截断厚尾动力系统的第一退出时间分析
First-Exit Time Analysis for Truncated Heavy-Tailed Dynamical Systems
AI总结：
本文分析了截断厚尾动力系统的第一退出时间，通过大偏差与元稳定性框架，揭示了相变层次和灾难原理，提出了厚尾系统的Freidlin-Wentzell理论扩展。
摘要：
本文研究了随机差分方程$X^η_{j+1}(x) = X^η_{j}(x) + ηa\big( X^η_{j}(x)\big) + ησ\big( X^η_{j}(x)\big)Z_{j+1}$及其截断变体$X^{η|b}_{j+1}(x) = X^{η|b}_{j}( x) + φ_b\big(ηa\big( X^{η|b}_{j}( x)\big) + ησ\big( X^{η|b}_{j}( x)\big) Z_{j+1}\big)$，其中$φ_b(x) = (x/|x|)\min\{|x|, b\}$，且噪声$Z_t$的分布是多变量正则变化的。截断运算符$φ_b(\cdot)$常用于厚尾系统中，例如深度学习中的随机梯度下降算法。通过发展一个连接大偏差与元稳定性的大框架，我们利用Wang和Rhee（2024）中对两个过程的局部均匀路径大偏差来获得首次退出时间和退出位置的联合分布的精确刻画。所得极限定理揭示了当截断阈值$b$变化时，出现离散的相变层次（即退出时间），并表现了灾难原理，即厚尾系统中的关键事件或元稳定性行为由少数组件的灾难性行为驱动，而其余系统则正常运作。这些发展导致了经典Freidlin-Wentzell理论的厚尾对应物。
英文摘要：
In this paper, we study the first-exit time of stochastic difference equation $X^η_{j+1}(x) = X^η_{j}(x) + ηa\big( X^η_{j}(x)\big) + ησ\big( X^η_{j}(x)\big)Z_{j+1}$ and its truncated variant $X^{η|b}_{j+1}(x) = X^{η|b}_{j}( x) + φ_b\big(ηa\big( X^{η|b}_{j}( x)\big) + ησ\big( X^{η|b}_{j}( x)\big) Z_{j+1}\big)$, where $φ_b(x) = (x/|x|)\min\{|x|, b\}$ and the law of the noise $Z_t$ is multivariate regularly varying. The truncation operator $φ_b(\cdot)$ is often introduced as a modulation mechanism in heavy-tailed systems, such as stochastic gradient descent algorithms in deep learning. By developing a framework that connects large deviations with metastability, we leverage the locally uniform sample-path large deviations for both processes in Wang and Rhee (2024) to obtain precise characterizations of the joint distributions of the first exit times and exit locations. The resulting limit theorem unveils a discrete hierarchy of phase transitions (i.e., exit times) as the truncation threshold $b$ varies, and manifests the catastrophe principle, whereby key events or metastable behaviors in heavy-tailed systems are driven by catastrophic behavior in a few components while the rest of the system behaves nominally. These developments lead to a comprehensive heavy-tailed counterpart of the classical Freidlin-Wentzell theory.
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