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 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 non-commutativity in
stochastic calculus, Levy processes, and the Malliavin calculus.
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