Rare events for stationary Gaussian functions

A Gaussian stationary process is a random function f:R-->R whose distribution is invariant under real shifts, and whose evaluation at any finite number of points is a centered Gaussian random vector. The mathematical study of these random functions goes back at least 80 years, with pioneering works by Kac, Rice and Wiener. Nonetheless, many basic questions about them turned out to have complicated answers, or remained open for many years. One prominent example is estimating the probability that the process does not cross a certain level (one-sided barrier) or a certain pair of levels (two-sided barrier) during a long period of time. A more advanced question (most interesting to physicists) is: when such a rare event occurs, how does it occur?

In this talk we will provide a quick introduction to Gaussian stationary processes, and describe how a spectral perspective, combined with tools from probability and  harmonic analysis, sheds new light on these long-lasting questions.