This mograph is a progressive introduction to n-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 n-commutative couples of random variables, n-commutative stochastic processes with independent increments (quantum Levy 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 n-commutativity in stochastic calculus, Levy processes, and the Malliavin calculus.
Uwe Franz is Professor at the Laboratoire de Mathematiques, UFR Sciences et Techniques, Universite de Franche-Comte, Besancon, France. Nicolas Privault is Associate Professor in the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, at Nanyang Technological University, Singapore.