On the multi-indices approach to path-wise stochastic analysis

Recently Linares-Otto-Tempelmayr have unveiled a very interesting algebraic structure which allows to define a new class of rough paths/regularity structures, with associated applications to stochastic PDEs or ODEs. This approach does not consider trees as combinatorial tools but their fertility, namely the function which associates to each integer k the number of vertices in the tree with exactly k children. In a joint work with J-D Jacques we have studied this algebraic structure and shown that it is related with a general and simple class of so-called post-Lie algebras. The construction has remarkable properties and I will try to present them in the simplest possible way.