The structure group in regularity structures: replacing combinatorics of trees by Lie geometry

We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$, which was introduced to tackle more classes of singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures of Hairer when it comes to the structure group $\mathsf{G}$ (the purpose of which is the re-centering of the model when passing from one to another base point). In particular, $\mathsf{G}\subset{\rm Aut}(\mathsf{T})$ arises from a Hopf algebra $\mathsf{T}^+$ and a comodule $\Delta\colon\mathsf{T}\rightarrow \mathsf{T}^+\otimes\mathsf{T}$, which are intertwined in a specific way.

In fact, this approach, where the dual $\mathsf{T}^*$ of the abstract model space $\mathsf{T}$ naturally embeds into a formal power series algebra, allows to interpret $\mathsf{G}^*\subset{\rm Aut}(\mathsf{T}^*)$ as a Lie group arising from a Lie algebra $\mathsf{L} \subset{\rm End}(\mathsf{T}^*)$ consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (nonlinearities, functions of space-time mod constants). These actions are shift of space-time, and tilt by space-time polynomials.

The Hopf algebra $\mathsf{T}^+$ arises from a coordinate representation of the universal enveloping algebra ${\rm U}(\mathsf{L})$ of the Lie algebra $\mathsf{L}$. The coordinates are determined by an underlying (however not entirely closed) pre-Lie algebra structure of $\mathsf{L}$. Strong finiteness properties, which are enforced by gradedness and the restrictive definition of $\mathsf{T}$, allow for this dualization in our infinite-dimensional setting.

We also argue that our structure is compatible with the tree-based structure in case of the generalized parabolic Anderson model. More precisely, we construct an endomorphism between the abstract model spaces (which is neither onto nor one to one) that lifts to an endomorphism between the Hopf structures.

This is joint work with Pablo Linares and Markus Tempelmayr.