Stochastic Models for Fractional Calculus

ISBN-13: 9783110258691 / Angielski / Twarda / 2011 / 301 str.

Mark M. Meerschaert; Alla Sikorskii

Opis

This monograph develops the basic theory of fractional calculus and anomalous diffusion, from the point of view of probability. We 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.

Spis treści

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 LePage 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