Stochastic averaging of fast-slow systems forced with alpha-stable noise

Abstract:

Climate dynamics problems are often characterized by their complexity and the existence of dynamics on many different time scales, ranging from days to centuries or longer. Because of their complexity, these problems frequently require approximations to make them both analytically and computationally tractable. Dynamical models with stochasticity and multiple time scales can be studied using stochastic averaging methods to accurately approximate the slow time scale variables of the system in the weak (statistical) sense and to reduce the dimension of the problem. Most stochastic averaging studies have focused on the case where the stochastic forcing terms are Gaussian (due to the Central Limit Theorem). However, several atmospheric dynamics studies indicate that heavy-tailed noise is more appropriate for modelling purposes in some cases. In these situations, established stochastic averaging methods cannot be used.

We present stochastic averaging approximations that can be used in situations where the stochastic forcing is heavy-tailed, specifically alpha-stable, rather than Gaussian, emphasizing the key ideas in the derivation of the approximations, numerical results, and potential applications to climate research.