Rough differential equations and reduced rough paths: a Lie bracket characterization

Abstract

This paper studies rough differential equations from the viewpoint of reduced rough paths in the H\"older regime \(\frac13<\alpha\le\frac12\). A reduced rough path retains the first level and the symmetric part of the second level, while discarding the antisymmetric L\'evy-area component. We identify the precise obstruction to determining rough differential equation solutions from this reduced information. For an RDE driven by vector fields \(F_1,\ldots,F_d\), we prove that any two rough paths with the same reduced projection produce the same solution for every common initial value if and only if \[ [F_i,F_j]=0, \quad 1\le i,j\le d. \] Thus the antisymmetric L\'evy area is irrelevant exactly in the commuting-vector-field case.