Limit dynamics for the stochastic FitzHugh–Nagumo system

The asymptotic behavior of the stochastic FitzHugh–Nagumo system with small excitability is concerned. It is proved that solutions of the stochastic FitzHugh–Nagumo system converge in probability to the unique solution of the limit system as the excitability tends to zero. In our approach the proof of tightness of the distributions of solutions in some appropriate functional space is a key step. Furthermore, we establish the existence of a global random attractor for the stochastic FitzHugh–Nagumo system, then construct a local random attractor for the limit system and prove the upper semicontinuity between global random attractors for the original system and the local random attractor for the limit system as the excitability goes to zero. As the semigroup is not compact, a novel part is to introduce the DD–αα-contracting to prove the existence of global random attractor for stochastic FitzHugh–Nagumo system.