Elements of Random Walk and Diffusion Processes

Oliver C. Ibe 지음

Wiley 2013년 08월 29일 출간

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Presents an important and unique introduction to random walk theoryRandom walk is a stochastic process that has proven to be a useful model in understanding discrete-state discrete-time processes across a wide spectrum of scientific disciplines. Elements of Random Walk and Diffusion Processes provides an interdisciplinary approach by including numerous practical examples and exercises with real-world applications in operations research, economics, engineering, and physics.Featuring an introduction to powerful and general techniques that are used in the application of physical and dynamic processes, the book presents the connections between diffusion equations and random motion. Standard methods and applications of Brownian motion are addressed in addition to Levy motion, which has become popular in random searches in a variety of fields. The book also covers fractional calculus and introduces percolation theory and its relationship to diffusion processes.With a strong emphasis on the relationship between random walk theory and diffusion processes, Elements of Random Walk and Diffusion Processes features:Basic concepts in probability, an overview of stochastic and fractional processes, and elements of graph theoryNumerous practical applications of random walk across various disciplines, including how to model stock prices and gambling, describe the statistical properties of genetic drift, and simplify the random movement of molecules in liquids and gasesExamples of the real-world applicability of random walk such as node movement and node failure in wireless networking, the size of the Web in computer science, and polymers in physicsPlentiful examples and exercises throughout that illustrate the solution of many practical problemsElements of Random Walk and Diffusion Processes is an ideal reference for researchers and professionals involved in operations research,

목차

Preface xiiiAcknowledgments xv1 Review of Probability Theory 11.1 Introduction 11.2 Random Variables 11.3 Transform Methods 51.4 Covariance and Correlation Coefficient 91.5 Sums of Independent Random Variables 101.6 Some Probability Distributions 111.7 Limit Theorems 16Problems 192 Overview of Stochastic Processes 212.1 Introduction 212.2 Classification of Stochastic Processes 222.3 Mean and Autocorrelation Function 222.4 Stationary Processes 232.5 Power Spectral Density 242.6 Counting Processes 252.7 Independent Increment Processes 252.8 Stationary Increment Process 252.9 Poisson Processes 262.10 Markov Processes 292.11 Gaussian Processes 382.12 Martingales 38Problems 413 One-Dimensional Random Walk 443.1 Introduction 443.2 Occupancy Probability 463.3 Random Walk as a Markov Chain 493.4 Symmetric Random Walk as a Martingale 493.5 Random Walk with Barriers 503.6 Mean-Square Displacement 503.7 Gambler&'s Ruin 523.8 Random Walk with Stay 563.9 First Return to the Origin 573.10 First Passage Times for Symmetric Random Walk 593.11 The Ballot Problem and the Reflection Principle 653.12 Returns to the Origin and the Arc-Sine Law 673.13 Maximum of a Random Walk 723.14 Two Symmetric Random Walkers 733.15 Random Walk on a Graph 733.16 Random Walks and Electric Networks 803.17 Correlated Random Walk 853.18 Continuous-Time Random Walk 903.19 Reinforced Random Walk 943.20 Miscellaneous Random Walk Models 983.21 Summary 100Problems 1004 Two-Dimensional Random Walk 1034.1 Introduction 1034.2 The Pearson Random Walk 1054.3 The Symmetric 2D Random Walk 1104.4 The Alternating Random Walk 11

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저자(글) Oliver C. Ibe

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OLIVER C. IBE, ScD, is Associate Professor in the Department of Electrical and Computer Engineering at the University of Massachusetts at Lowell. He has more than thirty years of experience in academia and the telecommunications industry in various technical and management capacities. Dr. Ibe's research interests include stochastic systems modeling, bioinformatics, and communication network performance modeling. He is the author of Converged Network Architectures: Delivering Voice over IP, ATM, and Frame Relay and Fundamentals of Stochastic Networks, both published by Wiley.