Brownian motion is a process of tremendous practical and theoretical significance. It originated (a) as a model of the phenomenon observed by Robert Brown in 1828 that “pollen grains suspended in water perform a continual swarming motion,” and (b) in Bachelier's (1900) work as a model of the stock market. These are just two of many systems that Brownian motion has been used to model. On the theoretical side, Brownian motion is a Gaussian Markov process with stationary independent increments. It lies in the intersection of three important classes of processes and is a fundamental example in each theory.

The first part of this chapter develops properties of Brownian motion. In Section 8.1, we define Brownian motion and investigate continuity properties of its paths. In Section 8.2, we prove the Markov property and a related 0-1 law. In Section 8.3, we define stopping times and prove the strong Markov property. In Section 8.4, we take a close look at the zero set of Brownian motion. In Section 8.5, we introduce some martingales associated with Brownian motion and use them to obtain information about its properties.

The second part of this chapter applies Brownian motion to some of the problems considered in Chapters 2 and 3. In Section 8.6, we embed random walks into Brownian motion to prove Donsker's theorem, a far reaching generalization of the central limit theorem.

In 1989 when the first edition of this book was completed, my sons David and Greg were 3 and 1, and the cover picture showed the Dow Jones at 2650. The past 20 years have brought many changes, but the song remains the same. The title of the book indicates that as we develop the theory, we will focus our attention on examples. Hoping that the book would be a useful reference for people who apply probability in their work, we have tried to emphasize the results that are important for applications, and have illustrated their use with roughly 200 examples. Probability is not a spectator sport, so the book contains almost 450 exercises to challenge readers and to deepen their understanding.

This fourth edition has two major changes (in addition to a new publisher):

1. (i) The book has been converted from TeX to LaTeX. The systematic use of labels should eventually eliminate problems with references to other points in the text. In addition, the picture environment and graphicx package has allowed the figures lost from the third edition to be reintroduced and a number of new ones to be added.

2. (ii) Four sections of the old appendix have been combined with the first three sections of Chapter 1 to make a new first chapter on measure theory, which should allow the book to be used by people who do not have this background without making the text tedious for those who have.

Abstract

Results on the behaviour of the rightmost particle in the nth generation in the branching random walk are reviewed and the phenomenon of anomalous spreading speeds, noticed recently in related deterministic models, is considered. The relationship between such results and certain coupled reaction-diffusion equations is indicated.

AMS subject classification (MSC2010) 60J80

Introduction

I arrived at the University of Oxford in the autumn of 1973 for postgraduate study. My intention at that point was to work in Statistics. The first year of study was a mixture of taught courses and designated reading on three areas (Statistics, Probability, and Functional Analysis, in my case) in the ratio 2:1:1 and a dissertation on the main area. As part of the Probability component, I attended a graduate course that was an exposition, by its author, of the material in Hammersley (1974), which had grown out of his contribution to the discussion of John's invited paper on subadditive ergodic theory (Kingman, 1973). A key point of Hammersley's contribution was that the postulates used did not cover the time to the first birth in the nth generation in a Bellman–Harris process. Hammersley (1974) showed, among other things, that these quantities did indeed exhibit the anticipated limit behaviour in probability. I decided not to be examined on this course, which was I believe a wise decisin but I was intrigued by the material. That interest turned out to be critical a few months later.

Abstract

We consider perfect simulation algorithms for locally stable point processes based on dominated coupling from the past, and apply these methods in two different contexts. A new version of the algorithm is developed which is feasible for processes which are neither purely attractive nor purely repulsive. Such processes include multiscale area-interaction processes, which are capable of modelling point patterns whose clustering structure varies across scales. The other topic considered is nonparametric regression using wavelets, where we use a suitable area-interaction process on the discrete space of indices of wavelet coefficients to model the notion that if one wavelet coefficient is non-zero then it is more likely that neighbouring coefficients will be also. A method based on perfect simulation within this model shows promising results compared to the standard methods which threshold coefficients independently.

Keywords coupling from the past (CFTP), dominated CFTP, exact simulation, local stability, Markov chain Monte Carlo, perfect simulation, Papangelou conditional intensity, spatial birth-and-death process

AMS subject classification (MSC2010) 62M30, 60G55, 60K35

Introduction

Markov chain Monte Carlo (MCMC) is now one of the standard approaches of computational Bayesian inference. A standard issue when using MCMC is the need to ensure that the Markov chain we are using for simulation has reached equilibrium. For certain classes of problem, this problem was solved by the introduction of coupling from the past (CFTP) (Propp and Wilson, 1996, 1998).

A nonlinear Markov evolution is a dynamical system generated by a measurevalued ordinary differential equation (ODE) with the specific feature that it preserves positivity. This feature distinguishes it from a general Banachspace-valued ODE and yields a natural link with probability theory, both in the interpretation of results and in the tools of analysis. However, nonlinear Markov evolution can be regarded as a particular case of measure-valued Markov processes. Even more important (and not so obvious) is the interpretation of nonlinear Markov dynamics as a dynamic law of large numbers (LLN) for general Markov models of interacting particles. Such an interpretation is both the main motivation for and the main theme of the present monograph.

The power of nonlinear Markov evolution as a modeling tool and its range of applications are immense, and include non-equilibrium statistical mechanics (e.g. the classical kinetic equations of Vlasov, Boltzmann, Smoluchovski and Landau), evolutionary biology (replicator dynamics), population and disease dynamics (Lotka–Volterra and epidemic models) and the dynamics of economic and social systems (replicator dynamics and games). With certain modifications nonlinear Markov evolution carries over to the models of quantum physics.

Abstract

The paper considers the situation of transmission over a memoryless noisy channel with feedback, which can be given a number of interpretations. The criteria for achieving the maximal rate of information transfer are well known, but examples of a simple and meaningful coding meeting these are few. Such a one is found for the Gaussian channel.

Keywords feedback channel, Gaussian channel

AMS subject classification (MSC2010) 94A24

Interrogation, transmission and coding

In this section we set out material which is classic in high degree, harking back to Shannon's seminal paper (1948), and presented in some form in texts such as those of Blahut (1987), Cover and Thomas (1991) and MacKay (2003). However, some exposition is necessary if we are to distinguish clearly between three versions of the model.

Suppose that an experimenter wishes to determine the value of a random variable U of which he knows only the probability distribution P(U). The formal argument is conveyed well enough for the moment if we suppose all distributions discrete and use the notation P(·) generically for such distributions. We may also abuse this convention by occasionally using P(U) to denote the function of U defined by the distribution.

The outcome of the experiment, if errorless, might be written x(U), where the form of the function x(U) reflects the design of the experiment. The experimenter will choose this, subject to practical constraints, so as to make the experiment as informative as possible.