Algebraic and probabilistic aspects of regularity structures

This thesis is concerned with a solution theory for quasilinear singular stochastic partial differential equations. We approach the theory of regularity structures, a tool to tackle singular stochastic PDEs, from a new perspective which is well suited for, but not restricted to, quasilinear equations. In the first part of this thesis, we revisit the algebraic aspects of the theory of regularity structures. Although we approach regularity structures from a different perspective than originally done, we show that the same (Hopf-) algebraic structure is underlying. Trees do not play any role in our construction, hence the Hopf algebras underlying rough paths and regularity structures are not at our disposal. Instead, our alternative point of view gives a new (Lie-) geometric interpretation of the structure group, arising from simple actions on the nonlinearity of the equation and a parametrization of the solution manifold. In the second part of this thesis, we revisit the probabilistic aspects of the theory of regularity structures. We construct and stochastically estimate the centered model, which captures the local behaviour of the solution manifold. This is carried out under a spectral gap assumption on the driving noise, and based on a novel application of Malliavin calculus in regularity structures. In deriving the renormalized equation we are guided by symmetries, so that natural invariances of the model are built in. In the third part of this thesis, we make again use of the Malliavin derivative to obtain a robust characterization of the model, which persists for rough noise even as a mollification is removed. This allows for a simple derivation of invariances of the model that are not present at the level of approximations. Furthermore, we give a convergence result of models, which together with the characterization establishes a universality result in the class of noise ensembles satisfying uniformly a spectral gap assumption