Random dynamics of fractional stochastic retarded FitzHugh–Nagumo systems on unbounded domains

Abstract In this paper, we study the dynamics of fractional nonautonomous stochastic FitzHugh–Nagumo systems with variable delay defined on $\mathbb{R}^{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . We first consider the existence and uniqueness of pullback random attractors as well as the time-dependent properties of pullback random attractors. Then we study the upper semicontinuity of pullback random attractors as the delay time tends to zero. In order to achieve the goals of this article, we need to overcome two difficulties: (i) The noncompactness of Sobolev embedding on unbounded domains; (ii) The effect of the variable delay term. However, we can find the effective methods to resolve the two difficulties. More precisely, we use the method of backward uniform tail-estimates of solutions to resolve the first difficulty, and we introduce an inverse function to solve the second difficulty.