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## Über dieses Buch

This book is intended as a text for a first course in stochastic processes at the upper undergraduate or graduate levels, assuming only that the reader has had a serious calculus course-advanced calculus would even be better-as well as a first course in probability (without measure theory). In guiding the student from the simplest classical models to some of the spatial models, currently the object of considerable research, the text is aimed at a broad audience of students in biology, engineering, mathematics, and physics. The first two chapters deal with discrete Markov chains-recurrence and tran­ sience, random walks, birth and death chains, ruin problem and branching pro­ cesses-and their stationary distributions. These classical topics are treated with a modem twist: in particular, the coupling technique is introduced in the first chap­ ter and is used throughout. The third chapter deals with continuous time Markov chains-Poisson process, queues, birth and death chains, stationary distributions. The second half of the book treats spatial processes. This is the main difference between this work and the many others on stochastic processes. Spatial stochas­ tic processes are (rightly) known as being difficult to analyze. The few existing books on the subject are technically challenging and intended for a mathemat­ ically sophisticated reader. We picked several interesting models-percolation, cellular automata, branching random walks, contact process on a tree-and con­ centrated on those properties that can be analyzed using elementary methods.

## Inhaltsverzeichnis

### I. Discrete Time Markov Chains

Abstract
What is in this chapter? We consider a sequence of random variables (Xn)n≥1 defined on the same probability space and having countably many possible values. We think of X n as being the state of a certain system at time n. Given that Xn-1 is in some state i then X n will be in some state j with a probability denoted by p(i, j); the transition probabilities p(i, j) are built in the model. The fact that given Xn-1 we may compute the distribution of X n (we do not need to know the X k for k < n - 1) is a particular case of the Markov property: the future depends on the present and on the past only through the present.
Rinaldo B. Schinazi

### II. Stationary Distributions of a Markov Chain

Abstract
What is in this chapter? Let X n be the state of a Markov chain at time n. Assume that X0, the initial state of the chain, is distributed according to a distribution π. That is, assume that the probability that X0 is in state i is π(i). Can we find a distribution π such that if X0 has distribution π then X n , for all times n, also has distribution π? Such a distribution is said to be stationary for the chain. This chapter deals with the existence of and the convergence to stationary distributions.
Rinaldo B. Schinazi

### III. Continuous Time Birth and Death Markov Chains

Abstract
What is in this chapter? In Chapters I and II we considered chains that change state at integer times. In this chapter, the change of state occurs after a random time. We are still considering chains with countably many states. Many of the results and techniques of the preceding chapter may be adapted to this new class of models.
Rinaldo B. Schinazi

### IV. Percolation

Abstract
What is in this chapter? Percolation is the first spatial model we will consider. Percolation models are very popular in a number of fields: a search in the CARL data base turned out more than 1500 articles related to percolation for the period 1988–1997.
Rinaldo B. Schinazi

### V. A Cellular Automaton

Abstract
What is in this chapter? Cellular automata are widely used models in mathematical physics and in theoretical biology. These systems start from a random state and then evolve using deterministic rules, with time being discrete. We concentrate on a specific model in this chapter. We define the initial configuration as follows. For each site in Z 2 we put a 1 with probability p or a 0 with probability 1 — p. This is done independently for each site. The rules of evolution for the cellular automaton are the following. If there is 1 at a site it remains there forever. If there is a 0 at a given site and if at least one neighbor in each of the orthogonal directions is a 1 then we replace the 0 by a 1 at the next update.
Rinaldo B. Schinazi

### VI. Continuous Time Branching Random Walk

Abstract
What is in this chapter? Branching random walks are among the simplest continuous time spatial processes. Consider a system of particles that undergo branching and random motion on a countable graph (such as Z d or a homogeneous tree) according to the two following rules. A particle at x waits an exponential random time with rate λp(x, y) > 0 and then gives birth to a particle at y. p(x, y) are the transition probabilities of a Markov chain and λ > 0 is a parameter. A particle waits an exponential time with rate 1 and then dies.
Rinaldo B. Schinazi

### VII. The Contact Process on a Homogeneous Tree

Abstract
What is in this chapter? The contact process has the same birth and death rates as a particular branching Markov chain for which births occur only on nearest neighbor sites. The difference between the two models is that there is at most one particle per site for the contact process while there is no bound in the number of particles per site for branching Markov chains. So, branching Markov chains and the contact process may be thought of as two extreme points in the same class of models.
Rinaldo B. Schinazi

### Backmatter

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