Stochastic PDEs, Regularity Structures, and Interacting Particle Systems

These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of "Regularity structures" developed recently by Hairer in arXiv:1303.5113 . This theory gives a way to study well-posedness for a class of stochastic PDEs that could not be treated previously. Prominent examples include the KPZ equation as well as the dynamic $\Phi^4_3$ model. Such equations can be expanded into formal perturbative expansions. Roughly speaking the theory of regularity structures provides a way to truncate this expansion after finitely many terms and to solve a fixed point problem for the "remainder". The key ingredient is a new notion of "regularity" which is based on the terms of this expansion.Comment: Fixed typo

A smooth introduction to the wavefront set

The wavefront set provides a precise description of the singularities of a distribution. Because of its ability to control the product of distributions, the wavefront set was a key element of recent progress in renormalized quantum field theory in curved spacetime, quantum gravity, the discussion of time machines or quantum energy inequalitites. However, the wavefront set is a somewhat subtle concept whose standard definition is not easy to grasp. This paper is a step by step introduction to the wavefront set, with examples and motivation. Many different definitions and new interpretations of the wavefront set are presented. Some of them involve a Radon transform.Comment: 29 pages, 7 figure

Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a $3$-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic H\"older spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.Comment: v3, 56 pages. Different points about renormalisation matters have been clarified. Typos correcte

A Cutoff Procedure and Counterterms for Differential Renormalization

Explicit divergences and counterterms do not appear in the differential renormalization method, but they are concealed in the neglected surface terms in the formal partial integration procedure used. A systematic real space cutoff procedure for massless $\phi^4$ theory is therefore studied in order to test the method and its compatibility with unitarity. Through 3-loop order, it is found that cutoff bare amplitudes are equal to the renormalized amplitudes previously obtained using the formal procedure plus singular terms which can be consistently cancelled by adding conventional counterterms to the Lagrangian. Renormalization group functions $\beta (g)$ and $\gamma (g)$ obtained in the cutoff theory also agree with previous results.Comment: 28 pages, CTP#2099-DTP/92/40-UBECMPF/92/13, 2 figures (not included). (Tex problems solved and information about number of pages and figures added