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Reduced $$\alpha$$-stable dynamics for multiple time scale systems forced with correlated additive and multiplicative Gaussian white noise. (English) Zbl 1390.34185
Summary: Stochastic averaging problems with Gaussian forcing have been the subject of numerous studies, but far less attention has been paid to problems with infinite-variance stochastic forcing, such as an $$\alpha$$-stable noise process. It has been shown that simple linear systems driven by correlated additive and multiplicative (CAM) Gaussian noise, which emerge in the context of reduced atmosphere and ocean dynamics, have infinite variance in certain parameter regimes. In this study, we consider the stochastic averaging of systems where a linear CAM noise process in the infinite variance parameter regime drives a comparatively slow process. We use (semi)-analytical approximations combined with numerical illustrations to compare the averaged process to one that is forced by a white $$\alpha$$-stable process, demonstrating consistent properties in the case of large time-scale separation. We identify the conditions required for the fast linear CAM process to have such an influence in driving a slower process and then derive an (effectively) equivalent fast, infinite-variance process for which an existing stochastic averaging approximation is readily applied. The results are illustrated using numerical simulations of a set of example systems.