Rough paths based on multi-indices

Following the approach of Otto, Sauer, Smith and Weber, we describe solutions to driven ODEs in terms of power series of derivatives of the nonlinearities. This gives a notion of rough path based on multi-indices: Its underlying Hopf algebra is dual to the Lie universal envelope of the free Novikov algebra. We also discuss how to introduce local counterterms (algebraic renormalization) and their associated (multi) pre-Lie algebra, leading to groups of translations of rough paths.