On the Lack of Gaussian Tail for Rough Line Integrals along Fractional Brownian Paths

We show that the tail probability of the rough line integral ${\int }_0^1\phi (X_t)d{Y}_t$, where $(X,Y)$ is a 2D fractional Brownian motion with Hurst parameter $H\in (1\mathrm{/}4,1\mathrm{/}2)$ and $\phi$ is a $C_b^{\mathrm{\infty }}$-function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a $\gamma$-Weibull tail with any exponent $\gamma >2H+1$. In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class $C_b^{\mathrm{\infty }}$. This also demonstrates that the well-known upper tail estimate proved by Cass-Litterer-Lyons in 2013 is essentially sharp.