Naomi Feldheim

Persistence of Gaussian stationary processes

Let f:R->R be a Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution.
What is the probability that f remains above a certain fixed line for a long period of time?
This simple question, which was posed by mathematicians and engineers more than 60 years ago (e.g. Rice, Slepian), has some surprising answers which were discovered only recently. I will describe how a spectral point of view leads to those results.
Based on joint works with O. Feldheim, F. Nazarov, S. Nitzan, B. Jaye and S. Mukherjee.