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Stochastic Tools in Mathematics and Science

Alexandre J. Chorin and Ole H. Hald
Springer Verlag
Publication Date: 
Number of Pages: 
Surveys and Tutorials in the Applied Mathematics Sciences 1
[Reviewed by
William J. Satzer
, on

This book grew from a course in stochastic methods for applied mathematics at the University of California at Berkeley, designed for first year graduate students. The department there had asked former students who had gone into nonacademic jobs what they actually did in their jobs. (What an idea!) They found that most of the former students were working with stochastic methods that had not appeared anywhere in their graduate curriculum. As the course evolved it developed a wider audience of science and engineering students. The resulting text is a high-level introduction to probabilistic modeling that covers basic stochastic methods and tools used in the sciences and engineering.

The authors’ intent is not to provide a comprehensive introduction to probability and its applications. Instead, they offer glimpses of striking applications of stochastic methods for mathematics students and fill in the background behind the tools already used by science and engineering students. This approach forces some inevitable compromises that result in simplified mathematical explanations and applications without a lot of detail. Nonetheless, this short book is attractive and challenging.

In six chapters the authors discuss background material on least squares and Fourier series, basic probability, Brownian motion, stationary stochastic processes, and statistical mechanics (both standard and time-dependent). The concept of conditional expectation — in a simple form — is exploited throughout for approximation, prediction, and renormalization. The basic plan in each chapter is to introduce the mathematical tools and then quickly to develop a few examples that show how the tools can be used. For example, the authors introduce Brownian motion and then connect it with the solution of the heat equation via random walk and the central limit theorem. They illustrate stationary stochastic processes with an extended example of fully developed turbulent fluid flow. Here the authors deduce the Kolmogorov-Obukov law for the “inertial range” spectrum of turbulence using stochastic methods and a good dose of dimensional analysis. (“Inertial range” refers to the region in wave number space where the inertial or nonlinear effects dominate viscous dissipation.)

Statistical mechanics, both standard and non-equilibrium, is treated in the last two chapters. In the context of the Ising model, the authors discuss Markov chain Monte Carlo and renormalization techniques. Non-equilibrium statistical mechanics is introduced via special cases (the Langevin equation and a coupled system of harmonic oscillators) before introducing the more general Mori-Zwanzig formalism.

Each chapter has a bibliography and a modest number of exercises. For a small book there are a lot of ideas here, and they are presented quickly. Almost any one of the example applications could be expanded into an independent project.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Preface v

Chapter 1. Preliminaries 1

1.1. Least Squares Approximation 1

1.2. Orthonormal Bases 7

1.3. Fourier Series 9

1.4. Fourier Transform 12

1.5. Exercises 16

1.6. Bibliography 16

Chapter 2. Probability 17

2.1. Definitions 17

2.2. Expected Values and Moments 20

2.3. Monte Carlo Methods 26

2.4. Parametric Estimation 29

2.5. The Central Limit Theorem 31

2.6. Conditional Probability and Conditional Expectation 34

2.7. Bayes’ Theorem 38

2.8. Exercises 40

2.9. Bibliography 42

Chapter 3. Brownian Motion 43

3.1. Definition of Brownian Motion 43

3.2. Brownian Motion and the Heat Equation 45

3.3. Solution of the Heat Equation by Random Walks 47

3.4. The Wiener Measure 50

3.5. Heat Equation with Potential 52

3.6. Physicists’ Notation for Wiener Measure 55

3.7. Another Connection Between Brownian Motion and the

Heat Equation 57

3.8. First Discussion of the Langevin Equation 59

3.9. Solution of a Nonlinear Differential Equation by Branching

Brownian Motion 64

3.10. A Brief Introduction to Stochastic ODEs 65

3.11. Exercises 67



3.12. Bibliography 69

Chapter 4. Stationary Stochastic Processes 71

4.1. Weak Definition of a Stochastic Process 71

4.2. Covariance and Spectrum 74

4.3. The Inertial Spectrum of Turbulence 76

4.4. Random Measures and Random Fourier Transforms 78

4.5. Prediction for Stationary Stochastic Processes 84

4.6. Data Assimilation 89

4.7. Exercises 92

4.8. Bibliography 95

Chapter 5. Statistical Mechanics 97

5.1. Mechanics 97

5.2. Statistical Mechanics 99

5.3. Entropy and Equilibrium 102

5.4. The Ising Model 105

5.5. Markov Chain Monte Carlo 107

5.6. Renormalization 111

5.7. Exercises 116

5.8. Bibliography 117

Chapter 6. Time-Dependent Statistical Mechanics 119

6.1. More on the Langevin Equation 119

6.2. A Coupled System of Harmonic Oscillators 122

6.3. Mathematical Addenda 124

6.4. The Mori-Zwanzig Formalism 129

6.5. More on Fluctuation-Dissipation Theorems 134

6.6. Scale Separation and Weak Coupling 135

6.7. Noninstantaneous Memory 137

6.8. Exercises 141

6.9. Bibliography 142

Index 145