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A Gaussian stationary function is a stationary process f, indexed on either R or Z, whose finite marginals are mean zero multi-normals. The persistence probability of such a process, denoted by P^x(T), is the probability that f remains above a fixed level x on an interval of length T.ÌýThis quantity has been extensively studied by Slepian, Rice, Rosenblatt, Majumdar, Schehr, DemboÌýand others for the last 60 years.Ìý

When the correlations decay fast, it is natural to expect that f wouldÌýdemonstrate an i.i.d. like behavior so that P^x(T) would be roughly exponential.ÌýIn recent years, a spectral point of view on the problem allowed us to show thatÌýindeed, the value of log P^x(T)/T as T tends to infinity is bounded between two constants,Ìýfor a much broader family of processes, some of which have non-decaying correlations.ÌýHowever, in general, the convergence of log P^x(T)/T to a persistence exponent remained open.

In this talk, I will present our very recent work, in which we establishÌýa nearly exact condition for the existence of the persistence exponent. We show thatÌýsuch an exponent exists if the spectral measure of the process has a continuousÌýÌýnon-vanishing density at the origin. Our methods involve establishing several continuity propertiesÌýof persistence probabilities, borrowing ideas from harmonic and convex analysis.

Joint work with Ohad Feldheim and Sumit Mukherjee.