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An Introduction to Singular Stochastic PDEs. N. Berglund |
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Contents
| 1 | Introduction | 1 | ||
| 1.1 | Stochastic partial differential equations | 1 | ||
| 1.2 | Singular SPDEs | 3 | ||
| 1.3 | Regularity structures | 6 | ||
| 1.4 | About this book | 7 | ||
| 2 | A system of interacting diffusions | 13 | ||
| 2.1 | Deterministic dynamics | 13 | ||
| 2.2 | Existence and uniqueness of solutions | 16 | ||
| 2.3 | Invariant measure and reversibility | 17 | ||
| 2.4 | Large deviations | 19 | ||
| 2.5 | Metastability | 20 | ||
| Bibliographical notes | 31 | |||
| 3 | Allen-Cahn SPDE in one space dimension | 33 | ||
| 3.1 | Deterministic dynamics | 34 | ||
| 3.2 | Space-time white noise | 37 | ||
| 3.3 | The stochastic heat equation | 41 | ||
| 3.4 | Existence and uniqueness of solutions | 47 | ||
| 3.5 | Invariant measure | 50 | ||
| 3.6 | Large devations | 54 | ||
| 3.7 | Metastability | 55 | ||
| Bibliographical notes | 61 | |||
| 4 | Allen-Cahn SPDE in two space dimensions | 63 | ||
| 4.1 | The need for renormalisation | 63 | ||
| 4.2 | Wick calculus | 64 | ||
| 4.3 | Existence and uniqueness of solutions | 74 | ||
| 4.4 | Invariant measure | 75 | ||
| 4.5 | Large deviations | 79 | ||
| 4.6 | Metastability | 80 | ||
| Bibliographical notes | 84 | |||
| 5 | Allen-Cahn SPDE in three space dimensions | 87 | ||
| 5.1 | Failure of the previous approaches | 87 | ||
| 5.2 | Perturbation theory for the Gibbs measure | 89 | ||
| 5.3 | Regularity structures: Introduction | 99 | ||
| 5.4 | Regularity structures: Algebraic aspects | 102 | ||
| 5.5 | Regularity structures: Analytic aspects | 118 | ||
| 5.6 | Existence and uniqueness of solutions | 146 | ||
| 5.7 | Large deviations | 151 | ||
| Bibliographical notes | 155 | |||
| A | Potential theory for reversible Markov chains | 157 | ||
| A.1 | First-hitting time and capacity | 157 | ||
| A.2 | Dirichlet principle | 160 | ||
| A.3 | Thomson principle | 161 | ||
| Bibliographical notes | 163 | |||
| B | Hopf algebras | 165 | ||
| B.1 | Definition and examples | 165 | ||
| B.2 | Group action | 169 | ||
| B.3 | Comodules | 173 | ||
| C | BPHZ renormalisation | 175 | ||
| C.1 | Hepp sectors | 176 | ||
| C.2 | Subdivergences | 178 | ||
| C.3 | Zimmermann's forest formula | 180 | ||
| C.4 | A renormalisation result | 182 | ||
| C.5 | Decorated graphs | 184 | ||
| C.6 | Other bounds involving singular kernels | 185 | ||
| D | Wiener chaos expansion | 189 | ||
| D.1 | Hermite polynolials revisited | 189 | ||
| D.2 | Wiener chaos | 190 | ||
| D.3 | Wick product | 192 | ||
| E | Hints and solutions for some exercises | 195 | ||
| E.1 | Exercises of Chapter 2 | 195 | ||
| E.2 | Exercises of Chapter 3 | 196 | ||
| E.3 | Exercises of Chapter 4 | 198 | ||
| E.4 | Exercises of Chapter 5 | 199 | ||
| References | 205 | |||
| Index | 217 | |||