分数布朗运动驱动的微分方程的尾部最优估计
Optimal Tail Estimates for Differential Equations Driven by Fractional Brownian Motion
浏览论文内容
中文总结 AI 辅助
本文研究分数布朗运动驱动的随机微分方程尾部概率的精确衰减速率，在一致椭圆性假设下建立了Young与粗糙情形的下尾估计，并通过反例和条件改进揭示了高斯下尾的适用边界。
中文摘要 AI 辅助
本文旨在研究尾部概率 $\mathbb{P}(|X_1-x_0|>R)$ 在大 $R$ 情形下的精确衰减速率，其中 $X_t$ 是初始条件为 $x_0$ 的、由分数布朗运动驱动的多维随机微分方程的解。首先，在一致椭圆性的假设下，我们在 Young 情形（$H\in(1/2,1)$）下建立了通用的 $(2H+1)$-Weibull 下尾估计，并在粗糙情形（$H\in(1/4,1/2)$）下建立了通用的高斯下尾估计。这两个估计在其各自的情形下均被证明是精确的。其次，我们通过构造显式例子证明了一个令人瞩目的事实：具有 $C_b^\infty$ 系数的均匀椭圆系统在 Young 情形下可能\textit{不具有}高斯下尾。因此，我们认为，\n\cite{BKT16} 中的多维高斯下界估计在不对向量场作进一步假设的情况下，可能无法以其当前的一般性形式成立。第三，我们给出了向量场的一个简单非退化条件，在该条件下可以在 Young 情形下建立高斯下尾。此外，鉴于上述反例的存在，我们证明了另一个令人瞩目的事实：任何 $2\times 2$ 周期的、均匀椭圆系统在 Young 情形下总是具有高斯下尾。最后，在粗糙情形下，我们为满足向量场上非交换条件的一大类系统建立了通用的 $(2H+1)$-Weibull 下尾。该结果对 \cite{BG24} 中提出的关于非交换系统著名的 Cass-Litterer-Lyons 估计的普遍性的猜想给出了部分肯定的回答。
英文摘要
The goal of the present paper is to investigate the exact decay rate of the tail probability $\mathbb{P}(|X_1-x_0|>R)$ for large $R$, where $X_t$ is the solution to a multidimensional stochastic differential equation driven by a fractional Brownian motion with initial condition $x_0$. In the first place, under the assumption of uniform ellipticity, we establish a general $(2H+1)$-Weibull lower tail estimate in Young's regime of $H\in(1/2,1)$ and a general Gaussian lower tail estimate in the rough regime of $H\in(1/4,1/2)$. These two estimates are seen to be sharp in their respective regimes. In the second place, we prove a striking fact by constructing explicit examples that a uniformly elliptic system with $C_b^\infty$-coefficients could \textit{fail} to have Gaussian lower tail in Young's regime. As a consequence, in our modest opinion, the multidimensional Gaussian lower estimate of \cite{BKT16} might not hold in its current form of generality without further assumptions on the vector fields. In the third place, we provide a simple nondegeneracy condition on the vector fields, under which a Gaussian lower tail can be established in Young's regime. In addition, given the existence of the aforementioned counterexamples, we prove another striking fact that any $2\times 2$ periodic, uniformly elliptic system always has Gaussian lower tail in Young's regime. Lastly, in the rough regime we establish a general $(2H+1)$-Weibull lower tail for a rich class of systems that satisfy a noncommutativity condition on the vector fields. This result provides a partially affirmative answer to a conjecture raised in \cite{BG24} on the genericness of the well-known Cass-Litterer-Lyons estimate for noncommutative systems.
发表机构
- University of Warwick（华威大学）
- University of Melbourne（墨尔本大学）
- International School for Advanced Studies (SISSA)（高级研究所）
机构由 AI 辅助整理，请以论文原文为准。
