The structure group for quasi-linear equations via universal enveloping algebras
The structure group for quasi-linear equations via universal enveloping algebras
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基于泛包络代数的拟线性方程的结构群
DOI：
10.1090/cams/16
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Markus Tempelmayr
中科院分区：
文献类型：
--
作者：
P. Linares;F. Otto;Markus Tempelmayr
我们用多指标代替树作为抽象模型空间的指标集来处理拟线性奇异随机偏微分方程。当涉及到由Hopf代数和模产生的结构群时，我们证明了这种方法与正则结构的公设是一致的。
We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group, which arises from a Hopf algebra and a comodule.
Our approach, where the dual of the abstract model space naturally embeds into a formal power series algebra, allows to interpret the structure group as a Lie group arising from a Lie algebra consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (non-linearities, functions of space-time mod constants).
We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the stochastic heat equation.