Stochastic Models for Fractional Calculus (eBook)

Mark M. Meerschaert, Alla Sikorskii (Autoren)

eBook Download: PDF

2012
301 Seiten
De Gruyter (Verlag)
978-3-11-025816-5 (ISBN)

Infos

This monograph develops the basic theory of fractional calculus and anomalous diffusion, from the point of view of probability. The reader will see how fractional calculus and anomalous diffusion can be understood at a deep and intuitive level, using ideas from probability. The book covers basic limit theorems for random variables and random vectors with heavy tails. Heavy tails are applied in finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering.

Mark M. Meerschaert and Alla Sikorskii, Michigan State University, East Lansing, Michigan, USA.

Autor

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Mark M. Meerschaert and Alla Sikorskii, Michigan State University, East Lansing, Michigan, USA.

Inhalt

Preface 6
Contents 10
Acknowledgments 8
1 Introduction 12
1.1 The traditional diffusion model 13
1.2 Fractional diffusion 21
2 Fractional Derivatives 32
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
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
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
5.1 R codes for fractional diffusion 128
5.2 Sample path simulations 137
6 Vector Fractional Diffusion 154
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
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
Index 300

Rezensionen

"[...] It is mainly a book for graduate students and researchers in probability and stochastic processes; however, it is written in a style which makes it accessible also to scientists from other fields. In particular, the examples and real-life applications presented make it attractive also for non-mathematicians. Overall, the book should become a standard reference for researchers who work in the area of fractional calculus, anomalous diffusion, and heavy tails. It will definitely inspire both theoreticians and practitioners to further develop the theory of anomalous diffusion."
Marcin Magdziarz, Mathematical Reviews

Produkt-Details

• Erscheint lt. Verlag: 23.12.2012
• Reihe/Serie: De Gruyter Studies in Mathematics
• Reihe/Serie: ISSN
• Zusatzinfo: 77 b/w ill.
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Anomalous diffusion • Fractional Calculus Model • Fractional Derivative • Fractional Diffusion Equation • fractional diffusion equations, random walks, statistical physics • Particle Jump • Probability • Random Walk • Satistical Physics • Tempered Fractional Derivative • Vector Fractional Derivative
• ISBN-10: 3-11-025816-1 / 3110258161
• ISBN-13: 978-3-11-025816-5 / 9783110258165