Large Deviations and Metastability Analysis for Heavy-Tailed Dynamical Systems
| Autor: | Wang, Xingyu, Rhee, Chang-Han |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | This paper introduces a novel framework that connects large deviations and metastability analysis in heavy-tailed stochastic dynamical systems. Employing this framework in the context of stochastic difference equations $X^\eta_{j+1}(x) = X^\eta_{j}(x) + \eta a\big( X^\eta_{j}(x)\big) + \eta \sigma\big( X^\eta_{j}(x)\big)Z_{j+1}$ and its variations with truncated dynamics, we first establish locally uniform sample path large deviations and then translate such asymptotics into a precise characterization of the joint distribution of the first exit time and exit location. As a result, we obtain the heavy-tailed counterparts of the classical Freidlin-Wentzell and Eyring-Kramers theorems. Our large deviations asymptotics are sharp enough to identify how rare events arise in heavy-tailed dynamical systems and characterize the catastrophe principle. Moreover, it also unveils the discrete hierarchy of phase transitions in the asymptotics of the first exit times and locations under truncated heavy-tailed noises. Our results in this paper open up the possibility of systematic analysis of the global dynamics of heavy-tailed stochastic processes. In the appendix, we also present the corresponding results for the L\'evy driven SDEs. Comment: 80 pages, 0 figure |
| Databáze: | arXiv |
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