| Preface | 6 |
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| Contents | 10 |
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| Acknowledgments | 8 |
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| 1 Introduction | 12 |
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| 1.1 The traditional diffusion model | 13 |
| 1.2 Fractional diffusion | 21 |
| 2 Fractional Derivatives | 32 |
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| 2.1 The Grünwald formula | 32 |
| 2.2 More fractional derivatives | 40 |
| 2.3 The Caputo derivative | 45 |
| 2.4 Time-fractional diffusion | 53 |
| 3 Stable Limit Distributions | 60 |
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| 3.1 Infinitely divisible laws | 60 |
| 3.2 Stable characteristic functions | 66 |
| 3.3 Semigroups | 70 |
| 3.4 Poisson approximation | 77 |
| 3.5 Shifted Poisson approximation | 80 |
| 3.6 Triangular arrays | 83 |
| 3.7 One-sided stable limits | 88 |
| 3.8 Two-sided stable limits | 92 |
| 4 Continuous Time Random Walks | 98 |
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| 4.1 Regular variation | 98 |
| 4.2 Stable Central Limit Theorem | 105 |
| 4.3 Continuous time random walks | 110 |
| 4.4 Convergence in Skorokhod space | 114 |
| 4.5 CTRW governing equations | 117 |
| 5 Computations in R | 128 |
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| 5.1 R codes for fractional diffusion | 128 |
| 5.2 Sample path simulations | 137 |
| 6 Vector Fractional Diffusion | 154 |
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| 6.1 Vector random walks | 154 |
| 6.2 Vector random walks with heavy tails | 164 |
| 6.3 Triangular arrays of random vectors | 169 |
| 6.4 Stable random vectors | 176 |
| 6.5 Vector fractional diffusion equation | 181 |
| 6.6 Operator stable laws | 188 |
| 6.7 Operator regular variation | 197 |
| 6.8 Generalized domains of attraction | 202 |
| 7 Applications and Extensions | 214 |
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| 7.1 LePage Series Representation | 214 |
| 7.2 Tempered stable laws | 218 |
| 7.3 Tempered fractional derivatives | 224 |
| 7.4 Pearson Diffusions | 228 |
| 7.5 Fractional Pearson diffusions | 244 |
| 7.6 Fractional Brownian motion | 252 |
| 7.7 Fractional random fields | 262 |
| 7.8 Applications of fractional diffusion | 268 |
| 7.9 Applications of vector fractional diffusion | 281 |
| Bibliography | 290 |
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| Index | 300 |
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