Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R

In this paper, we consider isotropic and stationary real Gaussian random fields defined on S2 × R and we investigate the asymptotic behavior, as T->+∞, of the empirical measure (excursion area) in S2 × [0,T] at any threshold, covering both cases when the field exhibits short and long memory, that is, integrable and non-integrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry's cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws.