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2014 | Book

A Short Probability Review Read chapter Read first chapter

Classical and Spatial Stochastic Processes

With Applications to Biology

Author: Rinaldo B. Schinazi

Publisher: Springer New York

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The revised and expanded edition of this textbook presents the concepts and applications of random processes with the same illuminating simplicity as its first edition, but with the notable addition of substantial modern material on biological modeling. While still treating many important problems in fields such as engineering and mathematical physics, the book also focuses on the highly relevant topics of cancerous mutations, influenza evolution, drug resistance, and immune response. The models used elegantly apply various classical stochastic models presented earlier in the text, and exercises are included throughout to reinforce essential concepts.

The second edition of Classical and Spatial Stochastic Processes is suitable as a textbook for courses in stochastic processes at the advanced-undergraduate and graduate levels, or as a self-study resource for researchers and practitioners in mathematics, engineering, physics, and mathematical biology.

Reviews of the first edition:

An appetizing textbook for a first course in stochastic processes. It guides the reader in a very clever manner from classical ideas to some of the most interesting modern results. … All essential facts are presented with clear proofs, illustrated by beautiful examples. … The book is well organized, has informative chapter summaries, and presents interesting exercises. The clear proofs are concentrated at the ends of the chapters making it easy to find the results. The style is a good balance of mathematical rigorosity and user-friendly explanation. —Biometric Journal

This small book is well-written and well-organized. ... Only simple results are treated ... but at the same time many ideas needed for more complicated cases are hidden and in fact very close. The second part is a really elementary introduction to the area of spatial processes. ... All sections are easily readable and it is rather tentative for the reviewer to learn them more deeply by organizing a course based on this book. The reader can be really surprised seeing how simple the lectures on these complicated topics can be. At the same time such important questions as phase transitions and their properties for some models and the estimates for certain critical values are discussed rigorously. ... This is indeed a first course on stochastic processes and also a masterful introduction to some modern chapters of the theory. —Zentralblatt Math

Table of Contents

Frontmatter
Chapter 1. A Short Probability Review
Abstract
In this chapter we will review probability techniques that will be useful in the sequel.
Rinaldo B. Schinazi
Chapter 2. Discrete Time Branching Process
Abstract
We introduce branching processes. They are a recurring theme throughout the book. In this chapter we use them to model drug resistance and cancer risk.
Rinaldo B. Schinazi
Chapter 3. The Simple Symmetric Random Walk
Abstract
A simple random walk is a discrete time stochastic process \((S_{n})_{n\geq 0}\) on the integers Z.
Rinaldo B. Schinazi
Chapter 4. Asymmetric and Higher Dimension Random Walks
Abstract
In this chapter we extend our study of random walks to more general cases. We state an important criterion for recurrence that we apply to random walks.
Rinaldo B. Schinazi
Chapter 5. Discrete Time Markov Chains
Abstract
Branching processes and random walks are examples of Markov chains. In this chapter we study general properties of Markov chains.
Rinaldo B. Schinazi
Chapter 6. Stationary Distributions for Discrete Time Markov Chains
Abstract
We continue the study of Markov chains initiated in Chap. 5. A stationary distribution is a stochastic equilibrium for the chain. We find conditions under which such a distribution exists. We are also interested in conditions for convergence to a stationary distribution.
Rinaldo B. Schinazi
Chapter 7. The Poisson Process
Abstract
In this chapter we introduce a continuous time stochastic process called the Poisson process. It is a good model in a number of situations and it has many interesting mathematical properties. There is a strong link between the exponential distribution and the Poisson process. This is why we start by reviewing the exponential distribution.
Rinaldo B. Schinazi
Chapter 8. Continuous Time Branching Processes
Abstract
We introduce continuous time branching processes. The main difference between discrete and continuous branching processes is that births and deaths occur at random times for continuous time processes. Continuous time branching processes have the Markov property if (and only if) birth and death times are exponentially distributed. We will use several properties of the exponential distribution.
Rinaldo B. Schinazi
Chapter 9. Continuous Time Birth and Death Chains
Abstract
In this chapter we look at a more general class of continuous time Markov processes. The main constraint is that the process changes state by one unit at a time. An important tool to study these processes is again differential equations.
Rinaldo B. Schinazi
Chapter 10. Percolation
Abstract
Percolation is the first spatial model we will consider. The model is very easy to define but not so easy to analyze. Elementary methods can however be used to prove a number of results. We will use combinatorics, discrete branching processes, and coupling techniques.
Rinaldo B. Schinazi
Chapter 11. A Cellular Automaton
Abstract
Cellular automata are widely used in mathematical physics and in theoretical biology. These systems start from a random state and evolve using deterministic rules. We concentrate on a specific model in this chapter. The techniques we use are similar to the ones used in percolation.
Rinaldo B. Schinazi
Chapter 12. A Branching Random Walk
Abstract
In this chapter we consider a continuous time spatial branching process. Births and deaths are as in the binary branching process. In addition we keep track of the spatial location of the particles. We use results about the binary branching process.
Rinaldo B. Schinazi
Chapter 13. The Contact Process on a Homogeneous Tree
Abstract
The contact process has the same birth and death rates as the branching random walk of the preceding chapter. The difference between the two models is that there is at most one particle per site for the contact process. The one particle per site condition makes offsprings of different particles dependent (unlike what happens for branching models). Exact computations become impossible. However, branching models are used to analyze the contact process.
Rinaldo B. Schinazi
Backmatter
Metadata
Title
Classical and Spatial Stochastic Processes
Author
Rinaldo B. Schinazi
Copyright Year
2014
Publisher
Springer New York
Electronic ISBN
978-1-4939-1869-0
Print ISBN
978-1-4939-1868-3
DOI
https://doi.org/10.1007/978-1-4939-1869-0