Preface ........................................................ ix
Introduction and Overview ...................................... xi
1 Gaussian Measures on Infinite-Dimensional Spaces ............. 1
V.I. Bogachev
0 Introduction .............................................. 1
0.1 Notation and terminology ............................. 2
1 Gaussian Measures on  d ................................... 3
2 Infinite-Dimensional Gaussian Distributions ............... 6
3 The Wiener Measure ....................................... 12
4 Radon Gaussian Measures .................................. 17
5 The Cameron-Martin Space and Measurable Linear
Operators ................................................ 19
6 Zero-one Laws and Dichotomies ............................ 32
7 The Ornstein-Uhlenbeck Semigroup ......................... 32
8 The Hermite-Chebyshev Polynomials ........................ 34
9 Sobolev Classes over Gaussian Measures ................... 42
10 Transformations of Gaussian Measures ..................... 51
11 Convexity ................................................ 68
12 Open Problems ............................................ 73
References .................................................. 75
2 Random Fields and Hypergroups ............................... 85
Herbert Heyer
0 Introduction ............................................. 85
1 Commutative Hypergroups .................................. 86
1.1 Definition and first examples ....................... 86
1.2 Some harmonic analysis .............................. 91
1.3 Basic constructions of hypergroups .................. 98
2 Random Fields over Hypergroups .......................... 110
2.1 Second order random fields ......................... 110
2.2 Translation and decomposition ...................... 118
2.3 Harmonizability .................................... 129
3 Generalized Random Fields over Hypergroups .............. 142
3.1 Segal algebras ..................................... 142
3.2 The extended Feichtinger algebra ................... 148
3.3 Covariance and duality ............................. 161
3.4 Suggestions for further research ................... 176
References ................................................. 179
3 A Concise Exposition of Large Deviations ................... 183
F. Hiai
0 Introduction ............................................ 183
1 Definitions and Generalities ............................ 185
2 The Cramer Theorem ...................................... 191
3 The Gärtner-Ellis Theorem ............................... 199
4 Varadhan's Integral Lemma ............................... 211
5 The Sanov Theorem ....................................... 218
6 Large Deviations for Random Matrices .................... 230
7 Quantum Large Deviations in Spin Chains ................. 245
8 Applications of Large Deviations ........................ 252
8.1 Boltzmann-Gibbs entropy and mutual information ..... 252
8.2 Free entropy and orbital free entropy ................. 257
References ................................................. 265
4 Quantum White Noise Calculus and Applications .............. 269
Un Cig Ji and Nobuaki Obata
1 Introduction ............................................ 269
2 Elements of Gaussian Analysis ........................... 273
2.1 Standard construction of countable Hilbert spaces .. 273
2.2 Gaussian space ..................................... 276
2.3 Fock spaces and the Wiener-Ito decomposition ....... 279
2.4 Underlying spaces .................................. 282
3 White Noise Distributions ............................... 285
3.1 Standard CKS-space ................................. 285
3.2 Brownian motion .................................... 291
3.3 The S-transform .................................... 292
3.4 Infinite dimensional holomorphic functions ......... 297
4 White Noise Operators ................................... 300
4.1 White noise operators and their symbols ............ 300
4.2 Quantum white noise ................................ 302
4.3 Integral kernel operators and Fock expansion ....... 306
4.4 Characterization of operator symbols ............... 310
4.5 Wick product and wick multiplication operators ..... 311
4.6 Multiplication operators ........................... 314
4.7 Convolution operators .............................. 315
5 Quantum Stochastic Gradients ............................ 321
5.1 Annihilation, creation and conservation processes .. 321
5.2 Classical stochastic gradient ...................... 322
5.3 Creation gradient .................................. 324
5.4 Annihilation gradient .............................. 327
5.5 Conservation gradient .............................. 329
6 Quantum Stochastic Integrals ............................ 331
6.1 The Hitsuda-Skorohod integral ...................... 331
6.2 Creation integral .................................. 331
6.3 Annihilation integral .............................. 334
6.4 Conservation integral .............................. 335
7 Quantum White Noise Derivatives ......................... 336
7.1 Quadratic functions of quantum white noise ......... 336
7.2 Quantum white noise derivatives .................... 337
7.3 Wick derivations ................................... 340
7.4 Quantum white noise differential equations of
Wick type .......................................... 342
7.5 The implementation problem ......................... 343
References ................................................. 348
5 Weak Radon-Nikodym Derivatives, Dunford-Schwartz Type
Integration, and Cramer and Karhunen Processes ............. 355
Yûichirô Kakihara
1 Introduction ............................................ 355
2 Hilbert Space Valued Measures ........................... 357
2.1 Radon-Nikodym property ............................. 357
2.2 Weak Radon-Nikodym derivatives ..................... 360
2.3 Existence and uniqueness ........................... 364
2.4 Orthogonally scattered measures and dilation ....... 374
2.5 Dunford-Schwar-te type integration ................. 377
2.6 Bimeasure integration .............................. 382
3 Hilbert-Schmidt Class Operator Valued Measures .......... 384
3.1 The space of Hilbert-Schmidt class operators as
a normal Hilbert module ............................ 384
3.2 The space  1(ξ) .................................... 386
3.3 Weak Radon-Nikodym derivatives ..................... 390
3.4 The spaces 1DS(ξ) and 1*(ξ) ....................... 396
3.5 The spaces 1DS(η)) and L2(Fη) ...................... 404
3.6 The spaces 1*(ξ) and 2*(Mξ) ....................... 406
4 Cramer and Karhunen Processes ........................... 414
4.1 Infinite dimensional second order stochastic
processes .......................................... 414
4.2 Cramer processes ................................... 416
4.3 Karhunen processes ................................. 420
4.4 Operator representation ............................ 423
References ................................................. 428
6 Entropy, SDE-LDP and Fenchel-Legendre-Orlicz Classes ....... 431
M.M. Rao
1 Introduction ............................................ 431
2 Error Estimation Problems from Probabilty Limit Theory .. 434
3 Higher Order SDE and Related Classes .................... 450
4 Entropy, Action/Rate Punctionals and LDP ................ 461
5 Vector Valued Processes and Multiparameter FLO Classes .. 483
6 Evaluations and Representations of Conditional Means .... 493
References ................................................. 498
7 Bispectral Density Estimation in Harmonizable Processes .... 503
H. Soedjak
1 Introduction ............................................ 503
2 Assumptions and a Resampling Procedure .................. 505
3 The Limit Distribution of the Estimator ................. 515
4 Final Remarks and Suggestions ........................... 558
References ................................................. 560
Contributors .................................................. 561
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