White noise on bialgebras.

*(English)*Zbl 0773.60100
Lecture Notes in Mathematics. 1544. Berlin: Springer-Verlag. vii, 146 p. (1993).

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.

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.

Reviewer: J.Franchi (Paris)

##### MSC:

60K40 | Other physical applications of random processes |

81S25 | Quantum stochastic calculus |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

60J99 | Markov processes |

60B99 | Probability theory on algebraic and topological structures |