Physical extremes of Gaussian processes

Theory of extremes of Gaussian processes is well developed for evaluation of asymptotic behavior of probability of high extremes of theirs trajectories, which is the probability of existence of an excursion of a trajectory above a high level. Given such an excursion, it is very short; the trajectory tends to come quickly back under the level. Since a real process in reliability or safety is modeled, the probability of long or “physical” excursions above a high level is of interest. The problem of estimation of such the probabilities could not be solved during long time. We will discuss how to evaluate the exact asymptotic behavior of the probability of high excursion for Gaussian stationary processes, how long is the typical excursion, how to evaluate the logarithmic asymptotic behavior of the probability of high excursion for Gaussian general processes, and, finally, how, in many interesting cases, one can evaluate the exact asymptotic behavior of the probability of physical excursions above a high level.