Probability on Real Lie Algebras (Cambridge Tracts in Mathematics, Series Number 206)

This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus. 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. Uwe Franz, Université De Franche-comté, Nicolas Privault, Nanyang Technological University, Singapore. Includes Bibliographical References And Index. Content: Introduction 1. Boson fock space 2. Real Lie algebras 3. Basic probability distributions on Lie algebras 4. Noncommutative random variables 5. Noncommutative stochastic integration 6. Random variables on real Lie algebras 7. Weyl calcuus on real Lie algebras 8. Levy processes on real Lie algebras 9. A guide to the Malliavin calculus 10. Noncommutative Girsanov theorem 11. Noncommutative integration by parts 12. Smoothness of densities on real Lie algebras Appendix Exercise solutions.