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81S25 Quantum theory -- General quantum mechanics and problems of quantization -- Quantum stochastic calculus

81R50 Quantum theory -- Groups and algebras in quantum theory -- Quantum groups and related algebraic methods

60Jxx Probability theory and stochastic processes -- Markov processes

60Bxx Probability theory and stochastic processes -- Probability theory on algebraic and topological structures En-ligne: Springerlink | Zentralblatt | MathSciNet

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This book presents some recent improvements in the theory of quantum (noncommutative) probability theory, enclosed in a self-contained course on this topic, including a presentation of the rich algebraic structure that is needed. It is based on several recent works, mainly by R. L. Hudson and K. R. Parthasarathy, L. Accardi, P. Glockner, H. Maassens, W. von Waldenfels, and the author himself.

The classical theory of processes with independent and stationary increments on a group becomes here a very general noncommutative theory of what is called “white noise” on bialgebras of Hopf algebras. (Hopf algebras generalize the “quantum groups” of some authors.) The main aim attained in this book is the characterization of such a general white noise as the solution of a quantum stochastic differential equation of R. L. Hudson and K. R. Parthasarathy. Additive white noise, infinitely divisible representations on Lie algebras, and Azéma noise are also characterized. (Zentralblatt)

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