# Elements of Random Walk and Diffusion Processes

# Elements of Random Walk and Diffusion Processes

ISBN: 978-1-118-61793-9

Aug 2013

276 pages

$80.99

## Description

**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.

Preface xiii

Acknowledgments xv

**1 Review of Probability Theory 1**

1.1 Introduction 1

1.2 Random Variables 1

1.3 Transform Methods 5

1.4 Covariance and Correlation Coefficient 9

1.5 Sums of Independent Random Variables 10

1.6 Some Probability Distributions 11

1.7 Limit Theorems 16

Problems 19

**2 Overview of Stochastic Processes 21**

2.1 Introduction 21

2.2 Classification of Stochastic Processes 22

2.3 Mean and Autocorrelation Function 22

2.4 Stationary Processes 23

2.5 Power Spectral Density 24

2.6 Counting Processes 25

2.7 Independent Increment Processes 25

2.8 Stationary Increment Process 25

2.9 Poisson Processes 26

2.10 Markov Processes 29

2.11 Gaussian Processes 38

2.12 Martingales 38

Problems 41

**3 One-Dimensional Random Walk 44**

3.1 Introduction 44

3.2 Occupancy Probability 46

3.3 Random Walk as a Markov Chain 49

3.4 Symmetric Random Walk as a Martingale 49

3.5 Random Walk with Barriers 50

3.6 Mean-Square Displacement 50

3.7 Gambler’s Ruin 52

3.8 Random Walk with Stay 56

3.9 First Return to the Origin 57

3.10 First Passage Times for Symmetric Random Walk 59

3.11 The Ballot Problem and the Reflection Principle 65

3.12 Returns to the Origin and the Arc-Sine Law 67

3.13 Maximum of a Random Walk 72

3.14 Two Symmetric Random Walkers 73

3.15 Random Walk on a Graph 73

3.16 Random Walks and Electric Networks 80

3.17 Correlated Random Walk 85

3.18 Continuous-Time Random Walk 90

3.19 Reinforced Random Walk 94

3.20 Miscellaneous Random Walk Models 98

3.21 Summary 100

Problems 100

**4 Two-Dimensional Random Walk 103**

4.1 Introduction 103

4.2 The Pearson Random Walk 105

4.3 The Symmetric 2D Random Walk 110

4.4 The Alternating Random Walk 115

4.5 Self-Avoiding Random Walk 117

4.6 Nonreversing Random Walk 121

4.7 Extensions of the NRRW 126

4.8 Summary 128

**5 Brownian Motion 129**

5.1 Introduction 129

5.2 Brownian Motion with Drift 132

5.3 Brownian Motion as a Markov Process 132

5.4 Brownian Motion as a Martingale 133

5.5 First Passage Time of a Brownian Motion 133

5.6 Maximum of a Brownian Motion 135

5.7 First Passage Time in an Interval 135

5.8 The Brownian Bridge 136

5.9 Geometric Brownian Motion 137

5.10 The Langevin Equation 137

5.11 Summary 141

Problems 141

**6 Introduction to Stochastic Calculus 143**

6.1 Introduction 143

6.2 The Ito Integral 145

6.3 The Stochastic Differential 146

6.4 The Ito’s Formula 147

6.5 Stochastic Differential Equations 147

6.6 Solution of the Geometric Brownian Motion 148

6.7 The Ornstein–Uhlenbeck Process 151

6.8 Mean-Reverting Ornstein–Uhlenbeck Process 155

6.9 Summary 157

**7 Diffusion Processes 158**

7.1 Introduction 158

7.2 Mathematical Preliminaries 159

7.3 Diffusion on One-Dimensional Random Walk 160

7.4 Examples of Diffusion Processes 164

7.5 Correlated Random Walk and the Telegraph Equation 167

7.6 Diffusion at Finite Speed 170

7.7 Diffusion on Symmetric Two-Dimensional Lattice Random Walk 171

7.8 Diffusion Approximation of the Pearson Random Walk 173

7.9 Summary 174

**8 Levy Walk 175**

8.1 Introduction 175

8.2 Generalized Central Limit Theorem 175

8.3 Stable Distribution 177

8.4 Self-Similarity 182

8.5 Fractals 183

8.6 Levy Distribution 185

8.7 Levy Process 186

8.8 Infinite Divisibility 186

8.9 Levy Flight 188

8.10 Truncated Levy Flight 191

8.11 Levy Walk 191

8.12 Summary 195

**9 Fractional Calculus and Its Applications 196**

9.1 Introduction 196

9.2 Gamma Function 197

9.3 Mittag–Leffler Functions 198

9.4 Laplace Transform 200

9.5 Fractional Derivatives 202

9.6 Fractional Integrals 203

9.7 Definitions of Fractional Integro-Differentials 203

9.8 Fractional Differential Equations 207

9.9 Applications of Fractional Calculus 210

9.10 Summary 224

**10 Percolation Theory 225**

10.1 Introduction 225

10.2 Graph Theory Revisited 226

10.3 Percolation on a Lattice 228

10.4 Continuum Percolation 235

10.5 Bootstrap (or k-Core) Percolation 237

10.6 Diffusion Percolation 237

10.7 First-Passage Percolation 239

10.8 Explosive Percolation 240

10.9 Percolation in Complex Networks 242

10.10 Summary 245

References 247

Index 253