Large Deviations and Metastability Analysis for Heavy-Tailed Dynamical Systems

Abstract

This paper introduces novel frameworks for large deviations and metastability analysis in heavy-tailed stochastic dynamical systems. We develop and apply these frameworks within the context of stochastic difference equation $X^\eta_{j+1}(x) = X^\eta_{j}(x) + \eta a\big( X^\eta_{j}(x)\big) + \eta \sigma\big( X^\eta_{j}(x)\big)Z_{j+1}$ and its variation with truncated dynamics $X^{\eta|b}_{j+1}(x) = X^{\eta|b}_{j}( x) + \varphi_b\big(\eta a\big( X^{\eta|b}_{j}( x)\big) + \eta \sigma\big( X^{\eta|b}_{j}( x)\big) Z_{j+1}\big)$, where $\varphi_b(x) = (x/\|x\|)\max\{\|x\|, b\}$. The truncation operator $\varphi_b(\cdot)$ is often introduced as a modulation mechanism in heavy-tailed systems, such as stochastic gradient descent algorithms in deep learning. Thus, it is crucial to successfully analyze both $X^{\eta}_{j}(x)$ and $X^{\eta|b}_{j}(x)$. We establish locally uniform sample-path large deviations for both processes and translate these asymptotics into precise characterizations of the joint distributions of the first exit times and exit locations. Our large deviations asymptotics are sharp enough to rigorously characterize \emph{the catastrophe principle} by establishing the distributional limit of the sample paths conditional on the rare events of interest, thereby revealing the most likely paths through which rare events arise in heavy-tailed dynamical systems. Moreover the resulting limit theorem unveils a discrete hierarchy of phase transitions (i.e., exit times) as the truncation threshold $b$ varies. Together, these developments serve as a heavy-tailed counterpart of the classical Freidlin-Wentzell theory. We also present the corresponding results for continuous-time processes in the appendix.