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Stochastic PDEs, regularity structures, and interacting particle systems. (English. French summary) Zbl 1421.81001
These lecture notes deals with stochastic partial differential equations (SPDEs), regularity structures and interacting particle systems, and aims to explain some aspects of the theory of “Regularity structure” developed by Hairer. In Lecture 1 the authors outline the scope of the theory of regularity structures. It is discussed the Kardar-Parisi-Zhang equation which is given by \(\partial_{t}h(t,x)=\partial^2_{x}h(t,x)+(\partial_{x}h(t,x))^2/2+\xi(t,x)\), where the term \(\xi(t,x)\) denotes a spacetime white noise. The authors focus more on the second example, namely the dynamic \(\Phi_{d}^4\) model given by \(\partial_{t}\phi(t,x)=\Delta\phi(t,x)-\phi^3(t,x)-m^2\phi(t,x)+\xi(t,x)\). They consider the scaling behaviour for SPDEs and subcriticality. Then they discuss the need for renormalisation and consider the approximation of renormalised SPDEs by interacting particle systems. Lecture 2 starts by describing how we will keep track of the regularity of spacetime functions and distributions. Then the authors give a review of classical solution techniques for semilinear SPDEs. It is explained how a lack of regularity causes problems for these theories, using \((\Phi_{d}^4)\) in \(d=2,3\) as an examples. Then they describe a perturbative approach to these equations. Lecture 3 starts by discussing the Da Prato-Debussche argument and then sketch why it fails for \(\Phi_{3}^4\). The authors introduce some of basic objects of the theory of regularity structures: regularity structures, models, and modelled distributions. In the last Lecture 4 the authors discuss regularity structures and models associated with controlled rough paths, \(\Phi_{2}^4\), and \(\Phi_{3}^4\). The construction of canonical models is presented, the convergence of the models and renormalization are discussed.

MSC:
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
35R60 PDEs with randomness, stochastic partial differential equations
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
82C22 Interacting particle systems in time-dependent statistical mechanics
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