On the regularity of the distribution of the maximum of one-parameter Gaussian processes

The main result in this paper states that if a one-parameter Gaussian process has C2k paths and satisfies a non-degeneracy condition, then the distribution of its maximum on a compact interval is of class Ck. The methods leading to this theorem permit also to give bounds on the successive derivatives of the distribution of the maximum and to study their asymptotic behaviour as the level tends to infinity.