We study the asymptotic behavior of solutions to a class of weakly dissipative fractional hyperbolic equations equations on \({\mathbb {R}}^N\) driven by colored and white noise. The equation has two significant features: the weak dissipativeness due to the fractional dispersive term, and the superlinear growth of the noise. Under certain conditions, the existence, uniqueness and temperedness of pullback random attractors are proved in \(H^s({\mathbb {R}}^N)\times H^s({\mathbb {R}}^N)\) for all \(s\in (0,1)\) and \(N\in {\mathbb {N}}\) when the nonlinear draft term satisfies a fractional subcritical (N, s)-growth condition. In addition, the upper semi-continuity(almost surely and in probability) of the random attractors of the equation with additive colored noise is established as the correlation time of the colored noise shrinks to zero. The idea of uniform tail-estimates and the method of spectral decomposition are combined to prove the pullback asymptotic compactness of the solutions in order to overcome the lack of compact Sobolev embeddings on \({\mathbb {R}}^N\) and the weakly dissipative structure of the equation. The results of the paper are even new when the random equation reduces to a deterministic one, and our methods are applicable when the fractional Laplace operators are replaced by the standard one.
