Limit theorems for geometrical characteristics of Gaussian excursion sets

Excursion sets of stationary random fields have attracted much attention in recent years. They have been applied to modeling complex geometrical structures in tomography, astrophysics and hydrodynamics. Given a random field (observed in some bounded window) and a specified level, it is natural to consider geometrical properties of the generated excursion set. Main examples of functionals studied include the volume, the surface area and the Euler characteristics. Starting from the classical Rice formula (1945), many results concerning calculation of moments of these geometrical functionals have been proven. There are much less results concerning the asymptotic behavior (since the observation window size grows to infinity in some sense), as random variables considered here depend non-smoothly on the realizations of the random field. Important results here are due to H.Cram${\rm\acute{e}}$r, Yu.K.Belyaev, V.I.Piterbarg, T.Slud, M.Kratz and other scientists. In the talk we discuss several recent achievements in this domain, concentrating on asymptotic normality and functional central limit theorems.