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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. 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.
"The book systematically presents the theory of pseudo-differential operators with symbols singular in dual variables. ... The book is interesting both for probabilists and for researchers in those areas of functional analysis where pseudo-differential operators arise. The interesting connections are emphasized by a multitude of examples. Additional notes at the end of each chapter provide nice historical insights and help to understand the role of the presented results within the big picture." (Alexander Schnurr, zbMATH 1331.35005, 2016)