Determine the canonical Lipschitz-atlas choice and its effect on manifold rough-path constructions

Determine which Lipschitz-b3 atlas should be chosen for a smooth manifold in the Lipschitz-b3-manifold approach to rough differential equations and determine how the choice of atlas affects the resulting rough-path and rough-differential-equation objects.

Background

The paper reviews an approach in which transition functions between coordinate patches are required to be Lipschitz-b3, with b greater than the regularity parameter of the driving rough path. This additional atlas structure provides the uniform estimates needed to transfer Euclidean rough differential equations between charts and glue local solutions.

The authors explicitly identify two unresolved issues in that approach: a smooth manifold is compatible with many distinct Lipschitz-b3 atlases, and it is unknown which atlas should be selected or whether different choices lead to equivalent resulting objects. The present paper avoids this ambiguity by using natural bundles and coordinate-invariant rough velocity fields instead of imposing a Lipschitz-b3 atlas.

References

It is not clear which Lipschitz-$\gamma$ atlas should be chosen given that a smooth manifold is compatible with many distinct ones and neither it is clear how this choice effects the resulting objects.

— Rough differential equations on manifolds via natural bundles  (2609.01190 - Bělohlávek et al., 1 Sep 2026) in Section 1, subsection “State of the art,” subsubsection “RDEs on manifolds”