Real stochastic processes are random real-valued functions on an underlying space (in this talk, Z^d or R^d). Gaussianity occurs when a process is obtained as a sum of many infinitesimal independent contributions, and stationarity occurs when the phenomenon in question is invariant under translations in time or in space. This makes stationary Gaussian processes (SGPs) an excellent model for noise and random signals, placing them amongst the most well studied stochastic processes.
Persistence of a stochastic process is the event of remaining above a fixed level on a large ball of radius T. For a Stationary Gaussian process, we ask two basic questions:
1. What is the asymptotic behavior of the persistence probability, as T grows?
2. Conditioned on the persistence event, what is the typical shape of the process (if there is one)?
These questions, posed by physicists and applied mathematicians decades ago, have been successfully addressed only in the last few years, by exploiting strong relations with harmonic analysis.
In this talk, we will describe old and new results, the main tools and ideas used to achieve them, and many open questions that remain.
