论文标题
Hopf和Pre-Lie代数在规律性结构中
Hopf and pre-Lie algebras in regularity structures
论文作者
论文摘要
这些讲义旨在介绍ARXIV中开发的规则性结构的代数理论：1303.5113，Arxiv：1610.08468和Arxiv：1711.10239。该理论的主要目的是建立一种系统的奇异SPDE重新归一化的方法。与互补的分析结果一起，这些作品为亚临界方案中的一类半线性抛物线奇异SPDE提供了一般解决方案理论。我们证明了如何使用相互作用的HOPF代数来描述“正”和“负”的重力降临化，以及如何计算SPDE中的非线性非线性。对于后者而言，至关重要的重要性是对非线性的前一个结构，而负性重态化组通过前型形态起作用。为了显示这些结果的主要方面，而没有引入许多符号和假设，我们将重点放在一个普通理论的特殊情况下，其中只有一个方程式和一个噪声。这些讲座的注释是2021年11月在维也纳ESI的大师班和车间“从重生化的更高结构”上的材料的扩展。
These lecture notes aim to present the algebraic theory of regularity structures as developed in arXiv:1303.5113, arXiv:1610.08468, and arXiv:1711.10239. The main aim of this theory is to build a systematic approach to renormalisation of singular SPDEs; together with complementary analytic results, these works give a general solution theory for a wide class of semilinear parabolic singular SPDEs in the subcritical regime. We demonstrate how "positive" and "negative" renormalisation can be described using interacting Hopf algebras, and how the renormalised non-linearities in SPDEs can be computed. For the latter, of crucial importance is a pre-Lie structure on non-linearities on which the negative renormalisation group acts through pre-Lie morphisms. To show the main aspects of these results without introducing many notations and assumptions, we focus on a special case of the general theory in which there is only one equation and one noise. These lectures notes are an expansion of the material presented at a minicourse with the same title at the Master Class and Workshop "Higher Structures Emerging from Renormalisation" at the ESI, Vienna, in November 2021.
