Regularity of the distribution of the maximum of a Gaussian process with regular paths

Let X be a real-valued Gaussian process with a one-dimensional parameter If the sample paths of X are of class C-2k, k integer and if the marginal dis tributions of X and its derivatives have no degeneracy, then the distributi on function of the maximum on a bounded interval is of class C-k. We give a recurrence method allowing to express the derivatives by means of certain implicit formulae. (C) Academie des Sciences/Elsevier, Paris.