Pablo Linares & Markus Tempelmayr — A tree-free construction of the structure group

We present a new approach to regularity structures, and in particular to the construction of the structure group, which replaces the tree-based framework of Hairer by a more Lie-geometric setting. We consider the space of pairs (a,p), where a is a placeholder for the nonlinearity and p is a polynomial which locally parameterizes the manifold of solutions to a given (stochastic) PDE, together with natural actions of shift by space-time vectors and tilt by space-time polynomials. Moreover, we provide a coordinate representation for the algebra of functions of (a,p) in terms of multi-indices, and lift the actions of shift and tilt to endomorphisms of this algebra. The infinitesimal generators of these actions form a Lie algebra, from which we trace a completely algebraic path towards the structure group via the universal enveloping algebra. We also connect to the tree-based approach in the driven ODE case (branched rough paths), showing that our multi-indices actually correspond to specific linear combinations of trees. Joint work with Felix Otto.