Laplace approximation for rough differential equation driven by fractional Brownian motion

We consider a rough differential equation indexed by a small parameter $\epsilon >0$. When the rough differential equation is driven by fractional Brownian motion with Hurst parameter $H$ ($1\mathrm{/}4<H<1\mathrm{/}2$), we prove the Laplace-type asymptotics for the solution as the parameter $\epsilon$ tends to zero.