A diagram-free approach to the stochastic estimates in regularity structures (Q6585701)

In this paper, a version of Hairer's regularity structures, based on a greedier index set than trees, is explored, as introduced by \textit{F. Otto} et al. in [``A priori bounds for quasi-linear SPDEs in the full sub-critical regime'', Preprint, \url{arXiv:2103.11039}] and algebraically characterized by \textit{P. Linares} et al. in [Commun. Am. Math. Soc. 3, 1--64 (2023; Zbl 1535.60156)]. More precisely, the renormalized model postulated by Otto et al. is constructed and stochastically estimated, with the use of Feynman diagrams being avoided, while the process is carried out in a fully automated, inductive manner. This is achieved for a class of quasi-linear parabolic PDEs driven by noise in the fully singular but renormalizable range. It is assumed that a spectral gap inequality holds for the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model complements its vanishing expectation, which arises from the BPHZ choice of renormalization. The gain in regularity at the level of the Malliavin derivative of the model is captured through its description as a modelled distribution. Symmetry is incorporated into the renormalization Ansatz and serves as a guiding principle. The approach taken is analytic and top-down, rather than combinatorial and bottom-up.