List of publications

List of publications by fields of interest

Beyond the Triangle:
Brownian Motion, Ito Calculus, and Fokker-Planck Equation - Fractional Generalizations,Sabir Umarov, Marjorie Hahn, Kei Kobayashi

*Published by World Scientific, ISBN 978-981-3230-91-0 (hardcover)
ISBN: 978-981-3230-99-6 (ebook)
*

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols, Sabir Umarov

*Published by Springer, ISBN 978-3-319-20770-4, 978-3-319-20771-1 (eBook)*

The book is devoted to the fundamental relationship between three objects: a stochastic process, stochastic differential equations driven by that process and their associated Fokker-Planck-Kolmogorov equations. This book discusses wide fractional generalizations of this fundamental triple relationship, where the driving process represents a time-changed stochastic process; the Fokker-Planck-Kolmogorov equation involves time-fractional order derivatives and spatial pseudo-differential operators; and the associated stochastic differential equation describes the stochastic behavior of the solution process. It contains recent results obtained in this direction. This book is important since the latest developments in the field, including the role of driving processes and their scaling limits, the forms of corresponding stochastic differential equations, and associated FPK equations, are systematically presented. Examples and important applications to various scientific, engineering, and economics problems make the book attractive for all interested researchers, educators, and graduate students.

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols, Sabir Umarov

The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of these theories to various initial and multi-point boundary value problems for pseudo-differential equations. Fractional Fokker-Planck-Kolmogorov equations associated with a large class of stochastic processes are presented.
Fractional Duhamel's principle is developed and presented in detail.
A complex version of the theory of pseudo-differential operators with meromorphic symbols based on the recently introduced complex Fourier transform is developed and applied for initial and boundary value problems for systems of complex differential and pseudo-differential equations. Each chapter contains Additonal Notes on historical facts, relevant results, open problems, and references for further reading.

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