Elements of Random Walk and Diffusion Processes

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