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Probability on Real Lie Algebras [Kõva köide]

(Nanyang Technological University, Singapore), (Université de Franche-Comté)
  • Formaat: Hardback, 302 pages, kõrgus x laius x paksus: 229x152x21 mm, kaal: 600 g, Worked examples or Exercises; 2 Line drawings, unspecified
  • Sari: Cambridge Tracts in Mathematics
  • Ilmumisaeg: 25-Jan-2016
  • Kirjastus: Cambridge University Press
  • ISBN-10: 110712865X
  • ISBN-13: 9781107128651
Teised raamatud teemal:
Probability on Real Lie AlgebrasSuurem pilt
  • Formaat: Hardback, 302 pages, kõrgus x laius x paksus: 229x152x21 mm, kaal: 600 g, Worked examples or Exercises; 2 Line drawings, unspecified
  • Sari: Cambridge Tracts in Mathematics
  • Ilmumisaeg: 25-Jan-2016
  • Kirjastus: Cambridge University Press
  • ISBN-10: 110712865X
  • ISBN-13: 9781107128651
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781107128651.html
Märksõnad:
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.

Muu info

This monograph is a progressive introduction to non-commutativity in probability theory.
Notation xi
Preface xiii
Introduction xv
1 Boson Fock space
1(9)
1.1 Annihilation and creation operators
1(3)
1.2 Lie algebras on the boson Fock space
4(2)
1.3 Fock space over a Hilbert space
6(4)
Exercises
9(1)
2 Real Lie algebras
10(17)
2.1 Real Lie algebras
10(2)
2.2 Heisenberg--Weyl Lie algebra
12(1)
2.3 Oscillator Lie algebra osc
13(1)
2.4 Lie algebra sl2(R)
14(6)
2.5 Affine Lie algebra
20(1)
2.6 Special orthogonal Lie algebras
21(6)
Exercises
26(1)
3 Basic probability distributions on Lie algebras
27(20)
3.1 Gaussian distribution on
27(4)
3.2 Poisson distribution on osc
31(5)
3.3 Gamma distribution on sl2(R)
36(11)
Exercises
44(3)
4 Noncommutative random variables
47(28)
4.1 Classical probability spaces
47(1)
4.2 Noncommutative probability spaces
48(6)
4.3 Noncommutative random variables
54(3)
4.4 Functional calculus for Hermitian matrices
57(2)
4.5 The Lie algebra so(3)
59(6)
4.6 Trace and density matrix
65(5)
4.7 Spin measurement and the Lie algebra so(3)
70(5)
Exercises
72(3)
5 Noncommutative stochastic integration
75(15)
5.1 Construction of the Fock space
75(5)
5.2 Creation, annihilation, and conservation operators
80(3)
5.3 Quantum stochastic integrals
83(3)
5.4 Quantum Ito table
86(4)
Exercises
88(2)
6 Random variables on real Lie algebras
90(13)
6.1 Gaussian and Poisson random variables on osc
90(4)
6.2 Meixner, gamma, and Pascal random variables on sl2(R)
94(2)
6.3 Discrete distributions on so(2) and so(3)
96(1)
6.4 The Lie algebra e(2)
97(6)
Exercises
99(4)
7 Weyl calculus on real Lie algebras
103(28)
7.1 Joint moments of noncommuting random variables
103(3)
7.2 Combinatorial Weyl calculus
106(1)
7.3 Heisenberg--Weyl algebra
107(7)
7.4 Functional calculus on real Lie algebras
114(3)
7.5 Functional calculus on the affine algebra
117(5)
7.6 Wigner functions on so(3)
122(6)
7.7 Some applications
128(3)
Exercises
130(1)
8 Levy processes on real Lie algebras
131(18)
8.1 Definition
131(3)
8.2 Schurmann triples
134(6)
8.3 Levy processes on and osc
140(2)
8.4 Classical processes
142(7)
Exercises
148(1)
9 A guide to the Malliavin calculus
149(29)
9.1 Creation and annihilation operators
149(6)
9.2 Wiener space
155(7)
9.3 Poisson space
162(6)
9.4 Sequence models
168(10)
Exercises
173(5)
10 Noncommutative Girsanov theorem
178(12)
10.1 General method
178(2)
10.2 Quasi-invariance on osc
180(3)
10.3 Quasi-invariance on sl2(R)
183(1)
10.4 Quasi-invariance on
184(1)
10.5 Quasi-invariance for Levy processes
185(5)
Exercises
189(1)
11 Noncommutative integration by parts
190(27)
11.1 Noncommutative gradient operators
190(2)
11.2 Affine algebra
192(5)
11.3 Noncommutative Wiener space
197(15)
11.4 The white noise case
212(5)
Exercises
216(1)
12 Smoothness of densities on real Lie algebras
217(14)
12.1 Noncommutative Wiener space
217(5)
12.2 Affine algebra
222(2)
12.3 Towards a Hormander-type theorem
224(7)
Exercises
230(1)
Appendix
231(18)
A.1 Polynomials
231(8)
A.2 Moments and cumulants
239(2)
A.3 Fourier transform
241(2)
A.4 Cauchy--Stieltjes transform
243(1)
A.5 Adjoint action
244(1)
A.6 Nets
245(1)
A.7 Closability of linear operators
246(1)
A.8 Tensor products
247(2)
Exercise solutions
249(22)
Chapter 1
249(1)
Chapter 2
250(3)
Chapter 3
253(3)
Chapter 4
256(3)
Chapter 5
259(1)
Chapter 6
260(6)
Chapter 7
266(1)
Chapter 8
266(1)
Chapter 9
267(2)
Chapter 10
269(1)
Chapter 11
270(1)
Chapter 12
270(1)
References 271(8)
Index 279
Uwe Franz is Professor at the Laboratoire de Mathématiques, UFR Sciences et Techniques, Université de Franche-Comté, Besançon, France. Nicolas Privault is Associate Professor in the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, at Nanyang Technological University, Singapore.