Gaussian stationary processes: a spectral perspective
A Gaussian stationary process is a random function f:R-->R or f:C-->C,
whose distribution is invariant under real shifts, and whose evaluation at
any finite number of points is a centered Gaussian random vector.
The mathematical study of these random functions goes back at least 75 years,
with pioneering works by Kac, Rice and Wiener.
Nonetheless, many basic questions about them, such as the fluctuations of their number of zeroes,
or the probability of having no zeroes in a large region, remained unanswered for many years.
In this talk, we will provide an introduction to Gaussian stationary process and
describe how a new spectral perspective, combined with tools from harmonic, real and
complex analysis, yields new results about such long-lasting questions.