A Course on Rough Paths: With an Introduction to Regularity Structures

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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 inZentralblatt MATH 04 02 1 Introduction.- 2 The space of rough paths.- 3 Brownian motion as a rough path.- 4 Integration against rough paths.- 5 Stochastic integration and Itô’s formula.- 6 Doob–Meyer type decomposition for rough paths.- 7 Operations on controlled rough paths.- 8 Solutions to rough differential equations.- 9 Stochastic differential equations.- 10 Gaussian rough paths.- 11 Cameron–Martin regularity and applications.- 12 Stochastic partial differential equations.- 13 Introduction to regularity structures.- 14 Operations on modelled distributions.- 15 Application to the KPZ equation.- References.- Index. 13 02 Peter K. Friz is presently Einstein Professor of Mathematics at TU and WIAS Berlin. His previous professional affiliations include Cambridge University and Merrill Lynch, and he holds a PhD from the Courant Institute of New York University. He has made contributions to the understanding of the Navier-Stokes equation as dynamical system, pioneered new asymptotic techniques in financial mathematics and has written many influential papers on the applications of rough path theory to stochastic analysis, ranging from the interplay of rough paths with Malliavin calculus to a (rough-) pathwise view on non-linear SPDEs. Jointly with N. Victoir he authored a monograph on stochastic processes as rough paths. Martin Hairer KBE FRS is currently Professor of Mathematics at Imperial College London. He has mostly worked in the fields of stochastic partial differential equations in particular, and in stochastic analysis and stochastic dynamics in general. He made fundamental advances in various directions such as the study of hypoelliptic and/or hypocoercive diffusions, the development of an ergodic theory for stochastic PDEs, the systematisation of the construction of Lyapunov functions for stochastic systems, the development of a general theory of ergodicity for non-Markovian systems, multiscale analysis techniques, etc. Most recently, he has worked on applying rough path techniques to the analysis of certain ill-posed stochastic PDEs and introduced the theory of regularity structures. For this work he was awarded the Fields Medal at the 2014 ICM in Seoul. 18 02 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 theMathematical 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 inZentralblatt MATH 19 02 Provides a self-contained introduction to rough path analysis with many exercises Includes applications to stochastic partial differential equations Covers the basics of the new theory of regularity structures 06 05 300 01 https://covers.springernature.com/boo... 01 01 https://www.springer.com/9783030415556 01 http://www.springer.com/ 01 Springer Nature Imprint SPR Springer 01 01 SIP Springer International Publishing 01 05 5251753 Springer International Publishing Cham CH 02 20200601 2020 01 WORLD 08 0 gr 01 235 mm 02 155 mm 03 03 9783319083315 15 9783319083315 01 ISBN-13 hyphenated 978-3-319-08331-5 BC 27 03 9783030415563 15 9783030415563 01 ISBN-13 hyphenated 978-3-030-41556-3 DG Springer International Publishing 01 ROW NP 10 20200601 02 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 84.99 AUD AU 20200116 01 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 77.26 AUD AU 20200116 02 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 59.00 CHF CH R 2.5 20200116 02 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 53.49 EUR DE R 7 20200116 02 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 52.74 EUR FR 20200116 02 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 51.99 EUR IT 20200116 02 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 54.49 EUR NL 20200116 02 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 54.99 EUR AT R 10 20200116 01 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 49.99 EUR ROW 20200116 01 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 44.99 GBP GB 20200116 01 Recommended Retail Price 01 BIC discount group code ASPVB0 02 Product discount group SPVB0 01 3560.00 INR IN 20200116 Springer International Publishing 01 US 02 Y NP 10 20200601 01 Recommended Retail Price 01 BIC discount group code ADGNY2 02 Product discount group DGNY2 01 49.99 EUR ROW 20200116 01 Recommended Retail Price 01 BIC discount group code ADGNY2 02 Product discount group DGNY2 01 59.99 USD US 20200116 Preface to the Second Edition Preface to the First Edition Contents Chapter 1 Introduction 1.1 What is it all about? 1.2 Analogies with other branches of mathematics 1.3 Regularity structures 1.4 Frequently used notations 1.5 Rough path theory works in infinite dimensions Chapter 2 The space of rough paths 2.1 Basic definitions 2.2 The space of geometric rough paths 2.3 Rough paths as Lie group valued paths 2.4 Geometric rough paths of low regularity 2.5 Exercises 2.6 Comments Chapter 3 Brownian motion as a rough path 3.1 Kolmogorov criterion for rough paths 3.2 Itô Brownian motion 3.3 Stratonovich Brownian motion 3.4 Brownian motion in a magnetic field 3.5 Cubature on Wiener Space 3.6 Scaling limits of random walks 3.7 Exercises 3.8 Comments Chapter 4 Integration against rough paths 4.1 Introduction 4.2 Integration of one-forms 4.3 Integration of controlled rough paths 4.4 Stability I: rough integration 4.5 Controlled rough paths of lower regularity 4.6 Stochastic sewing 4.7 Exercises 4.8 Comments Chapter 5 Stochastic integration and Itô's formula 5.1 Itô integration 5.2 Stratonovich integration 5.3 Itô's formula and Föllmer 5.4 Backward integration 5.5 Exercises 5.6 Comments Chapter 6 Doob–Meyer type decomposition for rough paths 6.1 Motivation from stochastic analysis 6.2 Uniqueness of the Gubinelli derivative and Doob–Meyer 6.3 Brownian motion is truly rough 6.4 A deterministic Norris' lemma 6.5 Brownian motion is Hölder rough 6.6 Exercises 6.7 Comments Chapter 7 Operations on controlled rough paths 7.1 Relation between rough paths and controlled rough paths 7.2 Lifting of regular paths. 7.3 Composition with regular functions. 7.4 Stability II: Regular functions of controlled rough paths 7.5 Itô's formula revisited 7.6 Controlled rough paths of low regularity 7.7 Exercises 7.8 Comments Chapter 8 Solutions to rough differential equations 8.1 Introduction 8.2 Review of the Young case: a priori estimates 8.3 Review of the Young case: Picard iteration 8.4 Rough differential equations: a priori estimates 8.5 Rough differential equations 8.6 Stability III: Continuity of the Itô–Lyons map 8.7 Davie's definition and numerical schemes 8.8 Lyons' original definition 8.9 Linear rough differential equations 8.10 Stability IV: Flows 8.11 Exercises 8.12 Comments Chapter 9 Stochastic differential equations 9.1 Itô and Stratonovich equations 9.2 The Wong–Zakai theorem 9.3 Support theorem and large deviations 9.4 Laplace method 9.5 Exercises 9.6 Comments Chapter 10 Gaussian rough paths 10.1 A simple criterion for Hölder regularity 10.2 Stochastic integration and variation regularity of the covariance 10.3 Fractional Brownian motion and beyond 10.4 Exercises 10.5 Comments Chapter 11 Cameron–Martin regularity and applications 11.1 Complementary Young regularity 11.2 Concentration of measure 11.2.1 Borell's inequality 11.2.2 Fernique theorem for Gaussian rough paths 11.2.3 Integrability of rough integrals and related topics 11.3 Malliavin calculus for rough differential equations 11.3.1 Bouleau–Hirsch criterion and Hörmander's theorem 11.3.2 Calculus of variations for ODEs and RDEs 11.3.3 Hörmander's theorem for Gaussian RDEs 11.4 Exercises 11.5 Comments Chapter 12 Stochastic partial differential equations 12.1 First order rough partial differential equations 12.1.1 Rough transport equation 12.1.2 Continuity equation and analytically weak formulation 12.2 Second order rough partial differential equations 12.2.1 Linear theory: Feynman–Kac 12.2.2 Mild solutions to semilinear RPDEs 12.2.3 Fully nonlinear equations with semilinear rough noise 12.2.4 Rough viscosity solutions 12.3 Stochastic heat equation as a rough path 12.3.1 The linear stochastic heat equation 12.4 Exercises 12.5 Comments Chapter 13 Introduction to regularity structures 13.1 Introduction 13.2 Definition of a regularity structure and first examples 13.2.1 The polynomial structure 13.2.2 The rough path structure 13.3 Definition of a model and first examples 13.3.1 The polynomial model 13.3.2 The rough path model 13.4 Proof of the reconstruction theorem 13.5 Exercises 13.6 Comments Chapter 14 Operations on modelled distributions 14.1 Differentiation 14.2 Products and composition by regular functions 14.3 Classical Schauder estimates 14.4 Multilevel Schauder estimates and admissible models 14.5 Rough volatility and robust Itô integration revisited 14.6 Exercises 14.7 Comments Chapter 15 Application to the KPZ equation 15.1 Formulation of the main result 15.2 Construction of the associated regularity structure 15.3 The structure group and positive renormalisation 15.4 Reconstruction for canonical lifts 15.5 Renormalisation of the KPZ equation 15.5.1 The renormalisation group 15.5.2 The renormalised equations 15.5.3 Convergence of the renormalised models 15.6 The KPZ equation and rough paths 15.7 Exercises 15.8 Comments References Index