Abstract

Coin tossing or Bernoulli?s random walk is the simplest stochastic process and has been studied extensively. The aim of this series of three lectures is to introduce a quantum generalization of the Bernoulli random walk (after Philippe Biane). In order to formulate this generalization, the classical theory is reinterpreted in the framework of a family of random variables associated with a commutative algebra with a linear functional (expectation). The quantum analogue is then formulated in the context of a non-commutative algebra of operators equipped with a trace functional. This leads to new notions of independence and limit theorems in free probability which will be discussed in the last lecture.



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