Flows driven by multi-indices Rough Paths
Carlo Bellingeri, Yvain Bruned, Yingtong Hou
TL;DR
This work develops a full solution theory for scalar RDEs driven by multi_indices rough paths by embedding the problem in a flow framework built from a log_ODE almost flow. It constructs a rich multi_index Hopf algebraic structure, defines elementary differentials that propagate through the algebra via a morphism property, and leverages the sewing lemma to obtain well-posedness results (local and global under suitable hypotheses). A central theme is the translation of rough paths, expressed through $T_\ell$, which preserves the rough-path product and induces corresponding translations of the vector field, enabling Itô_Stratonovich conversions and renormalization procedures within this unified algebraic setting. The approach yields explicit connections between Davie-type path solutions and flow solutions, and provides a rigorous mechanism to study how translations impact RDEs driven by multi_indices rough paths with applications to stochastic calculus and renormalized models. The framework thus bridges combinatorial Hopf-algebra methods, rough path analysis, and translation-based renormalization for a broad class of nonlinear differential equations.
Abstract
In this work, we introduce a solution theory for scalar-valued rough differential equations driven by multi-indices rough paths. To achieve this task, we will show how the flow approach using the log-ODE method introduced by Bailleul fits perfectly in this setting. In addition, we also describe the action of the translation of multi-indices rough paths at the level of rough differential equations.