Asymptotic expansion of the density for hypoelliptic rough differential equation

We study a rough differential equation driven by fractional Brownian motion with Hurst parameter $H$ $(1\mathrm{/}4<H\le 1\mathrm{/}2)$. Under H"ormander’s condition on the coefficient vector fields, the solution has a smooth density for each fixed time. Using Watanabe’s distributional Malliavin calculus, we obtain a short time full asymptotic expansion of the density under quite natural assumptions. Our main result can be regarded as a “fractional version” of Ben Arous’ famous work on the off-diagonal asymptotics.