A multiindex-based regularity structure for quasilinear SPDEs

We consider the approach to regularity structures introduced by Otto, Sauer, Smith and Weber for a class of quasilinear SPDEs. This approach replaces Hairer's tree-based description by a greedier index set built from derivatives of the nonlinearity. We provide a Hopf-algebraic construction of the structure group within this framework. Considering the infinitesimal generators of certain actions in the space of nonlinearities, we build a pre-Lie algebra; its universal envelope, after a proper choice of basis, is dual to a Hopf algebra from which we can build the group. This more Lie-algebraic approach connects to already-existing constructions in regularity structures, which are rather combinatorial. Based on joint work with Felix Otto and Markus Tempelmayr.