The textbook Applied Stochastic Analysis by Weinan E, Tiejun Li, and Eric Vanden-Eijnden is a well-thought-out treatment of a range of ideas central to stochastic analysis. The authors, noted experts in the field, use their expertise to show the reader the most important relevant mathematics research. Applied Stochastic Analysis might occupy a place on one’s bookshelf somewhere near J. R. Norris’s now-classic 1997 book Markov Chains.
Stochastic analysis has been remarkably successful at revealing the ways in which various random phenomena tend to organize. Energy analyses, limit theorems, Markovian invariant distributions, ergodic measures, and statistical mechanics all provide physicists with powerful tools for understanding the large-scale behavior of microscopically defined random models. Applied Stochastic Analysis covers those topics with clear, succinct, and complete proofs when possible and points to standard references for more involved proofs.
The book is divided into two broad sections. The first, Fundamentals, covers topics such as random variables, limit theorems, Markov chains, Monte Carlo methods, stochastic processes, and stochastic differential equations. The second section, Advanced Topics, has chapters on path integrals, random fields, rare events, statistical mechanics, and chemical reaction kinetics.
Most of the applied material promised in the title is contained in the second half of the book, which is oriented toward physical and chemical systems. Theory and applications have had a long interplay in the study of such systems. The authors review some basic results in statistical physics and chemical kinetics to give the reader an understanding of how stochastic tools can lead to meaningful conclusions and descriptions. Almost all readers will find a novel calculation or approach in the material.
One revealing perspective of a given graduate-level text is the last chapter, in which the authors usually open the throttle on a subject of their interest. In Applied Stochastic Analysis, the last chapter is an introduction to chemical kinetics. E, Li, and Vanden-Eijnden introduce the major ways that the formalism of stochastic processes can be used to create macroscopic dynamical models of interacting chemicals. The authors cover macroscopic ordinary differential equation models and then develop Poisson-driven stochastic differential equations to model individual molecule counts before moving on to cover diffusion limits. They then bring the theory of stationary distributions to bear, followed by a multiscale analysis. The time spent understanding the entire presentation is well worth it. Stewart Ethier and Thomas Kurtz’s definitive 1986 book Markov Processes: Characterization and Convergence develops a lot of machinery used in this chapter; Applied Stochastic Analysis shows why that machinery is important.
This book gives students of stochastics or mathematical physics a wonderfully solid starting point and is likely to be a favorite among physicists. By the end of it, readers should have a solid understanding of core tools in stochastic analysis.
Richard Sowers is professor of mathematics and industrial and enterprise systems engineering at the University of Illinois at Urbana-Champaign. His PhD is in applied mathematics.
