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

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.

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 |