In this talk we will study asymptotics of the persistence probability that a Gaussian Stationary Process(GSP) stays positive for a long time interval. If the GSP has non negative correlation function, it follows immediately from Slepian's lemma that there is a persistence exponent. However, the same question has remained entirely open for general GSPs, for almost half a century. In this talk we will discuss this question from a spectral perspective, by studying the underlying spectral measure whose Fourier transform is the correlation function. Developing tools along the way, we will establish the existence of the persistence exponent under mild natural conditions on the spectral measure. We will also demonstrate continuity of the persistence exponent under convergence of spectral measures, in an appropriate metric.

This talk is based on joint work with Naomi D. Feldheim and Ohad N. Feldheim.

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