A course on rough paths: with an introduction to regularity structures

Preface......Page 7
Contents......Page 10
 1.1 Controlled differential equations......Page 14
 1.2 Analogies with other branches of mathematics......Page 19
 1.3 Regularity structures......Page 21
 1.4 Frequently used notations......Page 22
 1.5 Rough path theory works in infinite dimensions......Page 24
 2.1 Basic definitions......Page 26
 2.2 The space of geometric rough paths......Page 29
 2.3 Rough paths as Lie-group valued paths......Page 30
 2.4 Geometric rough paths of low regularity......Page 32
 2.5 Exercises......Page 33
 2.6 Comments......Page 38
 3.1 Kolmogorov criterion for rough paths......Page 40
 3.2 Ito ˆBrownian motion......Page 44
 3.3 Stratonovich Brownian motion......Page 45
 3.4 Brownian motion in a magnetic field......Page 47
 3.5 Cubature onWiener Space......Page 52
 3.6 Scaling limits of random walks......Page 53
 3.7 Exercises......Page 55
 3.8 Comments......Page 59
 4.1 Introduction......Page 60
 -forms......Page 61
 4.3 Integration of controlled rough paths......Page 68
 4.4 Stability I: rough integration......Page 73
 4.5 Controlled rough paths of lower regularity......Page 74
 4.6 Exercises......Page 76
 4.7 Comments......Page 79
 5.1 Itˆo integration......Page 80
 5.2 Stratonovich integration......Page 82
 5.3 Itˆo’s formula and F¨ollmer......Page 83
 5.4 Backward integration......Page 88
 5.5 Exercises......Page 91
 5.6 Comments......Page 95
 6.1 Motivation from stochastic analysis......Page 96
 6.2 Uniqueness of the Gubinelli derivative and Doob–Meyer......Page 98
 6.3 Brownian motion is truly rough......Page 100
 6.4 A deterministic Norris’ lemma......Page 101
 6.5 Brownian motion is H¨older rough......Page 103
 6.7 Comments......Page 106
 7.1 Relation between rough paths and controlled rough paths......Page 108
 7.2 Lifting of regular paths.......Page 109
 7.3 Composition with regular functions.......Page 110
 7.4 Stability II: Regular functions of controlled rough paths......Page 111
 7.5 Itˆo’s formula revisited......Page 113
 7.6 Controlled rough paths of low regularity......Page 114
 7.7 Exercises......Page 115
 8.1 Introduction......Page 117
 8.2 Review of the Young case: a priori estimates......Page 118
 8.3 Review of the Young case: Picard iteration......Page 119
 8.4 Rough differential equations: a priori estimates......Page 121
 8.5 Rough differential equations......Page 124
 8.6 Stability III: Continuity of the Itˆo–Lyons map......Page 128
 8.7 Davie’s definition and numerical schemes......Page 129
 8.8 Lyons’ original definition......Page 131
 8.9 Stability IV: Flows......Page 132
 8.10 Exercises......Page 133
 8.11 Comments......Page 134
 9.1 Ito ˆand Stratonovich equations......Page 135
 9.2 The Wong–Zakai theorem......Page 136
 9.3 Support theorem and large deviations......Page 137
 9.4 Exercises......Page 138
 9.5 Comments......Page 139
 10.1 A simple criterion for H¨older regularity......Page 141
 10.2 Stochastic integration and variation regularity of the covariance......Page 143
 10.3 Fractional Brownian motion and beyond......Page 151
 10.4 Exercises......Page 154
 10.5 Comments......Page 159
 11.1 Complementary Young regularity......Page 160
 11.2 Concentration of measure......Page 165
 11.3 Malliavin calculus for rough differential equations......Page 171
 11.4 Exercises......Page 177
 11.5 Comments......Page 179
 12.1 Rough partial differential equations......Page 180
 12.2 Stochastic heat equation as a rough path......Page 191
 12.3 Exercises......Page 197
 12.4 Comments......Page 201
 13.1 Introduction......Page 202
 13.2 Definition of a regularity structure and first examples......Page 203
 13.3 Definition of a model and first examples......Page 208
 13.4 Wavelets and the reconstruction theorem......Page 215
 13.5 Exercises......Page 220
 13.6 Comments......Page 221
 14.1 Differentiation......Page 222
 14.2 Products and composition by regular functions......Page 223
 14.3 Schauder estimates and admissible models......Page 226
 14.4 Exercises......Page 230
 15.1 Formulation of the main result......Page 232
 15.2 Construction of the associated regularity structure......Page 235
 15.3 The structure group......Page 238
 15.4 Canonical lifts of regular functions......Page 240
 15.5 Renormalisation of the KPZ equation......Page 241
 15.6 The KPZ equation and rough paths......Page 249
References......Page 252
Index......Page 260