Stochastic Averaging of Dynamical Systems with Multiple Time Scales Forced with $\alpha$-Stable Noise

Abstract

Stochastic averaging allows for the reduction of the dimension and complexity of stochastic dynamical systems with multiple time scales, replacing fast variables with statistically equivalent stochastic processes in order to analyze variables evolving on the slow time scale. These procedures have been studied extensively for systems driven by Gaussian noise, but very little attention has been given to the case of $\alpha$-stable noise forcing which arises naturally from heavy-tailed stochastic perturbations. In this paper, we study nonlinear fast-slow stochastic dynamical systems in which the fast variables are driven by additive $\alpha$-stable noise perturbations and the slow variables depend linearly on the fast variables. Using a combination of perturbation methods and Fourier analysis, we derive stochastic averaging approximations for the statistical contributions of the fast variables to the slow variables. In the case that the diffusion term of the reduced model depends on the state of the slow variable, we show that this term is interpreted in terms of the Marcus calculus. For the case $\alpha = 2$, which corresponds to Gaussian noise, the results are consistent with previous results for stochastic averaging in the Gaussian case. Although the main results are derived analytically for $1 < \alpha < 2$, we provide evidence of their validity for $\alpha < 1$ with numerical examples. We numerically simulate both linear and nonlinear systems and the corresponding reduced models, demonstrating good agreement for their stationary distributions and temporal dependence properties.