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Peter Friz and Martin HairerA Course on Rough PathsWith an introduction to regularity structuresJune 2014Errata (last update: April 2015)Springer

To Waltraud and Rudolf FrizandTo Xue-Mei

PrefaceSince its original development in the mid-nineties by Terry Lyons, culminating inthe landmark paper [Lyo98], the theory of rough paths has grown into a mature andwidely applicable mathematical theory, and there are by now several monographsdedicated to the subject, notably Lyons–Qian [LQ02], Lyons et al [LCL07] andFriz–Victoir [FV10b]. So why do we believe that there is room for yet another bookon this matter? Our reasons for writing this book are twofold.First, the theory of rough paths has gathered the reputation of being difficult toaccess for “mainstream” probabilists because it relies on some non-trivial algebraicand / or geometric machinery. It is true that if one wishes to apply it to signalsof arbitrary roughness, the general theory relies on several objects (in particularon the Hopf-algebraic properties of the free tensor algebra and the free nilpotentgroup embedded in it) that are unfamiliar to most probabilists. However, in ouropinion, some of the most interesting applications of the theory arise in the contextof stochastic differential equations, where the driving signal is Brownian motion. Inthis case, the theory simplifies dramatically and essentially no non-trivial algebraicor geometric objects are required at all. This simplification is certainly not novel.Indeed, early notes by Lyons, and then of Davie and Gubinelli, all took place inthis simpler setting (which allows to incorporate Brownian motion and Lévy’s area).However, it does appear to us that all these ideas can nowadays be put together inunprecedented simplicity, and we made a conscious choice to restrict ourselves tothis simpler case throughout most of this book.The second and main raison d’être of this book is that the scope of the theoryhas expanded dramatically over the past few years and that, in this process, thepoint of view has slightly shifted from the one exposed in the aforementionedmonographs. While Lyons’ theory was built on the integration of 1-forms, Gubinelligave a natural extension to the integration of so-called “controlled rough paths”. As abenefit, differential equations driven by rough paths can now be solved by fixed pointarguments in linear Banach spaces which contain a sufficiently accurate (secondorder) local description of the solution.This shift in perspective has first enabled the use of rough paths to provide solutiontheories for a number of classically ill-posed stochastic partial differential equationsvii

viiiPrefacewith one-dimensional spatial variables, including equations of Burgers type andthe KPZ equation. More recently, the perspective which emphasises linear spacescontaining sufficiently accurate local descriptions modelled on some (rough) input,spurred the development of the theory of “regularity structures” which allows togive consistent interpretations for a number of ill-posed equations, also in higherdimensions. It can be viewed as an extension of the theory of controlled rough paths,although its formulation is somewhat different. In the last chapters of this book, wegive a short and rather informal (i.e. very few proofs) introduction to that theory,which in particular also sheds new light on some of the definitions of the theory ofrough paths.This book does not have the ambition to provide an exhaustive description of thetheory of rough paths, but rather to complement the existing literature on the subject.As a consequence, there are a number of aspects that we chose not to touch, or to doso only barely. One omission is the study of rough paths of arbitrarily low regularity:we do provide hints at the general theory at the end of several chapters, but these areself-contained and can be skipped without impacting the understanding of the restof the book. Another serious omission concerns the systematic study of signatures,that is the collection of all iterated integrals over a fixed interval associated to asufficiently regular path, providing an intriguing nonlinear characterisation.We have used several parts of this book for lectures and mini-courses. In particular,over the last years, the material on rough paths was given repeatedly by the firstauthor at TU Berlin (Chapters 1-12, in the form of a 4h/week, full semester lecture foran audience of beginning graduate students in stochastics) and in some mini-courses(Vienna, Columbia, Rennes, Toulouse; e.g. Chapters 1-5 with a selection of furthertopics). The material of Chapters 13-15 originates in a number of minicourses bythe second author (Bonn, ETHZ, Toulouse, Columbia, XVII Brazilian School ofProbability, 44th St. Flour School of Probability, etc). The “KPZ and rough paths”summer school in Rennes (2013) was a particularly good opportunity to try out muchof the material here in joint mini-course form – we are very grateful to the organisersfor their efforts. Chapters 13-15 are, arguably, a little harder to present in a classroom.Jointly with Paul Gassiat, the first author gave this material as full lecture at TUBerlin (with examples classes run by Joscha Diehl, and more background materialon Schwartz distributions, Hölder spaces and wavelet theory than what is foundin this book); we also started to use consistently colours on our handouts. We feltthe resulting improvement in readability was significant enough to try it out alsoin the present book and take the opportunity to thank Jörg Sixt from Springer formaking this possible, aside from his professional assistance concerning all otheraspects of this book project. We are very grateful for all the feedback we receivedfrom participants at all theses courses. Furthermore, we would like to thank BruceDriver, Paul Gassiat, Massimilliano Gubinelli, Terry Lyons, Etienne Pardoux, JeremyQuastel and Hendrik Weber for many interesting discussions on how to present thismaterial. In addition, Khalil Chouk, Joscha Diehl and Sebastian Riedel kindly offeredto partially proofread the final manuscript.At last, we would like to acknowledge financial support: PKF was supported bythe European Research Council under the European Union’s Seventh Framework

PrefaceixProgramme (FP7/2007-2013) / ERC grant agreement nr. 258237 and DFG, SPP 1324.MH was supported by the Leverhulme trust through a leadership award and by theRoyal Society through a Wolfson research award.Berlin and Coventry,June 2014Peter K. FrizMartin Hairer

Contents1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Controlled differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Analogies with other branches of mathematics . . . . . . . . . . . . . . . . . . 61.3 Regularity structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Frequently used notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Rough path theory works in infinite dimensions . . . . . . . . . . . . . . . . . 112The 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131316171920253Brownian 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2727313234394042464Integration against rough paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2 Integration of 1-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3 Integration of controlled rough paths . . . . . . . . . . . . . . . . . . . . . . . . . .4.4 Stability I: rough integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5 Controlled rough paths of lower regularity . . . . . . . . . . . . . . . . . . . . . .474748556061xi

xiiContents4.64.7Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665Stochastic 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .676769707578826Doob–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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83838587889093937Operations on controlled rough paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.1 Relation between rough paths and controlled rough paths . . . . . . . . . 957.2 Lifting of regular paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.3 Composition with regular functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.4 Stability II: Regular functions of controlled rough paths . . . . . . . . . . 987.5 Itô’s formula revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.6 Controlled rough paths of low regularity . . . . . . . . . . . . . . . . . . . . . . . 1017.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028Solutions to rough differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.2 Review of the Young case: a priori estimates . . . . . . . . . . . . . . . . . . . . 1068.3 Review of the Young case: Picard iteration . . . . . . . . . . . . . . . . . . . . . 1078.4 Rough differential equations: a priori estimates . . . . . . . . . . . . . . . . . . 1098.5 Rough differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.6 Stability III: Continuity of the Itô–Lyons map . . . . . . . . . . . . . . . . . . . 1168.7 Davie’s definition and numerical schemes . . . . . . . . . . . . . . . . . . . . . . 1178.8 Lyons’ original definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.9 Stability IV: Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218.11 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Contentsxiii9Stochastic differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.1 Itô and Stratonovich equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.2 The Wong–Zakai theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.3 Support theorem and large deviations . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12710Gaussian rough paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910.1 A simple criterion for Hölder regularity . . . . . . . . . . . . . . . . . . . . . .

A Course on Rough Paths With an introduction to regularity structures June 2014 Errata (last update: April 2015) Springer. To Waltraud and Rudolf Friz and To Xue-Mei. Preface Since its original development in the mid-nineties by Terry Lyons, culminating in the landmark paper [Lyo98], the theory of rough paths has grown into a mature and widely applicable mathematical theory, and there are by .

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