Approximation du temps local des surfaces gaussiennes
Résumé
Let X be a centered stationary Gaussian stochastic process with a d-dimensional parameter ($d \ge 2$), F its spectral measure, $$\int\limits_{R^d } {||x||^2 F(dx) = + \infty } $$ (VerbarxVerbar denotes the Euclidean norm of x). We consider regularizations of the trajectories of X by means of convolutions of the form X_{\varepsilon}(t)=(\Psi_ {varepsilon}*X)(t) where $\Psi_{\varepsilon}$ stands for an approximation of unity (as $\varepsilon$ tends to zero) satisfying certain regularity conditions. The aim of this paper is to recover the local time of X at a given level u, as a limit of appropriate normalizations of the geometric measure of the u-level set of the regular approximating processes $X_{\varepsilon}$. A part of the difficulties comes from the fact that the geometric behavior of the covariance of the Gaussian process $X_{\varepsilon}$ can be a complex one as $\varepsilon$ approaches O. The results are on $L^{2}$-convergence and include bounds for the speed of convergence. $L^{p}$ results may be obtained in similar ways, but almost sure convergence or simultaneous convergence for the various values of u do not seem to follow from our methods. In Sect. 3 we have included examples showing a diversity of geometric behaviors, especially in what concerns the dependence on the thickness of the set in which the covariance of the original process X is irregular. Some technical results of analytic nature are included as appendices in Sect. 4.