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Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding.

fractional diffusion --- continuous time random walks --- reaction–diffusion equations --- reaction kinetics --- multidimensional scaling --- fractals --- fractional calculus --- financial indices --- entropy --- Dow Jones --- complex systems --- Skellam process --- subordination --- Lévy measure --- Poisson process of order k --- running average --- complexity --- chaos --- logistic differential equation --- liouville-caputo fractional derivative --- local discontinuous Galerkin methods --- stability estimate --- Mittag-Leffler functions --- Wright functions --- fractional relaxation --- diffusion-wave equation --- Laplace and Fourier transform --- fractional Poisson process complex systems --- distributed-order operators --- viscoelasticity --- transport processes --- control theory --- fractional order PID control --- PMSM --- frequency-domain control design --- optimal tuning --- Gaussian watermarks --- statistical assessment --- false positive rate --- semi-fragile watermarking system --- fractional dynamics --- fractional-order thinking --- heavytailedness --- big data --- machine learning --- variability --- diversity --- telegrapher’s equations --- fractional telegrapher’s equation --- continuous time random walk --- transport problems --- fractional conservations laws --- variable fractional model --- turbulent flows --- fractional PINN --- physics-informed learning --- n/a --- reaction-diffusion equations --- Lévy measure --- telegrapher's equations --- fractional telegrapher's equation

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Stochastic processes have wide relevance in mathematics both for theoretical aspects and for their numerous real-world applications in various domains. They represent a very active research field which is attracting the growing interest of scientists from a range of disciplines.This Special Issue aims to present a collection of current contributions concerning various topics related to stochastic processes and their applications. In particular, the focus here is on applications of stochastic processes as models of dynamic phenomena in research areas certain to be of interest, such as economics, statistical physics, queuing theory, biology, theoretical neurobiology, and reliability theory. Various contributions dealing with theoretical issues on stochastic processes are also included.

arithmetic progressions --- weighted quadratic variation --- fractional differential-difference equations --- small deviations --- periodic intensity functions --- realized volatility --- rate of convergence --- host-parasite interaction --- first Chebyshev function --- regularly varying functions --- Cohen and Grossberg neural networks --- mixture of Gaussian laws --- diffusion model --- transition densities --- re-service --- Strang–Marchuk splitting approach --- random delays --- nematode infection --- first-passage-time --- total variation distance --- forecast combinations --- products of primes --- discrete time stochastic model --- multiplicative noises --- slowly varying functions --- growth curves --- stochastic process --- loan interest rate regulation --- birth-death process --- non-Markovian queue --- catastrophes --- exogenous factors --- seasonal environment --- repairs --- proportional hazard rates --- structural breaks --- transient probabilities --- first passage time (FPT) --- bounds --- double-ended queues --- mixed Gaussian process --- stochastic order --- time between inspections --- busy period --- diffusion --- continuous-time Markov chains --- general bulk service --- time-non-homogeneous birth-death processes --- stand-by server --- reliability --- sensor networks --- random impulses --- scale family of distributions --- maximum likelihood estimation --- multi-state network --- totally positive of order 2 --- lognormal diffusion process --- fractional birth-death processes --- exact asymptotics --- stochastic orders --- time-non-homogeneous jump-diffusion processes --- asymptotic distribution --- inverse first-passage problem --- nonhomogeneous Poisson process --- two-dimensional signature --- multiple vacation --- first-passage time --- mean square stability --- fractional queues --- differential entropy --- random parameter matrices --- Wasserstein distance --- breakdown and repair --- fusion estimation