论文标题
关于沿分数布朗路径的粗线积分缺乏高斯尾巴
On the Lack of Gaussian Tail for Rough Line Integrals along Fractional Brownian Paths
论文作者
论文摘要
我们表明，粗线积分$ \ int_ {0}^{1} ϕ（x_ {t}）dy_ {t} $的尾巴概率，其中$（x，y）$是2D分数布朗运动，带有hurst parameter $ h \ in（1/4,4,1/2）$ and $ c_ $ c _}在其导数上，轻度的非固定条件不能比任何指数$γ> 2H+1 $ $γ$ -Weibull尾巴更快地衰减。特别是，这会产生由FBM驱动的微分方程的简单示例，即使假定基础向量字段是$ c_ {b}^{\ infty} $，其解决方案也无法具有高斯尾巴。这也表明，卡斯·勒莱斯（Cass Litterer-Laneons）在2013年证明的众所周知的上尾估计基本上是敏锐的。
We show that the tail probability of the rough line integral $\int_{0}^{1}ϕ(X_{t})dY_{t}$, where $(X,Y)$ is a 2D fractional Brownian motion with Hurst parameter $H\in(1/4,1/2)$ and $ϕ$ is a $C_{b}^{\infty}$-function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a $γ$-Weibull tail with any exponent $γ>2H+1$. 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 $C_{b}^{\infty}$. This also demonstrates that the well-known upper tail estimate proved by Cass-Litterer-Lyons in 2013 is essentially sharp.
