Minimality and convexity of pullback measure attractors for fractional FitzHugh–Nagumo equations with superlinear noise

For a measure process, we introduce the new concept of a pullback measure attractor, which is the minimal compact invariant set and pullback attracts all norm-bounded sets of probability measures. We then establish two abstract existence theorems for pullback measure attractors. Applying to the fractional stochastic FitzHugh–Nagumo equations on an unbounded domain driven by superlinear space–time noise, we prove that the dual measure process has a minimal pullback measure attractor, which is further proved to be backward norm-bounded and convex. New methods of two-stages and direct-sum decomposition play key roles in the proofs of the continuity, forward invariance, past absorption, and pullback asymptotic tightness for the dual process.