High Dimensional Nonlinear Diffusion Stochastic Processes

This work is devoted to high-dimensional (or large-scale) diffusion stochastic processes (DSPs) with nonlinear coefficients. These processes are closely associated with nonlinear Ito's stochastic ordinary differential equations (ISODEs) and with the space-discretized versions of nonlinear Ito's stochastic partial integro-differential equations. The latter models include Ito's stochastic partial differential equations (ISPDEs). The book presents the new analytical treatment which can serve as the basis of a combined, analytical-numerical approach to the greater computational efficiency in engineering problems. A few examples discussed in the book include: the high-dimensional DSPs; the modification of the well-known stochastic-adaptive-interpolation method by means of bases of function spaces; ISPDEs as the tool to consistently model non-Markov phenomena; the ISPDE system for semiconductor devices; the corresponding classification of charge transport in macroscale, mesoscale and microscale semiconductor regions based on the wave-diffusion equation; the fully time-domain nonlinear-friction aware analytical model for the velocity covariance of particle of uniform fluid, simple or dispersed; the specific time-domain analytics for the long, non-exponential "tails" of the velocity in case of the hard-sphere fluid. These examples demonstrate not only the capabilities of the developed techniques but also emphasize the usefulness of the complex-system-related approaches to solve some problems which have not been solved with the traditional, statistical-physics methods yet. From this viewpoint, the book can be regarded as a kind of complement to such books as "Introduction to the Physics of Complex Systems: the Mesoscopic Approach to Fluctuations, Nonlinearity and Self-Organization" by Serra, Andretta, Compiani and Zanarini, "Stochastic Dynamical Systems: Concepts, Numerical Methods, Data Analysis" and "Statistical Physics: An Advanced Approach with Applications" by Honerkamp, which deal with physics of complex systems, some of the corresponding analysis methods and an innvoative, stochastics-based vision of theoretical physics. To facilitate the reading by non-mathematicians, the introductory chapter outlines the basic notions and results of theory of Markov and diffusion stochastic processes without involving the measure-theoretical approach. This presentation is based on probability densities commonly used in engineering and applied sciences.