Fractional calculus is a rapidly growing field of research, at the interface between probability, differential equations, and mathematical physics. It is used to model anomalous diffusion, in which a cloud of particles spreads in a different manner than traditional diffusion. This monograph develops the basic theory of fractional calculus and anomalous diffusion, from the point of view of probability.
In this book, we will see how fractional calculus and anomalous diffusion can be understood at a deep and intuitive level, using ideas from probability. It covers basic limit theorems for random variables and random vectors with heavy tails. This includes regular variation, triangular arrays, infinitely divisible laws, random walks, and stochastic process convergence in the Skorokhod topology. The basic ideas of fractional calculus and anomalous diffusion are closely connected with heavy tail limit theorems. Heavy tails are applied in finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering.
The goal of this book is to prepare graduate students in probability for research in the area of fractional calculus, anomalous diffusion, and heavy tails. Many interesting problems in this area remain open. This book will guide the motivated reader to understand the essential background needed to read and unerstand current research papers, and to gain the insights and techniques needed to begin making their own contributions to this rapidly growing field.
Table des matières
Preface
1 Introduction
1.1 The traditional diffusion model
1.2 Fractional diffusion
2 Fractional Derivatives
2.1 The Grünwald formula
2.2 More fractional derivatives
2.3 The Caputo derivative
2.4 Time-fractional diffusion
3 Stable Limit Distributions
3.1 Infinitely divisible laws
3.2 Stable characteristic functions
3.3 Semigroups
3.4 Poisson approximation
3.5 Shifted Poisson approximation
3.6 Triangular arrays
3.7 One-sided stable limits
3.8 Two-sided stable limits
4 Continuous Time Random Walks
4.1 Regular variation
4.2 Stable Central Limit Theorem
4.3 Continuous time random walks
4.4 Convergence in Skorokhod space
4.5 CTRW governing equations
5 Computations in R
5.1 R codes for fractional diffusion
5.2 Sample path simulations
6 Vector Fractional Diffusion
6.1 Vector random walks
6.2 Vector random walks with heavy tails
6.3 Triangular arrays of random vectors
6.4 Stable random vectors
6.5 Vector fractional diffusion equation
6.6 Operator stable laws
6.7 Operator regular variation
6.8 Generalized domains of attraction
7 Applications and Extensions
7.1 Le Page Series Representation
7.2 Tempered stable laws
7.3 Tempered fractional derivatives
7.4 Pearson Diffusion
7.5 Classification of Pearson diffusions
7.6 Spectral representations of the solutions of Kolmogorov equations
7.7 Fractional Brownian motion
7.8 Fractional random fields
7.9 Applications of fractional diffusion
7.10 Applications of vector fractional diffusion
Bibliography
Index
A propos de l’auteur
Mark M. Meerschaert and Alla Sikorskii, Michigan State University, East Lansing, Michigan, USA.Plus d’ebooks du même auteur(s) / Éditeur
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