An Introduction to Singular Stochastic PDEs.

Allen-Cahn Equations, Metastability, and Regularity Structures

N. Berglund

EMS Press, EMS Series of Lectures in Mathematics (2022)

ISBN: 978-3-98547-014-3

Publisher's website

Contents

1Introduction1 1.1 Stochastic partial differential equations 1 1.2 Singular SPDEs 3 1.3 Regularity structures 6 1.4 About this book 7 2A system of interacting diffusions13 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 3Allen-Cahn SPDE in one space dimension33 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 4Allen-Cahn SPDE in two space dimensions63 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 5Allen-Cahn SPDE in three space dimensions87 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 APotential theory for reversible Markov chains157 A.1 First-hitting time and capacity 157 A.2 Dirichlet principle 160 A.3 Thomson principle 161 Bibliographical notes 163 BHopf algebras165 B.1 Definition and examples 165 B.2 Group action 169 B.3 Comodules 173 CBPHZ renormalisation175 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 DWiener chaos expansion189 D.1 Hermite polynolials revisited 189 D.2 Wiener chaos 190 D.3 Wick product 192 EHints and solutions for some exercises195 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 References205 Index217