In the project, we have developed a general theory of solutions for quasilinear parabolic SPDEs in the full sub-critical regime and identified the main symmetry one wants to impose on solutions of these equations, namely, the chain rule symmetry. This symmetry allows us to get local counter-terms in the renormalised equations. It also provides global well-posedness for the space-time white noise.
Such an achievement has been possible thanks to a full characterisation of the chain rule. This characterisation is obtained via ideas from homological algebra and operad theory. The main new insight is to describe the space of invariant terms homologically, using a suitable perturbation of the differential of the operadic twisting of some well-chosen operad.
Another main achievement is a thorough understanding of multi-indices, a new combinatorial structure that has emerged for studying scalar-valued low regularity dynamics. We have investigated the appearance of this new combinatorial set in Quantum Field Theory and Numerical Analysis. We have results saying that they may not exist other combinatorial sets lying between multi-indices and decorated trees, making these two sets unique for studying singular dynamics.
We have also made important progress in understanding discretisation of singular SPDEs, proposing a semi-general convergence theorem for the generalized KPZ discrete model in regularity structures.
Concerning dispersive equations, we have identified the main algebraic structure used for deriving normal forms, which is the arborification coming from the mould calculus introduced by Jean Ecalle.
This is an important step toward a better combinatorial understanding of dispersive PDEs with random initial data and Wave turbulence.
