Massive excursions of Gaussian isotropic fields. Method of moments

The asymptotic behavior of probability of large massive excursions above a high level is evaluated for Gaussian isotropic twice differentiable Gaussian fields. A massive excursion is the excursion with base diameter exceeding a fixed positive number. For proofs, we introduce and study a vector Gaussian process with components that are independent copies of the initial field, and consider the point process of exits of trajectories of this process from an appropriate infinitely expanding set. By studying the asymptotic behavior of the first and second moments of distribution of this point process, the desired asymptotic behavior is obtained. Moreover, general results on the logarithmic (rough) asymptotic behavior of the large massive excursion probabilities are put forward.