Stochastic models for fractional calculus

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Fractional calculus is an emerging research area at the intersection of probability, differential equations, and mathematical physics, particularly useful for modeling anomalous diffusion, where particle spread deviates from traditional diffusion patterns. This monograph presents the foundational theory of fractional calculus and anomalous diffusion through a probabilistic lens. It delves into fundamental limit theorems for random variables and vectors with heavy tails, covering topics such as regular variation, triangular arrays, infinitely divisible laws, random walks, and stochastic process convergence in the Skorokhod topology. The concepts of fractional calculus and anomalous diffusion are intricately linked to heavy tail limit theorems, which find applications across various fields, including finance, insurance, physics, geophysics, cell biology, ecology, medicine, and computer engineering. Aimed at preparing graduate students in probability for research in fractional calculus, anomalous diffusion, and heavy tails, this book addresses many unresolved problems in the field. It equips motivated readers with the essential background to comprehend current research papers and develop the insights and techniques necessary to contribute to this rapidly advancing area of study.