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A Course in Rough Paths

Peter K. Friz and Martin Hairer
Publication Date: 
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[Reviewed by
William J. Satzer
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The idea of a rough path was introduced by Lyons in 1998. The theory that he developed — and that continues to evolve — lies at the intersection of analysis, control theory and stochastic calculus. At its heart is a powerful tool to analyze stochastic differential equations that are too “rough” to have solutions in the class of functions or distributions manageable by the methods of classical analysis.

Martin Hairer used the theory of rough paths to develop a solution to the KPZ equation (a model for random interface growth in physics named after Kardar, Parisi and Zhang). Hairer then proposed a significant generalization that included what he called regularity structures. For this and related work on stochastic differential equations he was awarded a Fields Medal in 2014.

The concept of a regularity structure incorporates an algebraic framework that allows one to describe functions or distributions using a kind of local Taylor expansion (a jet of sorts) around each point. But instead of the classical polynomials that ordinarily appear as jets and are appropriate for describing smooth functions, these new objects are individually built for each problem.

The full theory of regularity structures includes, among other things, something like a graded commutative Hopf algebra with a free tensor algebra having a free nilpotent group embedded in it. This additional level of algebraic complication makes an already difficult subject even less accessible to those (probabilists, control theorists and physicists, among others) who would otherwise be drawn to it.

In the current book Hairer and Friz restrict the driving signal to Brownian motion. This significantly simplifies the theory and eliminates the need for the more complicated algebraic structures. Many of the most interesting applications can be handled in this context. The major part of the book deals with various aspects of the analysis of rough paths — integration against rough paths, Brownian motion and Gaussian rough paths, and connections to related prior work of Itô, Gubinelli, Doob-Meyer and others. The most challenging material — on regularity structures — is in the final three chapters.

This is a difficult book. The publisher’s blurb claims that it is largely self-contained and accessible to readers with a decent analysis background and an interest in stochastic analysis. A realistic minimum background would include familiarity with the Itô calculus and at least a modest acquaintance with stochastic differential equations. For those interested in getting started in this area, one of Lyons’ early papers (such as “Differential Equations Driven by Rough Signals”) would probably be a good place to start.

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.