Elements of Random Walk and Diffusion Processes

Oliver C. Ibe

• Publisher: Wiley
• Year: 2013
• Print ISBN: 9781118618097
• eBook ISBN: 9781118617939
• Edition: 1
• Topic: Mathematics
• Subtopic: Operations
• Index: Mathematics

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.

CHAPTER 1

REVIEW OF PROBABILITY THEORY

1.1 INTRODUCTION

The concepts of experiments and events are very important in the study of probability. In probability, an experiment is any process of trial and observation. An experiment whose outcome is uncertain before it is performed is called a random experiment. When we perform a random experiment, the collection of possible elementary outcomes is called the sample space of the experiment, which is usually denoted by Ω. We define these outcomes as elementary outcomes because exactly one of the outcomes occurs when the experiment is performed. The elementary outcomes of an experiment are called the sample points of the sample space and are denoted by w_i, i = 1, 2, …. If there are n possible outcomes of an experiment, then the sample space is Ω = {w₁, w₂, …, w_n}. An event is the occurrence of either a prescribed outcome or any one of a number of possible outcomes of an experiment. Thus, an event is a subset of the sample space.

1.2 RANDOM VARIABLES

Consider a random experiment with sample space Ω. Let w be a sample point in Ω. We are interested in assigning a real number to each w ∈ Ω. A random variable, X(w), is a single-valued real function that assigns a real number, called the value of X(w), to each sample point w ∈ Ω. That is, it is a mapping of the sample space onto the real line.

Generally, a random variable is represented by a single letter X instead of the function X(w). Therefore, in the remainder of the book we use X to denote a random variable. The sample space Ω is called the domain of the random variable X. Also, the collection of all numbers that are values of X is called the range of the random variable X.

Let X be a random variable and x a fixed real value. Let the event A_x define the subset of Ω that consists of all real sample points to which the random variable X assigns the number x. That is,

Since A_x is an event, it will have a probability, which we define as follows:

We can define other types of events in terms of a random variable. For fixed numbers x, a, and b, we can define the following:

These events have probabilities that are denoted by the following:

• P[X ≤ x] is the probability that X takes a value less than or equal to x.
• P[X > x] is the probability that X takes a value greater than x; this is equal to 1 − P[X ≤ x].
• P[a < X < b] is the probability that X takes a value that strictly lies between a and b.

1.2.1 Distribution Functions

Let X be a random variable and x be a n...

Table of contents

• COVER
• WILEY SERIES IN OPERATIONS RESEARCH AND MANAGEMENT SCIENCE
• TITLE PAGE
• COPYRIGHT PAGE
• PREFACE
• ACKNOWLEDGMENTS
• CHAPTER 1: REVIEW OF PROBABILITY THEORY
• CHAPTER 2: OVERVIEW OF STOCHASTIC PROCESSES
• CHAPTER 3: ONE-DIMENSIONAL RANDOM WALK
• CHAPTER 4: TWO-DIMENSIONAL RANDOM WALK
• CHAPTER 5: BROWNIAN MOTION
• CHAPTER 6: INTRODUCTION TO STOCHASTIC CALCULUS
• CHAPTER 7: DIFFUSION PROCESSES
• CHAPTER 8: LEVY WALK
• CHAPTER 9: FRACTIONAL CALCULUS AND ITS APPLICATIONS
• CHAPTER 10: PERCOLATION THEORY
• REFERENCES
• INDEX