High-Dimensional Nonlinear Diffusion Stochastic Processes Modelling for Engineering Applications

This book is the first one 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 greater computational efficiency in engineering problems. A few examples discussed in the book include: the high-dimensional DSPs described with the ISODE systems for semiconductor circuits; the nonrandom model for stochastic resonance (and other noise-induced phenomena) in 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 veiwpoint, 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 innovative, stochastics-based vision of theoretical physics. To facilitate the reading by nonmathematicians, 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. Contents: Introductory ChapterDiffusion ProcessesInvariant Diffusion ProcessesStationary Diffusion ProcessesItô's Stochastic Partial Differential Equations as Non-Markov Models Leading to High-Dimensional Diffusion ProcessesItô's Stochastic Partial Differential Equations for Electron Fluids in SemiconductorsDistinguishing Features of Engineering ApplicationsAnalytical-Numerical Approach to Engineering Problems and Common Analytical TechniquesAppendices:Example of Markov Processes: Solutions of the Cauchy Problems for Ordinary Differential Equation SystemSignal-to-Noise RatioExample of Application of Corollary 1.2: Nonlinear Friction and Unbounded Stationary Probability Density of the Particle Velocity in Uniform FluidProofs of the Theorems in Chapter 2 and Other DetailsProofs of the Theorems in Chapter 4Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and Chemical-Reaction/Generation-Recombination NoiseExample: Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in Three-Dimensional SpaceResources for Engineering Parallel Computing under Windows 95 Readership: Nonmathematicians (e.g., theoretical physicists, engineers in industry, specialists in models for finance or biology, computing scientists), mathematicians, undergraduate and postgraduate students of the corresponding specialties, managers in applied sciences and engineering dealing with the advancements in the related fields, any specialists who use diffusion stochastic processes to model high-dimensional (or large-scale) nonlinear stochastic systems. Keywords:Diffusion Stochastic Process;Nonlinear Coefficients;High-Dimensional Diffusion Process;Analytical-Numerical Methods;Nonlinear Differential Equations;Noise-Induced Phenomena;Ito's Stochastic PDE;Function Banach Spaces;Nonlinear Stochastic Models;Wave-Diffusion;Hidden Randomness;Common Hydrodynamic EquationsReviews:“The book is clearly written and organized, with self-contained contents which provide the interested reader with all the mathematical tools needed for their correct understanding and application … it provides a large variety of new ideas and problems to be solved, which both pure and applied mathematicians should be strongly motivated to tackle.”Mathematical Reviews, 2002