Short time kernel asymptotics for rough differential equation driven by fractional Brownian motion

Abstract

We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter $H ~(1/3 < H \le 1/2)$ under the ellipticity assumption at the starting point. In such a case, the law of the solution at a fixed time has a kernel, i.e., a density function with respect to Lebesgue measure. In this paper we prove a short time off-diagonal asymptotic expansion of the kernel under mild additional assumptions. Our main tool is Watanabe’s distributional Malliavin calculus.