# Reduced {\alpha}-stable dynamics for multiple time scale systems forced with correlated additive and multiplicative Gaussian white noise

## TL;DR

This paper investigates how a slow process driven by correlated additive and multiplicative noise can be approximated as being influenced by an alpha-stable noise, extending stochastic averaging to infinite-variance forcing scenarios.

## Contribution

It introduces a method to approximate a slow process driven by CAM noise with an alpha-stable process, enabling stochastic averaging in infinite-variance contexts.

## Key findings

- Slow process behaves as driven by alpha-stable noise under large time-scale separation.
- Conditions identified for CAM process to influence the slow process.
- Numerical results demonstrate the approximation's validity.

## Abstract

Stochastic averaging problems with Gaussian forcing have been studied thoroughly for many years, but far less attention has been paid to problems where the stochastic forcing has infinite variance, such as an {\alpha}-stable noise forcing. It has been shown that simple linear processes driven by correlated additive and multiplicative (CAM) Gaussian noise, which emerge in the context of atmosphere and ocean dynamics, have infinite variance in certain parameter regimes. In this paper, we study a stochastic averaging problem where a linear CAM noise process in a particular parameter regime is used to drive a comparatively slow process. It is shown that the slow process exhibits properties consistent with being forced by a white {\alpha}-stable noise 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. These results are illustrated using a set of representative numerical results.