Provides comprehensive coverage of the most recent developments in the theory of non-Archimedean pseudo-differential equations and its application to stochastics and mathematical physics--offering current methods of construction for stochastic processes in the field of p-adic numbers and related structures. Develops a new theory for parabolic equations over non-Archimedean fields in relation to Markov processes.
Author(s): Anatoly Kochubei
Edition: 1st
Year: 2001
Language: English
Pages: 324
Preface......Page 9
Contents......Page 14
1.1 Field Extensions......Page 17
1.2 Finite Fields......Page 19
1.3 Local Fields......Page 21
1.4 Polynomials And Quadratic Forms......Page 27
1.5 Integration And Harmonic Analysis......Page 29
1.6 Distributions......Page 32
1.7 Homogeneous Distributions......Page 36
1.8 Gaussian Integrals......Page 39
1.9 Adeles......Page 46
2.1 The Riesz Kernel......Page 49
2.2 The Fractional Differentiation Operator......Page 52
2.3.1. Local Generator.......Page 55
2.3.2. The Case P = Oo.......Page 57
2.4 The Fractional Differential Operator On Ideles And Zeroes Of Riemann's Zeta Function......Page 60
2.5.1. Hypersingular Integral Representation.......Page 63
2.5.2. Fundamental Solution.......Page 65
2.5.3. Green Function.......Page 68
2.6 Coordinate Representation......Page 71
2.7 Fundamental Solutions Of Elliptic Operators......Page 75
2.8 Green Functions Of Elliptic Operators......Page 86
2.9 Hyperbolic Equations......Page 89
2.10.1. Classes Of Functions.......Page 92
2.10.2. The Inhomogeneous Equation.......Page 97
2.10.3. The Cauchy Problem.......Page 100
2.11 On Some Models Of P-adic Quantum Theory......Page 102
2.12 Comments......Page 104
3.1 Fractional Differentiation Operator......Page 105
3.2.1. General Properties.......Page 108
3.2.2. The Case Of A Radial Potential.......Page 110
3.2.3. Essential Spectrum.......Page 116
3.3.1. Structure Of Open Subsets.......Page 120
3.3.2. Model Examples.......Page 122
3.3.3. Invariant Subspaces.......Page 124
3.3.4. Self-adjointriess.......Page 126
3.4.2. Deficiency Index.......Page 134
3.4.3. Self-adjoint Extensions.......Page 141
3.5 Multiplicative Fractional Differentiation......Page 142
3.6 Comments......Page 144
4.1.1. Fundamental Solution.......Page 146
4.1.2. Heat Potentials And Cauchy Problem.......Page 150
4.2 Probabilistic Interpretation......Page 156
4.3 Stabilization......Page 159
4.4 General Uniqueness Theorem......Page 163
4.5.1. Parametrix.......Page 165
4.5.2. Heat Potentials.......Page 167
4.5.3. Cauchy Problem.......Page 168
4.5.4. Non-negative Solutions.......Page 174
4.6.1. The Process And Its Generator.......Page 175
4.7 Heat Equation On The Group Of Units......Page 180
4.8.1. Motivation.......Page 183
4.8.2. Hyper-singular Integrals.......Page 184
4.8.3. Parametrix.......Page 189
4.8.4. Heat Potential.......Page 197
4.8.5. Cauchy Problem.......Page 203
4.9 Comments......Page 204
5.1 Notations And Preliminaries......Page 205
5.2 Stochastic Integrals Of Deterministic Functions......Page 207
5.3.1. The Main Lemma.......Page 214
5.3.2. Definition And Properties.......Page 215
5.4.1. Existence And Uniqueness.......Page 217
5.4.2. Markov Processes.......Page 220
5.5 Rotation-invariant Processes With Independent Increments......Page 224
5.6 The Generator And Its Spectrum......Page 230
5.7 Recurrence And Hitting Probabilities......Page 235
5.8 Processes On Adeles......Page 240
5.9.1. Normalized Sums.......Page 243
5.9.2. Probability Distributions Of Stable Type.......Page 244
5.9.3. Degenerate Cases.......Page 252
5.10 Comments......Page 254
6.1.1. Definitions.......Page 256
6.1.2. Gaussian Random Variables.......Page 257
6.2.1. Main Notions.......Page 269
6.3.1. A-algebras.......Page 271
6.3.2. Fourier Transforms.......Page 272
6.3.3. Gaussian Measure.......Page 274
6.4.1. An Image Of The Fourier Transform.......Page 278
6.4.2. Fractional Differentiation Operator.......Page 281
6.4.3. Hyper-singular Integral Representation.......Page 283
6.5 Spectrum......Page 285
6.6.1. An Auxiliary Equation.......Page 287
6.6.2. Heat Measure.......Page 289
6.7 Hypersingular Integral Representation......Page 293
6.8 Comments......Page 297
7.1.1. Preliminaries.......Page 298
7.1.2. Construction Of The Process.......Page 300
7.1.3. Some Properties.......Page 301
7.2.1. Definition.......Page 304
7.2.2. The Canonical Measure On The Space Of Sequences.......Page 305
7.2.3. White Noise With P-adic Time.......Page 306
7.2.4. An Antiderivative Of The White Noise.......Page 307
7.2.5. The Wiener Process.......Page 310
7.3 Comments......Page 313
Bibliography......Page 314
