Multi-scale analysis of a gaussian Markovian process near a singular point.

Let X be some gaussian process. We suppose X to be markovian of order one, which means the topology of its Reproducing Kernel Hilbert Space H-X is given by a symmetric Dirichlet form A of order two. A point y is said to be singular when the leading coefficient of the form A vanishes or becomes infinite. In this paper, we analyze the regularity of the trajectories of such a process X near any isolated singular point y. In order to do this, X is decomposed on a wavelet basis of H-X. The behavior of X is given after a precise study of the wavelets near y. Using the same ideas the form A can also be identified in the vicinity of a singular point.