An Introduction to Singular Stochastic PDEs ; Allen-Cahn Equations, Metastability, and Regularity Structures.

Format:

Book

Author/Creator:

Berglund, Author.

Series:

EMS series of lectures in mathematics.

EMS Series of Lectures in Mathematics (elm) 33 2523-5184

Language:

English

Subjects (All):

Stochastic partial differential equations.

Stability.

Structural stability--Mathematical models.

Structural stability.

stability.

Physical Description:

1 online resource (230 pages)

Place of Publication:

EMS Press 2022

Berlin : EMS Press, 2022.

System Details:

text file PDF

Summary:

Stochastic partial differential equations (SPDEs) model the evolution in time of spatially extended systems subject to a random driving. Recent years have witnessed tremendous progress in the theory of so-called singular SPDEs. These equations feature a singular, distribution-valued driving term, a typical example being spacetime white noise, which makes them ill-posed as such. In many cases, it is however possible to make sense of these equations by applying a so-called renormalisation procedure, initially introduced in quantum field theory. This book gives a largely self-contained exposition of the subject of regular and singular SPDEs in the particular case of the Allen-Cahn equation, which models phase separation. Properties of the equation are discussed successively in one, two and three spatial dimensions, allowing to introduce new difficulties of the theory in an incremental way. In addition to existence and uniqueness of solutions, aspects of long-time dynamics such as invariant measures and metastability are discussed. A large part of the last chapter, about the three-dimensional case, is dedicated to the theory of regularity structures, which has been developed by Martin Hairer and co-authors in the last years, and allows to describe a large class of singular SPDEs. The book is intended for graduate students and researchers in mathematics and physics with prior knowledge in stochastic processes or stochastic calculus.

Contents:

Introduction

A system of interacting diffusions

Allen-Cahn SPDE in one space dimension

Allen-Cahn SPDE in two space dimensions

Allen-Cahn SPDE in three space dimensions.

Notes:

Includes bibliographical references (pages 205-215) and index.

Description based on publisher supplied metadata and other sources.

ISBN:

3-9854751-4-8