Extremes and crossings for differentiable stationary processes with application to Gaussian processes in Rm and Hilbert space

AbstractLet {ω(t)}t⩾0 be a stochastically differentiable stationary process in Rm and let Au⊆Rm satisfy limu↑u2P{ω(0) ∈ Au} = 0. We give a method to find the asymptotic behaviour of P{⋃0⩽t⩽h{ω(t) ∈ Au}} as u ↑u2. We use our method to study hitting probabilities for small sets with application to Gaussian processes and to study suprema of processes in R with application to (the norm of) Gaussian processes in Hilbert space