Overview

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.

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9783110258691

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Reviews

[...] 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

Author Information

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

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