Complete the pathwise regularity proof for the Gaussian tensor-field rough-path example

Complete the proof that the Gaussian tensor-field rough-path construction satisfies the asserted pathwise temporal regularity, either by applying the Kolmogorov continuity theorem to the step-3 nilpotent Lie group over $H^\gamma$ or by extending the rough-path Kolmogorov continuity theorem to level-3 rough paths.

Background

The paper presents a Gaussian vector-field example and constructs iterated tensor-field integrals. Moment estimates and Sobolev embedding establish the claimed regularity heuristically for the relevant roughness regimes. The authors explicitly acknowledge that the pathwise time-regularity argument has not been fully completed and identify two possible tools that could fill the gap: a group-valued Kolmogorov theorem or an extension of the rough-path Kolmogorov theorem to level 3.

References

To avoid technical details, we did not really complete the proof that the above example gives the correct pathwise time-regularity. To fill this gap, one can either employ the Kolmogorov continuity theorem applied to $G3(H{\gamma})$ -- the step-3 nilpotent Lie group over $H{\gamma}$ -- or extend the rough path Kolmogorov continuity theorem Theorem 3.1 to level 3 rough paths.

— Higher Order Unbounded Rough Drivers  (2609.05353 - Nilssen et al., 4 Sep 2026) in Remark immediately following the Gaussian noise vector-field example, Section 4, subsection “Tensor field rough paths”