On the lack of Gaussian tail for rough line integrals along fractional Brownian paths

Abstract

We show that the tail probability of the rough line integral $$\int _{0}^{1}\phi (X_{t})dY_{t}$$ ∫ 0 1 ϕ ( X t ) d Y t , where (X, Y) is a 2D fractional Brownian motion with Hurst parameter $$H\in (1/4,1/2)$$ H ∈ ( 1 / 4 , 1 / 2 ) and $$\phi $$ ϕ is a $$C_{b}^{\infty }$$ C b ∞ -function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a $$\gamma $$ γ -Weibull tail with any exponent $$\gamma >2H+1$$ γ > 2 H + 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 $$C_{b}^{\infty }$$ C b ∞ . This also demonstrates that the well-known upper tail estimate proved by Cass–Litterer–Lyons in 2013 is essentially sharp.