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 proce ss has C-2k paths and satisfies a non-degeneracy condition, then the distri bution of its maximum on a compact interval is of class C-k. The methods le ading to this theorem permit also to give bounds on the successive derivati ves of the distribution of the maximum and to study their asymptotic behavi our as the level tends to infinity.