标题： 随机偏微分方程的预李和后李代数的构造 显示英文标题

标题： Construction of pre-and post-Lie algebras for stochastic PDEs

摘要： 我们给出并研究了一种装饰了边和顶点的根树上的pre-Lie代数结构的构造，其接枝积通过映射 $\phi$ 作用于所创建边的装饰以及执行接枝的顶点，同时影响它们。 我们证明，这种构造给出一个pre-Lie代数当且仅当映射 $\phi$ 满足一个交换关系，称为树兼容性。 我们展示了如何通过与另一个post-Lie代数的半直积来扩展这个pre-Lie代数结构为一个post-Lie代数。 我们还定义了几个获得树兼容映射的构造，并给出了例子，包括当顶点装饰空间是 $2$-维且边装饰空间是有限维时所有树容许映射的描述。 他们使用这种构造的一个特殊例子是Bruned、Hairer和Zambotti用于随机偏微分方程的研究：当没有噪声时，我们证明基础的树兼容映射是更简单映射的指数，并推导出一个与经典根树pre-Lie代数的显式同构；当存在噪声时，我们得到的基础树兼容映射是直接和形式。 我们也用我们的形式主义得到了Bruned和Katsetsiadis描述的post-Lie代数。 显示英文摘要

摘要： We give and study a construction of pre-Lie algebra structures on rooted trees whose edges and vertices are decorated, with a grafting product acting, through a map $\phi$, both on the decoration of the created edge and on the vertex that holds the grafting. We show that this construction gives a pre-Lie algebra if, and only if, the map $\phi$ satisfies a commutation relation, called tree-compatibility. We show how to extend this pre-Lie algebra structure to a post-Lie one by a semi-direct extension with another post-Lie algebra. We also define several constructions to obtain tree-compatible maps, and give examples, including a description of all tree-admissible maps when the space of decorations of the vertices is $2$-dimensional and the space of decorations of the edges is finite-dimensional. A particular example of such a construction is used by Bruned, Hairer and Zambotti for the study of stochastic partial differential equations: when no noise is involved, we show that the underlying tree-compatible map is the exponential of a simpler one and deduce an explicit isomorphism with a classical pre-Lie algebra of rooted trees; when a noise is involved, we obtain the underlying tree-compatible map as a direct sum. We also obtain with our formalism the post-Lie algebras described by Bruned and Katsetsiadis.