## White noise on bialgebrasStochastic processes with independent increments on a group are generalized to the concept of "white noise" on a Hopf algebra or bialgebra. The main purpose of the book is the characterization of these processes as solutions of quantum stochastic differential equations in the sense of R.L. Hudsonand K.R. Parthasarathy. The notes are a contribution to quantum probability but they are also related to classical probability, quantum groups, and operator algebras. The Az ma martingales appear as examples of white noise on a Hopf algebra which is a deformation of the Heisenberg group. The book will be of interest to probabilists and quantum probabilists. Specialists in algebraic structures who are curious about the role of their concepts in probablility theory as well as quantum theory may find the book interesting. The reader should havesome knowledge of functional analysis, operator algebras, and probability theory. |

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### Contents

Introduction | 1 |

Basic concepts and first results | 12 |

Symmetric white noise on Bose Fock space | 41 |

Copyright | |

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### Common terms and phrases

a-invariant a)-bialgebra Accardi additive white noise algebra homomorphism Bose Fock space coalgebra cocommutative commutation factor comultiplication condition conditionally positive hermitian conditionally positive linear converges convolution product counit define denote double-module algebra double-module vector space elements Fock space g-white given graded vector space group-like hermitian linear functional Hilbert space Hopf algebra increment property infinitely divisible involutive kernels left invariant derivation left white noise Lemma Lie algebras linear mapping linear operator linear subspace matrix maximal quadratic component module algebra module vector space Moreover multiplicative white noise positive linear functional pre-Hilbert space Proceedings Proof Prop Proposition quantum probability space quantum random variables quantum random vector quantum stochastic differential quantum stochastic integral satisfies sesquilinear form stochastic differential equation stochastic process subcoalgebra symmetric bialgebra symmetric white noise tensor algebra tensor product Theorem Theory unitary white noise vanishes white noise Z2-graded