论文标题
在高维空间中的Ornstein-Uhlenbeck过程的平均第一次退出时间
Mean first exit times of Ornstein-Uhlenbeck processes in high-dimensional spaces
论文作者
论文摘要
$ d $ dimensional ornstein-uhlenbeck工艺（OUP）描述了$ d $二二维，球形对称，二次潜力的粒子的轨迹。 OUP由一个漂移项组成，由常数$θ\ geq 0 $加权和扩散系数加权$σ> 0 $。在没有漂移（即$θ= 0 $）的情况下，OUP简单成为标准的布朗尼运动（BM）。本文关注的是，从有限的半径$ l $的球估算OUP的平均第一外效果时间（MFET），用于$ d \ gg 0 $。我们证明，对于$ d \ to \ infty $，渐近的OUP（平均）不再比BM退出。换句话说，OUP的均值漂移（按$θ\ geq 0 $缩放）对其MFET均无影响。这一发现可能令人惊讶，因为对于\ Mathbb {n} $中的小$ d \，OUP退出时间明显大于bm的空间，该边距取决于$θ$。由于它允许漂移被忽略，因此可以简化许多领域中高维出时间问题的分析。 最后，我们使用Andronov-Vitt--pontryagin公式对OUP的非反应MFET的简短证明可能具有独立的利益。
The $d$-dimensional Ornstein--Uhlenbeck process (OUP) describes the trajectory of a particle in a $d$-dimensional, spherically symmetric, quadratic potential. The OUP is composed of a drift term weighted by a constant $θ\geq 0$ and a diffusion coefficient weighted by $σ> 0$. In the absence of drift (i.e. $θ= 0$), the OUP simply becomes a standard Brownian motion (BM). This paper is concerned with estimating the mean first-exit time (MFET) of the OUP from a ball of finite radius $L$ for large $d \gg 0$. We prove that, asymptotically for $d \to \infty$, the OUP takes (on average) no longer to exit than BM. In other words, the mean-reverting drift of the OUP (scaled by $θ\geq 0$) has asymptotically no effect on its MFET. This finding might be surprising because, for small $d \in \mathbb{N}$, the OUP exit time is significantly larger than BM by a margin that depends on $θ$. As it allows for the drift to be ignored, it might simplify the analysis of high-dimensional exit-time problems in numerous areas. Finally, our short proof for the non-asymptotic MFET of OUP, using the Andronov--Vitt--Pontryagin formula, might be of independent interest.
