Abstract:This paper studies rough differential equations from the viewpoint of reduced rough paths in the Hölder 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 Lévy-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 Lévy area is irrelevant exactly in the commuting-vector-field case.
From: Xing Gao [view email]
[v1]
Sun, 30 Nov 2025 00:02:16 UTC (17 KB)
[v2]
Sun, 12 Jul 2026 23:20:18 UTC (8 KB)
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