  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.

Probability;Dynamical Systems