Persistence of Gaussian stationary processes

Consider a real Gaussian stationary process, either on $$\mathbb{Z}$$ or on $$\mathbb{R}$$. That is, a stochastic process, invariant under translations, whose finite marginals are centred multi-variate Gaussians. The persistence of such a process on $$[0,T]$$ is the probability that it remains positive throughout this interval.

The relation between the decay of the persistence as T tends to infinity and the covariance function of the process has been investigated since the 1950s with motivations stemming from probability, engineering and mathematical physics. Nonetheless, until recently, good estimates were known only for particular cases, or when the covariance kernel of the process is either non-negative or summable.

In the talk we discuss a new spectral point of view on persistence which greatly simplifies its analysis. This is then used to obtain better bounds in a very general setting.

Joint work with Naomi Feldheim and Shahaf Nitzan.