An introduction to singular stochastic PDEs. Allen-Cahn equations, metastability, and regularity structures. (English) Zbl 1511.35001

Berglund, Nils

EMS Series of Lectures in Mathematics. Berlin: European Mathematical Society (EMS) (ISBN 978-3-98547-014-3/pbk; 978-3-98547-514-8/ebook). x, 220 p. (2022).

The present book focuses on parabolic stochastic partial differential equations (SPDEs) forced by space-time white noise. They are generally of the form \[ \partial_t \phi(t,x) = \Delta \phi(t,x) + F \big( \phi(t,x), \nabla \phi(t,x) \big) + \sqrt{2 \varepsilon} \, \xi(t,x). \] Particular examples, which are presented in Chapter 1, are the dynamic \(\Phi^4\) model, the stochastic Allen-Cahn equation, the KPZ equation and the parabolic Anderson model. Many of these examples are so-called singular SPDEs. This means that they are mathematically ill-defined, which is due to irregularity of the space-time white noise \(\xi\).
The book provides an introduction to singular SPDEs by focusing on the Allen-Cahn equation, which is given by \[ \partial_t \phi(t,x) = \Delta \phi(t,x) + \phi(t,x) - \phi(t,x)^3 + \sqrt{2 \varepsilon} \, \xi(t,x). \] In Chapter 2, the author considers a system of coupled stochastic differential equations (SDEs) obtained by discretising in space the one-dimensional Allen-Cahn equation. In Chapter 3 the one-dimensional Allen-Cahn SPDE is studied. Afterwards, in Chapter 4 the two-dimensional Allen-Cahn SPDE is treated. In contrast to the one-dimensional case, a renormalisation technique is required here. Finally, in Chapter 5 the three-dimensional Allen-Cahn SPDE is studied. Here the previous approaches fail, and the theory of regularity structures plays a crucial role.
In all these chapters, the author deals – besides proving existence and uniqueness – with results concerning the long-time behaviour of solutions. This includes invariant measures, speed of convergence, large deviations and metastability.

Reviewer: Stefan Tappe (Freiburg)

MSC:

35-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35R60	PDEs with randomness, stochastic partial differential equations
35K58	Semilinear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

stochastic partial differential equations; singular stochastic partial differential equations; renormalisation; regularity structures; Allen-Cahn equation; metastability