The averaging principle of multiscale fractional stochastic nonautonomous FitzHugh–Nagumo systems on RN

This paper is concerned with the averaging principle for the multi-scale fractional stochastic nonautonomous FitzHugh–Nagumo system on the whole Euclid space, where the drift term is a nonlinear function that has an arbitrary polynomial growth rate in its last argument, and the diffusion term is a family of globally Lipschitz continuous functions. Taking the Hilbert space of square-integrable functions as a state space, we first prove the existence and uniqueness of a periodic evolution system of measures for the fast equation when the slow component of the solution is fixed, and then establish the exponential ergodicity for the stochastic systems under certain conditions. By using the time discretization and variational methods, we finally show the strong rate of convergence for the slow component of the solution operator. The results of this paper are also valid when the fractional Laplace operator (−Δ)s with s ∈ (0, 1) becomes the standard Laplace operator −Δ.