the reader to become familiar with more advanced topics, including the notion of couples of noncommutative random variables via the use of Wigner densities, in relation with quantum optics.

The following chapters are more advanced in nature, and are targeted to the graduate and research levels. They include the results of recent research on quantum Lévy processes and the noncommutative Malliavin [75] calculus. The Malliavin calculus is introduced in both the commutative and noncommutative settings and contributes to a better understanding of the smoothness properties of Wigner densities.

While this text is predominantly based on research literature, part of the material has been developed for teaching in the course “Special topics in statistics” at the Nanyang Technological University, Singapore, in the second semester of academic year 2013–2014. We thank the students and participants for useful questions and suggestions.

We thank Souleiman Omar Hoche and Michaël Ulrich for their comments, suggestions, and corrections of an earlier version of these notes. During the writing of this book, UF was supported by the ANR Project OSQPI (ANR-11- BS01-0008) and by the Alfried Krupp Wissenschaftskolleg in Greifswald. NP acknowledges the support of NTU MOE Tier 2 Grant No. M4020140.