Title: Persistence and entropic repulsion for stationary Gaussian processes
Abstract:
A real stationary Gaussian process (SGP) is a shift-invariant distribution over continuous functions f on R^d, whose finite marginals are multi-normal. Such a function is characterized by its spectral measure, that is, a probability measure on R^d whose Fourier transform yields the covariance kernel r(t) = cov(f(0),f(t)).
Persistence of a stochastic process is the event of remaining above a fixed level on a large ball of radius T (a "hard wall" event). For a SGP, we ask two basic questions:
- What is the asymptotic behavior of the persistence probability, as T grows?
- 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 tools from real, complex and harmonic analysis.
After a survey of some past results, we shall focus on the regime for which entropic repulsion occurs. This is the phenomenon that, conditioned on a hard wall, the process is "pushed away" from the wall by a macroscopic amount, and fluctuates around some deterministic shape. We show that this phenomenon occurs universally for the class of SGPs with spectral blow-up at the origin.
Based on joint work with Ohad Feldheim and Stephen Muirhead.