Smoothness of the density for solutions to Gaussian rough differential equations

We consider stochastic differential equations of the form dYt=V(Yt)dXt+V0(Yt)dt driven by a multi-dimensional Gaussian process.Under the assumption that the vector fields V0 and V=(V1,.,Vd) satisfy Hörmander.s bracket condition, we demonstrate that Yt admits a smooth density for any t.(0,T], provided the driving noise satisfies certain nondegeneracy assumptions.Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes.Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter H>1/4, the Ornstein.Uhlenbeck process and the Brownian bridge returning after time T.