The structure group in regularity structures: avoiding trees

Felix Otto (MPI MiS Leipzig)

Nov 16. 2021, 09:00 — 10:00

We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$, which was introduced in with work with Sauer, Smith, and Weber, to tackle quasi-linear singular SPDE. In this talk, we show that this approach is consistent with the postulates of regularity structures of Hairer when it comes to the structure group $\mathsf{G}$. 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.   \smallskip   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.   \smallskip   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 (incomplete) 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.   \smallskip   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.

Further Information
ESI Boltzmann Lecture Hall
Slides Preiss
Associated Event:
Higher Structures Emerging from Renormalisation (Graduate School)
Pierre Clavier (U of Haut-Alsace)
Kurusch Ebrahimi-Fard (NTNU, Trondheim)
Peter K. Friz (TU Berlin)
Harald Grosse (U of Vienna)
Dominique Manchon (U Clerment Auvergne)
Sylvie Paycha (U of Potsdam)
Sylke Pfeiffer (U of Potsdam)