Elements of Random Walk and Diffusion Processes.

By: Ibe, Oliver CMaterial type: TextTextSeries: Wiley Series in Operations Research and Management Science SerPublisher: Somerset : John Wiley & Sons, Incorporated, 2013Copyright date: ©2013Edition: 1st edDescription: 1 online resource (278 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781118629857Subject(s): Diffusion processes | Random walks (Mathematics)Genre/Form: Electronic books.Additional physical formats: Print version:: Elements of Random Walk and Diffusion ProcessesDDC classification: 519.282 LOC classification: QA274.73 -- .I24 2013ebOnline resources: Click to View
Contents:
Intro -- Elements of Random Walk and Diffusion Processes -- Copyright -- Contents -- Preface -- Acknowledgments -- 1 Review of Probability Theory -- 1.1 Introduction -- 1.2 Random Variables -- 1.2.1 Distribution Functions -- 1.2.2 Discrete Random Variables -- 1.2.3 Continuous Random Variables -- 1.2.4 Expectations -- 1.2.5 Moments of Random Variables and the Variance -- 1.3 Transform Methods -- 1.3.1 The Characteristic Function -- 1.3.2 Moment-Generating Property of the Characteristic Function -- 1.3.3 The s-Transform -- 1.3.4 Moment-Generating Property of the s-Transform -- 1.3.5 The z-Transform -- 1.3.6 Moment-Generating Property of the z-Transform -- 1.4 Covariance and Correlation Coefficient -- 1.5 Sums of Independent Random Variables -- 1.6 Some Probability Distributions -- 1.6.1 The Bernoulli Distribution -- 1.6.2 The Binomial Distribution -- 1.6.3 The Geometric Distribution -- 1.6.4 The Poisson Distribution -- 1.6.5 The Exponential Distribution -- 1.6.6 Normal Distribution -- 1.7 Limit Theorems -- 1.7.1 Markov Inequality -- 1.7.2 Chebyshev Inequality -- 1.7.3 Laws of Large Numbers -- 1.7.4 The Central Limit Theorem -- Problems -- 2 Overview of Stochastic Processes -- 2.1 Introduction -- 2.2 Classification of Stochastic Processes -- 2.3 Mean and Autocorrelation Function -- 2.4 Stationary Processes -- 2.4.1 Strict-Sense Stationary Processes -- 2.4.2 Wide-Sense Stationary Processes -- 2.5 Power Spectral Density -- 2.6 Counting Processes -- 2.7 Independent Increment Processes -- 2.8 Stationary Increment Process -- 2.9 Poisson Processes -- 2.9.1 Compound Poisson Process -- 2.10 Markov Processes -- 2.10.1 Discrete-Time Markov Chains -- 2.10.2 State Transition Probability Matrix -- 2.10.3 The k-Step State Transition Probability -- 2.10.4 State Transition Diagrams -- 2.10.5 Classification of States -- 2.10.6 Limiting-State Probabilities.
2.10.7 Doubly Stochastic Matrix -- 2.10.8 Continuous-Time Markov Chains -- 2.10.9 Birth and Death Processes -- 2.11 Gaussian Processes -- 2.12 Martingales -- 2.12.1 Stopping Times -- Problems -- 3 One-Dimensional Random Walk -- 3.1 Introduction -- 3.2 Occupancy Probability -- 3.3 Random Walk as a Markov Chain -- 3.4 Symmetric Random Walk as a Martingale -- 3.5 Random Walk with Barriers -- 3.6 Mean-Square Displacement -- 3.7 Gambler's Ruin -- 3.7.1 Ruin Probability -- 3.7.2 Alternative Derivation of Ruin Probability -- 3.7.3 Duration of a Game -- 3.8 Random Walk with Stay -- 3.9 First Return to the Origin -- 3.10 First Passage Times for Symmetric Random Walk -- 3.10.1 First Passage Time via the Generating Function -- 3.10.2 First Passage Time via the Reflection Principle -- 3.10.3 Hitting Time and the Reflection Principle -- 3.11 The Ballot Problem and the Reflection Principle -- 3.11.1 The Conditional Probability Method -- 3.12 Returns to the Origin and the Arc-Sine Law -- 3.13 Maximum of a Random Walk -- 3.14 Two Symmetric Random Walkers -- 3.15 Random Walk on a Graph -- 3.15.1 Proximity Measures -- 3.15.2 Directed Graphs -- 3.15.3 Random Walk on an Undirected Graph -- 3.15.4 Random Walk on a Weighted Graph -- 3.16 Random Walks and Electric Networks -- 3.16.1 Harmonic Functions -- 3.16.2 Effective Resistance and Escape Probability -- 3.17 Correlated Random Walk -- 3.18 Continuous-Time Random Walk -- 3.18.1 The Master Equation -- 3.19 Reinforced Random Walk -- 3.19.1 Polya's Urn Model -- 3.19.2 ERRW and Polya's Urn -- 3.19.3 ERRW Revisited -- 3.20 Miscellaneous Random Walk Models -- 3.20.1 Geometric Random Walk -- 3.20.2 Gaussian Random Walk -- 3.20.3 Random Walk with Memory -- 3.21 Summary -- Problems -- 4 Two-Dimensional Random Walk -- 4.1 Introduction -- 4.2 The Pearson Random Walk -- 4.2.1 Mean-Square Displacement.
4.2.2 Probability Distribution -- 4.3 The Symmetric 2D Random Walk -- 4.3.1 Stirling's Approximation of Symmetric Walk -- 4.3.2 Probability of Eventual Return for Symmetric Walk -- 4.3.3 Mean-Square Displacement -- 4.3.4 Two Independent Symmetric 2D Random Walkers -- 4.4 The Alternating Random Walk -- 4.4.1 Stirling's Approximation of Alternating Walk -- 4.4.2 Probability of Eventual Return for Alternating Walk -- 4.5 Self-Avoiding Random Walk -- 4.6 Nonreversing Random Walk -- 4.7 Extensions of the NRRW -- 4.7.1 The Noncontinuing Random Walk -- 4.7.2 The Nonreversing and Noncontinuing Random Walk -- 4.8 Summary -- 5 Brownian Motion -- 5.1 Introduction -- 5.2 Brownian Motion with Drift -- 5.3 Brownian Motion as a Markov Process -- 5.4 Brownian Motion as a Martingale -- 5.5 First Passage Time of a Brownian Motion -- 5.6 Maximum of a Brownian Motion -- 5.7 First Passage Time in an Interval -- 5.8 The Brownian Bridge -- 5.9 Geometric Brownian Motion -- 5.10 The Langevin Equation -- 5.11 Summary -- Problems -- 6 Introduction to Stochastic Calculus -- 6.1 Introduction -- 6.2 The Ito Integral -- 6.3 The Stochastic Differential -- 6.4 The Ito's Formula -- 6.5 Stochastic Differential Equations -- 6.6 Solution of the Geometric Brownian Motion -- 6.7 The Ornstein-Uhlenbeck Process -- 6.7.1 Solution of the Ornstein-Uhlenbeck SDE -- 6.7.2 First Alternative Solution Method -- 6.7.3 Second Alternative Solution Method -- 6.8 Mean-Reverting Ornstein-Uhlenbeck Process -- 6.9 Summary -- 7 Diffusion Processes -- 7.1 Introduction -- 7.2 Mathematical Preliminaries -- 7.3 Diffusion on One-Dimensional Random Walk -- 7.3.1 Alternative Derivation -- 7.4 Examples of Diffusion Processes -- 7.4.1 Brownian Motion -- 7.4.2 Brownian Motion with Drift -- 7.5 Correlated Random Walk and the Telegraph Equation -- 7.6 Diffusion at Finite Speed.
7.7 Diffusion on Symmetric Two-Dimensional Lattice Random Walk -- 7.8 Diffusion Approximation of the Pearson Random Walk -- 7.9 Summary -- 8 Levy Walk -- 8.1 Introduction -- 8.2 Generalized Central Limit Theorem -- 8.3 Stable Distribution -- 8.4 Self-Similarity -- 8.5 Fractals -- 8.6 Levy Distribution -- 8.7 Levy Process -- 8.8 Infinite Divisibility -- 8.8.1 The Infinite Divisibility of the Poisson Process -- 8.8.2 Infinite Divisibility of the Compound Poisson Process -- 8.8.3 Infinite Divisibility of the Brownian Motion with Drift -- 8.9 Levy Flight -- 8.9.1 First Passage Time of Levy Flights -- 8.9.2 Leapover Properties of Levy Flights -- 8.10 Truncated Levy Flight -- 8.11 Levy Walk -- 8.11.1 Levy Walk as a Coupled CTRW -- 8.11.2 Truncated Levy Walk -- 8.12 Summary -- 9 Fractional Calculus and Its Applications -- 9.1 Introduction -- 9.2 Gamma Function -- 9.3 Mittag-Leffler Functions -- 9.4 Laplace Transform -- 9.5 Fractional Derivatives -- 9.6 Fractional Integrals -- 9.7 Definitions of Fractional Integro-Differentials -- 9.7.1 Riemann-Liouville Fractional Derivative -- 9.7.2 Caputo Fractional Derivative -- 9.7.3 Grunwald-Letnikov Fractional Derivative -- 9.8 Fractional Differential Equations -- 9.8.1 Relaxation Differential Equation of Integer Order -- 9.8.2 Oscillation Differential Equation of Integer Order -- 9.8.3 Relaxation and Oscillation Fractional Differential Equations -- 9.9 Applications of Fractional Calculus -- 9.9.1 Fractional Brownian Motion -- 9.9.2 Multifractional Brownian Motion -- 9.9.3 Fractional Random Walk -- 9.9.4 Fractional (or Anomalous) Diffusion -- 9.9.5 Fractional Gaussian Noise -- 9.9.6 Fractional Poisson Process -- 9.10 Summary -- 10 Percolation Theory -- 10.1 Introduction -- 10.2 Graph Theory Revisited -- 10.2.1 Complete Graphs -- 10.2.2 Random Graphs -- 10.3 Percolation on a Lattice.
10.3.1 Cluster Formation and Phase Transition -- 10.3.2 Percolation Probability and Critical Exponents -- 10.4 Continuum Percolation -- 10.4.1 The Boolean Model -- 10.4.2 The Random Connection Model -- 10.5 Bootstrap (or k-Core) Percolation -- 10.6 Diffusion Percolation -- 10.6.1 Bootstrap Percolation versus Diffusion Percolation -- 10.7 First-Passage Percolation -- 10.8 Explosive Percolation -- 10.9 Percolation in Complex Networks -- 10.9.1 Average Path Length -- 10.9.2 Clustering Coefficient -- 10.9.3 Degree Distribution -- 10.9.4 Percolation and Network Resilience -- 10.10 Summary -- References -- Index.
Summary: Presents an important and unique introduction to random walk theory Random 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 theory Numerous 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 gases Examples 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 physics Plentiful examples and exercises throughout that illustrate the solution of many practical problems Elements of Random Walk and Diffusion Processes is an ideal reference for researchers and professionalsSummary: involved in operations research, economics, engineering, mathematics, and physics. The book is also an excellent textbook for upper-undergraduate and graduate level courses in probability and stochastic processes, stochastic models, random motion and Brownian theory, random walk theory, and diffusion process techniques.
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Intro -- Elements of Random Walk and Diffusion Processes -- Copyright -- Contents -- Preface -- Acknowledgments -- 1 Review of Probability Theory -- 1.1 Introduction -- 1.2 Random Variables -- 1.2.1 Distribution Functions -- 1.2.2 Discrete Random Variables -- 1.2.3 Continuous Random Variables -- 1.2.4 Expectations -- 1.2.5 Moments of Random Variables and the Variance -- 1.3 Transform Methods -- 1.3.1 The Characteristic Function -- 1.3.2 Moment-Generating Property of the Characteristic Function -- 1.3.3 The s-Transform -- 1.3.4 Moment-Generating Property of the s-Transform -- 1.3.5 The z-Transform -- 1.3.6 Moment-Generating Property of the z-Transform -- 1.4 Covariance and Correlation Coefficient -- 1.5 Sums of Independent Random Variables -- 1.6 Some Probability Distributions -- 1.6.1 The Bernoulli Distribution -- 1.6.2 The Binomial Distribution -- 1.6.3 The Geometric Distribution -- 1.6.4 The Poisson Distribution -- 1.6.5 The Exponential Distribution -- 1.6.6 Normal Distribution -- 1.7 Limit Theorems -- 1.7.1 Markov Inequality -- 1.7.2 Chebyshev Inequality -- 1.7.3 Laws of Large Numbers -- 1.7.4 The Central Limit Theorem -- Problems -- 2 Overview of Stochastic Processes -- 2.1 Introduction -- 2.2 Classification of Stochastic Processes -- 2.3 Mean and Autocorrelation Function -- 2.4 Stationary Processes -- 2.4.1 Strict-Sense Stationary Processes -- 2.4.2 Wide-Sense Stationary Processes -- 2.5 Power Spectral Density -- 2.6 Counting Processes -- 2.7 Independent Increment Processes -- 2.8 Stationary Increment Process -- 2.9 Poisson Processes -- 2.9.1 Compound Poisson Process -- 2.10 Markov Processes -- 2.10.1 Discrete-Time Markov Chains -- 2.10.2 State Transition Probability Matrix -- 2.10.3 The k-Step State Transition Probability -- 2.10.4 State Transition Diagrams -- 2.10.5 Classification of States -- 2.10.6 Limiting-State Probabilities.

2.10.7 Doubly Stochastic Matrix -- 2.10.8 Continuous-Time Markov Chains -- 2.10.9 Birth and Death Processes -- 2.11 Gaussian Processes -- 2.12 Martingales -- 2.12.1 Stopping Times -- Problems -- 3 One-Dimensional Random Walk -- 3.1 Introduction -- 3.2 Occupancy Probability -- 3.3 Random Walk as a Markov Chain -- 3.4 Symmetric Random Walk as a Martingale -- 3.5 Random Walk with Barriers -- 3.6 Mean-Square Displacement -- 3.7 Gambler's Ruin -- 3.7.1 Ruin Probability -- 3.7.2 Alternative Derivation of Ruin Probability -- 3.7.3 Duration of a Game -- 3.8 Random Walk with Stay -- 3.9 First Return to the Origin -- 3.10 First Passage Times for Symmetric Random Walk -- 3.10.1 First Passage Time via the Generating Function -- 3.10.2 First Passage Time via the Reflection Principle -- 3.10.3 Hitting Time and the Reflection Principle -- 3.11 The Ballot Problem and the Reflection Principle -- 3.11.1 The Conditional Probability Method -- 3.12 Returns to the Origin and the Arc-Sine Law -- 3.13 Maximum of a Random Walk -- 3.14 Two Symmetric Random Walkers -- 3.15 Random Walk on a Graph -- 3.15.1 Proximity Measures -- 3.15.2 Directed Graphs -- 3.15.3 Random Walk on an Undirected Graph -- 3.15.4 Random Walk on a Weighted Graph -- 3.16 Random Walks and Electric Networks -- 3.16.1 Harmonic Functions -- 3.16.2 Effective Resistance and Escape Probability -- 3.17 Correlated Random Walk -- 3.18 Continuous-Time Random Walk -- 3.18.1 The Master Equation -- 3.19 Reinforced Random Walk -- 3.19.1 Polya's Urn Model -- 3.19.2 ERRW and Polya's Urn -- 3.19.3 ERRW Revisited -- 3.20 Miscellaneous Random Walk Models -- 3.20.1 Geometric Random Walk -- 3.20.2 Gaussian Random Walk -- 3.20.3 Random Walk with Memory -- 3.21 Summary -- Problems -- 4 Two-Dimensional Random Walk -- 4.1 Introduction -- 4.2 The Pearson Random Walk -- 4.2.1 Mean-Square Displacement.

4.2.2 Probability Distribution -- 4.3 The Symmetric 2D Random Walk -- 4.3.1 Stirling's Approximation of Symmetric Walk -- 4.3.2 Probability of Eventual Return for Symmetric Walk -- 4.3.3 Mean-Square Displacement -- 4.3.4 Two Independent Symmetric 2D Random Walkers -- 4.4 The Alternating Random Walk -- 4.4.1 Stirling's Approximation of Alternating Walk -- 4.4.2 Probability of Eventual Return for Alternating Walk -- 4.5 Self-Avoiding Random Walk -- 4.6 Nonreversing Random Walk -- 4.7 Extensions of the NRRW -- 4.7.1 The Noncontinuing Random Walk -- 4.7.2 The Nonreversing and Noncontinuing Random Walk -- 4.8 Summary -- 5 Brownian Motion -- 5.1 Introduction -- 5.2 Brownian Motion with Drift -- 5.3 Brownian Motion as a Markov Process -- 5.4 Brownian Motion as a Martingale -- 5.5 First Passage Time of a Brownian Motion -- 5.6 Maximum of a Brownian Motion -- 5.7 First Passage Time in an Interval -- 5.8 The Brownian Bridge -- 5.9 Geometric Brownian Motion -- 5.10 The Langevin Equation -- 5.11 Summary -- Problems -- 6 Introduction to Stochastic Calculus -- 6.1 Introduction -- 6.2 The Ito Integral -- 6.3 The Stochastic Differential -- 6.4 The Ito's Formula -- 6.5 Stochastic Differential Equations -- 6.6 Solution of the Geometric Brownian Motion -- 6.7 The Ornstein-Uhlenbeck Process -- 6.7.1 Solution of the Ornstein-Uhlenbeck SDE -- 6.7.2 First Alternative Solution Method -- 6.7.3 Second Alternative Solution Method -- 6.8 Mean-Reverting Ornstein-Uhlenbeck Process -- 6.9 Summary -- 7 Diffusion Processes -- 7.1 Introduction -- 7.2 Mathematical Preliminaries -- 7.3 Diffusion on One-Dimensional Random Walk -- 7.3.1 Alternative Derivation -- 7.4 Examples of Diffusion Processes -- 7.4.1 Brownian Motion -- 7.4.2 Brownian Motion with Drift -- 7.5 Correlated Random Walk and the Telegraph Equation -- 7.6 Diffusion at Finite Speed.

7.7 Diffusion on Symmetric Two-Dimensional Lattice Random Walk -- 7.8 Diffusion Approximation of the Pearson Random Walk -- 7.9 Summary -- 8 Levy Walk -- 8.1 Introduction -- 8.2 Generalized Central Limit Theorem -- 8.3 Stable Distribution -- 8.4 Self-Similarity -- 8.5 Fractals -- 8.6 Levy Distribution -- 8.7 Levy Process -- 8.8 Infinite Divisibility -- 8.8.1 The Infinite Divisibility of the Poisson Process -- 8.8.2 Infinite Divisibility of the Compound Poisson Process -- 8.8.3 Infinite Divisibility of the Brownian Motion with Drift -- 8.9 Levy Flight -- 8.9.1 First Passage Time of Levy Flights -- 8.9.2 Leapover Properties of Levy Flights -- 8.10 Truncated Levy Flight -- 8.11 Levy Walk -- 8.11.1 Levy Walk as a Coupled CTRW -- 8.11.2 Truncated Levy Walk -- 8.12 Summary -- 9 Fractional Calculus and Its Applications -- 9.1 Introduction -- 9.2 Gamma Function -- 9.3 Mittag-Leffler Functions -- 9.4 Laplace Transform -- 9.5 Fractional Derivatives -- 9.6 Fractional Integrals -- 9.7 Definitions of Fractional Integro-Differentials -- 9.7.1 Riemann-Liouville Fractional Derivative -- 9.7.2 Caputo Fractional Derivative -- 9.7.3 Grunwald-Letnikov Fractional Derivative -- 9.8 Fractional Differential Equations -- 9.8.1 Relaxation Differential Equation of Integer Order -- 9.8.2 Oscillation Differential Equation of Integer Order -- 9.8.3 Relaxation and Oscillation Fractional Differential Equations -- 9.9 Applications of Fractional Calculus -- 9.9.1 Fractional Brownian Motion -- 9.9.2 Multifractional Brownian Motion -- 9.9.3 Fractional Random Walk -- 9.9.4 Fractional (or Anomalous) Diffusion -- 9.9.5 Fractional Gaussian Noise -- 9.9.6 Fractional Poisson Process -- 9.10 Summary -- 10 Percolation Theory -- 10.1 Introduction -- 10.2 Graph Theory Revisited -- 10.2.1 Complete Graphs -- 10.2.2 Random Graphs -- 10.3 Percolation on a Lattice.

10.3.1 Cluster Formation and Phase Transition -- 10.3.2 Percolation Probability and Critical Exponents -- 10.4 Continuum Percolation -- 10.4.1 The Boolean Model -- 10.4.2 The Random Connection Model -- 10.5 Bootstrap (or k-Core) Percolation -- 10.6 Diffusion Percolation -- 10.6.1 Bootstrap Percolation versus Diffusion Percolation -- 10.7 First-Passage Percolation -- 10.8 Explosive Percolation -- 10.9 Percolation in Complex Networks -- 10.9.1 Average Path Length -- 10.9.2 Clustering Coefficient -- 10.9.3 Degree Distribution -- 10.9.4 Percolation and Network Resilience -- 10.10 Summary -- References -- Index.

Presents an important and unique introduction to random walk theory Random 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 theory Numerous 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 gases Examples 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 physics Plentiful examples and exercises throughout that illustrate the solution of many practical problems Elements of Random Walk and Diffusion Processes is an ideal reference for researchers and professionals

involved in operations research, economics, engineering, mathematics, and physics. The book is also an excellent textbook for upper-undergraduate and graduate level courses in probability and stochastic processes, stochastic models, random motion and Brownian theory, random walk theory, and diffusion process techniques.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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