Malliavin Differentiability and Hypoellipticity for Degenerate Diffusions via the Theory of Rough Paths

This monograph establishes a rigorous framework for the hypoellipticity of degenerate diffusions driven by low-regularity Gaussian signals, specifically within the context of Rough Path Theory. We consider stochastic differential equations (SDEs) driven by Gaussian processes with Hurst parameter H (1/4, 1/2], which fail to be semimartingales, thereby rendering classical Itô calculus inapplicable. By constructing the solution map as a continuous functional on the space of geometric rough paths, we prove the Malliavin differentiability of the solution flow. The core contribution is the derivation of a quantitative non-degeneracy estimate for the Malliavin covariance matrix under a weak Hörmander condition on the vector fields. This is achieved by adapting Norris's lemma to the rough path setting, ultimately ensuring the existence and smoothness of the density of the solution with respect to the Lebesgue measure. The results unify the deterministic stability of rough paths with the probabilistic regularity of the Malliavin calculus.