A Course On Rough Paths

With many updates and additional exercises, the second edition of this book continues to provide readers with a gentle introduction to rough path analysis and regularity structures, theories that have yielded many new insights into the analysis of stochastic differential equations, and, most recently, stochastic partial differential equations. Rough path analysis provides the means for constructing a pathwise solution theory for stochastic differential equations which, in many respects, behaves like the theory of deterministic differential equations and permits a clean break between analytical and probabilistic arguments. Together with the theory of regularity structures, it forms a robust toolbox, allowing the recovery of many classical results without having to rely on specific probabilistic properties such as adaptedness or the martingale property. Essentially self-contained, this textbook puts the emphasis on ideas and short arguments, rather than aiming for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis and probability courses, with little more than Itô-integration against Brownian motion required for most of the text. From the reviews of the first edition: "Can easily be used as a support for a graduate course ... Presents in an accessible way the unique point of view of two experts who themselves have largely contributed to the theory" - Fabrice Baudouin in the Mathematical Reviews "It is easy to base a graduate course on rough paths on this ... A researcher who carefully works her way through all of the exercises will have a very good impression of the current state of the art" - Nicolas Perkowski in Zentralblatt MATH

Each year young mathematicians congregate in Saint Flour, France, and listen to extended lecture courses on new topics in Probability Theory. The goal of these notes, representing a course given by Terry Lyons in 2004, is to provide a straightforward and self supporting but minimalist account of the key results forming the foundation of the theory of rough paths.

Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.

In 'A Course Called America', Tom Coyne plays his way across the United States in search of the great American golf course. Packed with fascinating tales from American golf history, comic road misadventures, illuminating insight into course design, and many a memorable round with local golfers, this book is an epic narrative travelogue brimming with heart and soul.

Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial • Applications • Further Reading

A compact introduction to this active and powerful area of research, combining basic theory, core techniques, and recent applications.

"Everything I look for in a fantasy." -- George R. R. Martin All paths lead to war. . . Marcus' hero days are behind him. He knows too well that even the smallest war still means somebody's death. When his men are impressed into a doomed army, staying out of a battle he wants no part of requires some unorthodox steps. Cithrin is an orphan, ward of a banking house. Her job is to smuggle a nation's wealth across a war zone, hiding the gold from both sides. She knows the secret life of commerce like a second language, but the strategies of trade will not defend her from swords. Geder, sole scion of a noble house, has more interest in philosophy than in swordplay. A poor excuse for a soldier, he is a pawn in these games. No one can predict what he will become. Falling pebbles can start a landslide. A spat between the Free Cities and the Severed Throne is spiraling out of control. A new player rises from the depths of history, fanning the flames that will sweep the entire region onto The Dragon's Path -- the path to war. The Dagger and the Coin The Dragon's Path The King's Blood The Tyrant's Law The Widow's House The Spider's War