White noise on bialgebras / Michael Schürmann.Material type: TextSeries: Lecture notes in mathematics (Springer-Verlag): 1544.Publisher: Berlin ; New York : Springer-Verlag, ©1993Description: 1 online resource (146 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783540476146; 3540476148Subject(s): Quantum theory -- Mathematics | Stochastic analysis | Théorie quantique -- Mathématiques | Analyse stochastique | Quantum theory -- Mathematics | Stochastic analysis | Kwantummechanica | Stochastische analyse | Weißes Rauschen | Hopf-Algebra | BialgebraGenre/Form: Electronic books. Additional physical formats: Print version:: White noise on bialgebras.DDC classification: 530.1/2/015192 LOC classification: QC174.17.M35 | S3 1993QA3 | .L28 no. 1544Other classification: 31.73 Online resources: Click here to access online
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Includes bibliographical references (pages 138-142) and index.
Stochastic 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.
1. Basic concepts and first results. 1.2. Quantum probabilistic notions. 1.3. Independence. 1.4. Commutation factors. 1.5. Invariance of states. 1.6. Additive and multiplicative white noise. 1.7. Involutive bialgebras. 1.9. White noise on involutive bialgebras -- 2. Symmetric white noise on Bose Fock space. 2.1. Bose Fock space over L[superscript 2](R[subscript+], H). 2.2. Kernels and operators. 2.3. The basic formula. 2.4. Quantum stochastic integrals and quantum Ito's formula. 2.5. Coalgebra stochastic integral equations -- 3. Symmetrization. 3.1. Symmetrization of bialgebras. 3.2. Schoenberg correspondence. 3.3. Symmetrization of white noise -- 4. White noise on Bose Fock space. 4.1. Group-like elements and realization of white noise. 4.2. Primitive elements and additive white noise. 4.3. Azema noise and quantum Wiener and Poisson processes. 4.4. Multiplicative and unitary white noise.
4.5. Cocommutative white noise and infinitely divisible representations of groups and Lie algebras -- 5. Quadratic components of conditionally positive linear functionals. 5.1. Maximal quadratic components. 5.2. Infinitely divisible states on the Weyl algebra -- 6. Limit theorems. 6.1. A coalgebra limit theorem. 6.2. The underlying additive noise as a limit. 6.3. Invariance principles.
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