## High-dimensional Nonlinear Diffusion Stochastic Processes: Modelling for Engineering ApplicationsThis 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 OC tailsOCO 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 OC Introduction to the Physics of Complex Systems. The Mesoscopic Approach to Fluctuations, Nonlinearity and Self-OrganizationOCO by Serra, Andretta, Compiani and Zanarini, OC Stochastic Dynamical Systems. Concepts, Numerical Methods, Data AnalysisOCO and OC Statistical Physics: An Advanced Approach with ApplicationsOCO 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 Chapter; Diffusion Processes; Invariant Diffusion Processes; Stationary Diffusion Processes; It''s Stochastic Partial Differential Equations as Non-Markov Models Leading to High-Dimensional Diffusion Processes; It''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; Appendices: 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. 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." |

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### Contents

1 | |

Diffusion Processes | 63 |

Invariant Diffusion Processes | 85 |

Stationary Diffusion Processes | 107 |

Itos Stochastic Partial Differential | 141 |

Itos Stochastic Partial Differential | 163 |

Distinguishing Features | 197 |

### Common terms and phrases

analytical analytical-numerical approach Appendix approximation Arnold assertion assumption asymptotic Banach spaces basis Bellomo Chapter computational considered corresponding covariance Definition Demir dependence derivatives described determined deterministic differential equations diffusion functions domain Q drift and diffusion drift function elementary event engineering entries evaluate example expression feature fluid formula function g holds independent initial condition initial-value problem instance integral ISODE system ISPDE ISPIDE Ito's stochastic Lebesgue integral linear macroscopic Mamontov and Willander Markov process mathematical matrix H(t,x means method momentum-relaxation noise nonlinear Note ODE system parameter particle physical present book probability density quantity random variable Remark respect right-hand side scalar semiconductor SF-ISPDE Soize solution specific spectral density stationary DSP Stochastic Differential Equations stochastic process stochastic resonance t_<tt t>to techniques Theorem 2.1 theory tion transition probability density treatment uniformly valid variance matrix velocity well-known Wiener process

### References to this book

Generalized Kinetic Models in Applied Sciences: Lecture Notes on ... Luisa Arlotti Limited preview - 2003 |