Construction of pre-and post-Lie algebras for stochastic PDEs
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916778258464768 |
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| author | Foissy, Loïc |
| author_facet | Foissy, Loïc |
| contents | 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 $ϕ$, 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 $ϕ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03767 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of pre-and post-Lie algebras for stochastic PDEs Foissy, Loïc Combinatorics 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 $ϕ$, 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 $ϕ$ 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. |
| title | Construction of pre-and post-Lie algebras for stochastic PDEs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.03767 |