Algebraic renormalisation of regularity structures
The paper under review gives a systematic description of a canonical renormalization procedure of stochastic PDEs containing nonlinearities involving generalized functions. This theory is based on the construction of a new class of regularity structures, which can well describe the sufficiently large subgroups of their automorphism groups. The construction is based on bialgebras. More precisely, there are two interacting Hopf algebras acting together on a vector space in which two twisted antipodes play a fundamental role. Kuo-Tsai Chen discovered that for any finite alphabet \(A\), the family of iterated integrals of a smooth path has a number of interesting algebraic properties. Note that, the family endowed with the suitable shuffle product and the deconcatenation coproduct, is a Hopf algebra. These two remarkable properties can therefore serve as an important tool if one wants to consider non-smooth paths and still build a consistent calculus. For example, Terry Lyons' definition of a geometric rough path and Gubinelli's concept of a branched rough path. The main aim of the reviewed article is to define an algebraic framework allowing to build regularity structures which, on the one hand, extend the ones built in [\textit{M. Hairer}, Invent. Math. 198, No. 2, 269--504 (2014; Zbl 1332.60093)] and, on the other hand, admit sufficiently many automorphisms to cover the renormalisation procedures of all subcritical stochastic PDEs that have been studied to date. Moreover, the construction is not restricted to the Gaussian setting and applies to any choice of the driving noise with minimal integrability conditions. It shows that the BPHZ renormalisation procedure belongs to the renormalisation group of the regularity structure associated to any class of subcritical semilinear stochastic PDEs. It is interesting to remark that the structure concerned is an extension of that previously described in related papers, which is itself an extension of the Connes-Kreimer Hopf algebra of rooted trees. It is also closely related to incidence Hopf algebras associated to families of posets. The paper under review together with a series of earlier works, complete the general theory initiated in [loc. cit.].
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