Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {Zs ≥ ζ (s),s ∈ S}, where Z is a continuous Gaussian process with stationary increments, ζ is a function that belongs to the reproducing kernel Hilbert space R of process Z, and S ⊂ R is compact. The main problem considered in this paper is identifying the function β∗ ∈ R satisfying β
∗(s) ≥ ζ (s) on S and having minimal R-norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ (s) = s for s ∈ [0, 1] and Z is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.