High-dimensional Nonlinear Diffusion Stochastic Processes: (56 Series on Advances in Mathematics for Applied Sciences)

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.

Table of Contents:
Introductory Chapter; Diffusion Processes; Invariant Diffusion Processes; Stationary Diffusion Processes; Ito's Stochastic Partial Differential Equations as Non-Markov Models Leading to High-Dimensional Diffusion Processes; Ito's Stochastic Partial Differential Equations for Electron Fluids in Semiconductors; Distinguishing Features of Engineering Applications; Analytical-Numerical Approach to Engineering Problems and Common Analytical Techniques; Appendlces: Example of Markov Processes: Solutions of the Cauchy Problems for Ordinary Differential Equation System; Signal-to-Noise Ratio; Example of Application of Corollary 1.2: Nonlinear Friction and Unbounded Stationary Probability Density of the Particle Velocity in Uniform Fluid; Proofs of the Theorems in Chapter 2 and Other Details; Proofs of the Theorems in Chapter 4; Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and Chemical-Reaction/Generation-Recombination Noise; Example: Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in Three-Dimensional Space; Resources for Engineering Parallel Computing under Windows 95.