A Gaussian stationary function is a shift-invariant 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^L(T),  
is the probability that f remains above a fixed level L on an interval of length T.
The ball probability, denoted by B^L(T), is the probability that f remains within [-L,L] on an interval of length T.
Both were extensively studied by Slepian, Rice, Rosenblatt, Majumdar, Dembo and others for the last 60 years.

Using classical Gaussian tools, it is easy to show that the ball probability has asymptotic exponential behavior;
However, the persistence probability may behave quite differently.
In recent years, a spectral point of view allowed us to give conditions under which log P^L(T)/T is bounded between two constants.
Until recently, the convergence of log P^L(T)/T to a persistence exponent remained open.

In this talk we will present a nearly exact condition for the existence of the persistence exponent.
We show that such an exponent exists if the spectral measure of the process has density at the origin.
Our methods involve establishing several continuity properties of persistence and ball probabilities,
borrowing ideas from harmonic and convex analysis.

Joint work with Ohad Feldheim and Sumit Mukherjee (https://arxiv.org/abs/2112.04820).