On the probability that a stationary Gaussian process with spectral gap remains non-negative on a long interval

Abstract

Let $f$ be a zero-mean continuous stationary Gaussian process on $\mathbb{R}$ whose spectral measure vanishes in a $\delta$-neighborhood of the origin. Then the probability that $f$ stays non-negative on an interval of length $L$ is at most $e^{-c{\delta }^2{L}^2}$ with some absolute $c>0$ and the result is sharp without additional assumptions.

15 pages. To appear in IMRN (Inter. Math. Res. Notices)