A tourist's guide to regularity structures and singular stochastic PDEs

Abstract

We give a short essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of stochastic singular PDEs.

Paracontrolled calculus and regularity structures II

“…We refer the reader to the reviews [13,14] of Chandra & Weber and Corwin & Shen for some nontechnical introductions to the domain of semilinear singular stochastic PDEs. One can refer to the books [16,5] of Friz & Hairer and Berglund for mildly technical introductions to regularity structures, and to Bailleul & Hoshino's Tourist's Guide [3] for a thorough tour of the analytic and algebraic sides of the theory. Hairer's lecture notes [18,19] are centered on the problems of renormalisation in the setting of Feynmann graphs and in the setting of singular stochastic PDEs, respectively.…”

“…) defines a concrete regularity structure in the sense of [3]. The structure of the (ε, p)-dependent tuple…”

Post-Lie algebras in Regularity Structures

“…We recommend the reader the following references [30,6] for surveys on Regularity Structures. From [37], the key point for getting the solution is the construction of the integrals u t,z,τ that may require a renormalisation procedure.…”

“…A Butcher series formalism for the expansion (1.4) has been initiated in [5] and it has been used for understanding how the renormalisation of the iterated integrals changed the equation we started with by adding counter-terms. The method of [5] has been simplified in [6] and in [3,4] a simple proof of the renormalised equation has been given that also works in the context of non-translation invariance where renormalisation constants are replaced by functions. The aim of the present work is to push forward the formalism of [5] by introducing the notion of Regularity Structures Butcher series.…”

Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization

“…The methods of Hairer and Gubinelli apply to many other SPDEs besides the one for the ϕ 4 3 -model, we shall however concentrate ourselves in mentioning papers directly related to the ϕ 4 3 -model (for other applications see, e.g. [37] and [87]). In the original work of Hairer solutions for the SQE to the ϕ 4 3 -model on T 3 were found on space C α with α ∈ (−2/3, −1/2) and with initial conditions in the same space.…”