Asymptotic expansion of the density for hypoelliptic rough differential equation

Abstract

We study a rough differential equation driven by fractional Brownian motion with Hurst parameter $H$ $(1/4<H\le 1/2)$. Under Hörmander'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.

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