Probability on real Lie algebras / Uwe Franz, Université de Franche-Comté, Nicolas Privault, Nanyang Technological University, Singapore

Author: Franz, Uwe
Published: New York, NY : Cambridge University Press, 2016.
Copyright Date: ©2016
Physical Description: xix, 281 pages ; 24 cm.
Additional Creators: Privault, Nicolas

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Series

Cambridge tracts in mathematics ; 206

Contents

Machine generated contents note: 1.Boson Fock space -- 1.1.Annihilation and creation operators -- 1.2.Lie algebras on the boson Fock space -- 1.3.Fock space over a Hilbert space -- Exercises -- 2.Real Lie algebras -- 2.1.Real Lie algebras -- 2.2.Heisenberg--Weyl Lie algebra -- 2.3.Oscillator Lie algebra osc -- 2.4.Lie algebra sl2(R) -- 2.5.Affine Lie algebra -- 2.6.Special orthogonal Lie algebras -- Exercises -- 3.Basic probability distributions on Lie algebras -- 3.1.Gaussian distribution on -- 3.2.Poisson distribution on osc -- 3.3.Gamma distribution on sl2(R) -- Exercises -- 4.Noncommutative random variables -- 4.1.Classical probability spaces -- 4.2.Noncommutative probability spaces -- 4.3.Noncommutative random variables -- 4.4.Functional calculus for Hermitian matrices -- 4.5.The Lie algebra so(3) -- 4.6.Trace and density matrix -- 4.7.Spin measurement and the Lie algebra so(3) -- Exercises -- 5.Noncommutative stochastic integration -- 5.1.Construction of the Fock space -- 5.2.Creation, annihilation, and conservation operators -- 5.3.Quantum stochastic integrals -- 5.4.Quantum Ito table -- Exercises -- 6.Random variables on real Lie algebras -- 6.1.Gaussian and Poisson random variables on osc -- 6.2.Meixner, gamma, and Pascal random variables on sl2(R) -- 6.3.Discrete distributions on so(2) and so(3) -- 6.4.The Lie algebra e(2) -- Exercises -- 7.Weyl calculus on real Lie algebras -- 7.1.Joint moments of noncommuting random variables -- 7.2.Combinatorial Weyl calculus -- 7.3.Heisenberg--Weyl algebra -- 7.4.Functional calculus on real Lie algebras -- 7.5.Functional calculus on the affine algebra -- 7.6.Wigner functions on so(3) -- 7.7.Some applications -- Exercises -- 8.Levy processes on real Lie algebras -- 8.1.Definition -- 8.2.Schurmann triples -- 8.3.Levy processes on and osc -- 8.4.Classical processes -- Exercises -- 9.A guide to the Malliavin calculus -- 9.1.Creation and annihilation operators -- 9.2.Wiener space -- 9.3.Poisson space -- 9.4.Sequence models -- Exercises -- 10.Noncommutative Girsanov theorem -- 10.1.General method -- 10.2.Quasi-invariance on osc -- 10.3.Quasi-invariance on sl2(R) -- 10.4.Quasi-invariance on -- 10.5.Quasi-invariance for Levy processes -- Exercises -- 11.Noncommutative integration by parts -- 11.1.Noncommutative gradient operators -- 11.2.Affine algebra -- 11.3.Noncommutative Wiener space -- 11.4.The white noise case -- Exercises -- 12.Smoothness of densities on real Lie algebras -- 12.1.Noncommutative Wiener space -- 12.2.Affine algebra -- 12.3.Towards a Hormander-type theorem -- Exercises -- Appendix -- A.1.Polynomials -- A.2.Moments and cumulants -- A.3.Fourier transform -- A.4.Cauchy--Stieltjes transform -- A.5.Adjoint action -- A.6.Nets -- A.7.Closability of linear operators -- A.8.Tensor products -- Exercise solutions -- Chapter 1 -- Chapter 2 -- Chapter 3 -- Chapter 4 -- Chapter 5 -- Chapter 6 -- Chapter 7 -- Chapter 8 -- Chapter 9 -- Chapter 10 -- Chapter 11 -- Chapter 12.

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ISBN

9781107128651 hardcover alkaline paper
110712865X hardcover alkaline paper

Bibliography Note

Includes bibliographical references and index.

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