Malliavin Calculus for rough stochastic differential equations

In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and L\ᵉ (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a H\"ormander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.