Mario Wschebor (1939-2011) was born in Uruguay and there, also in Venezuela, he left an important trace in the development of mathematics. His main area of research was the theory of stochastic processes. Being the theory of Gaussian processes and fields a subject in which he obtained important achievements. Particular mention deserves his study of the level sets of such processes. It is important to highlight the various generalizations he made of the Kac-Rice formula. Such a formula allows to obtain, among other things, the moments of the measure of the level set. In the last period of his career, he addressed problems related to the determination of the moments of the number of roots and the moments of the measure of nodal sets for random polynomial systems. Also, he tackled the study of delicate problems in algorithmic complexity. His work opened important research ways that we will present in this session.
Mean number and correlation function of critical points of isotropic Gaussian fields
Céline Delmas, INRA, Francia
Let X = fX(t) : t 2 RNg be an isotropic Gaussian random field with real values. In a first part we study the mean number of critical points of X with index k, above a level, using random matrices tools. We obtain an exact expression for the probability density of the eigenvalue of rank k of a N-GOE matrix. We deduce exact expressions for the mean number of critical points with a given index and their distribution as a function of their index. In a second part we study attraction or repulsion between these critical points again as a function of their index. A measure is the correlation function.
We prove attraction between critical points when N > 2, neutrality for N = 2 and repulsion for N = 1. We prove that the attraction between critical points that occurs when the dimension is greater than 2 is due to attraction between critical points with adjacent indexes. We prove a strong repulsion between maxima and minima. We study the correlation function between maxima (or minima).
Keywords. Critical points, Gaussian fields, GOE matrices, Kac-Rice formula, Point processes, Random matrices.
Rice formula: finiteness of moments and jumps
Federico Dalmao, UDELAR, Uruguay
In this talk, we consider the problem of assessing the finiteness of the moments of the number of crossings of a stochastic process, or the volume of the level set of a random field, and its interplay with Rice formulas.
Finally, we consider Rice formulas for processes including jumps in their paths. These results are part of a working paper by J.M. Azaïs and C. Delmas.
Testing Gaussian Process with applications to super-resolution
Yohann De Castro, Université d'Orsay, France
This talk introduces exact testing procedures on the mean of a Gaussian Process X derived from the outcomes of `1-minimization over the space of complex valued measures. The process X can be thought as the sum of two terms: first, the convolution
between some kernel and a target atomic measure (mean of the process); second, a random perturbation by an additive centered Gaussian process. The first testing procedure considered is based on a dense sequence of grids on the index set of X and we establish that it converges (as the grid step tends to zero) to a randomized testing procedure: the decision of the test depends on the observation X and also on an independent random variable. The second testing procedure is based on the maxima and the Hessian of X in a grid-less manner. We show that both testing procedures can be performed when the variance is unknown (and the correlation function of X is known). These testing procedures can be used for the problem of deconvolution over the space of complex valued measures, and applications in frame of the Super Resolution theory are presented. As a byproduct, numerical investigations may demonstrate that our grid-less method is more powerful (it detects sparse alternatives) than tests based on very thin grids.
Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets
Jean-Marc Azaïs, Université de Toulouse
For a smooth vectorial stationary Gaussian random field $X:\Omega\times\R^d\to\R^d$, we give necessary and sufficient conditions to have a finite second momento for the number of roots of $X(t)-u$. The results are obtained by using a method of proof inspired on the one obtained by D. Geman for stationary Gaussian processes long time ago. Afterwards the same method is applied to the number of critical points of a scalar random field and also to the level set of a vectorial process $X:\Omega\times\R^D\to\R^d$ with $D>d$.