发表机构
Technical University of Darmstadt; Indian Statistical Institute（达姆施塔特工业大学; 印度统计研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究推导了带漂移布朗运动的二次加性泛函的小偏差渐近式，证明其在受限L²范数条件下弱收敛到Ornstein-Uhlenbeck过程，且极限动力学与原始漂移无关。
AI 中文摘要
我们研究带漂移的布朗运动在二次加性泛函Z_T=∫₀^T W_s²ds的稀有事件条件下的长时间行为，其中(W_t)_{t≥0}是漂移为μ的布朗运动。更准确地说，我们考虑当T→∞时，给定事件Z_T≤θT的W的条件分布，允许漂移μ依赖于T。我们推导了Z_T的精确、一致的小偏差渐近式，包括精确的前置因子，并利用这些结果证明该条件过程弱收敛到Ornstein-Uhlenbeck过程。值得注意的是，极限动力学与原始漂移无关。
英文摘要
We study the long-time behaviour of Brownian motion with drift under a rare-event conditioning of the quadratic additive functional $Z_T=\int_0^T W_s^2ds$, where $(W_t)_{t\geq 0}$ is a Brownian motion with drift $μ$. More precisely, we consider the conditional law of $W$ given the event $Z_T\le θT$, as $T\to\infty$, allowing the drift $μ$ to depend on $T$. We derive sharp, uniform small-deviation asymptotics for $Z_T$, including the exact prefactor, and use them to show that the conditioned process converges weakly to an Ornstein--Uhlenbeck process. Remarkably, the limiting dynamics are independent of the original drift.
Comments31 pages
