, Universidad Adolfo Ib\u00e1\u00f1ez<\/em><\/strong><\/h5>\nA general framework for the analysis of kernel-based tests<\/strong><\/p>\nKernel-based tests provide a simple yet effective framework that utilizes the theory of reproducing kernel Hilbert spaces to design non-parametric testing procedures. In this presentation, we introduce new theoretical tools for studying the asymptotic behavior of kernel-based tests across various data scenarios and testing problems. Unlike current approaches, our methods circumvent the use of U and V-statistics expansions, which often result in lengthy and tedious computations and asymptotic approximations, especially in complex scenarios. I<\/span>nstead, we work directly with random functionals on the Hilbert space to analyse kernel-based tests. By harnessing the use of random functionals, our framework leads to a much cleaner analysis, involving less tedious computations. Additionally, it offers the advantage of accommodating pre-existing knowledge regarding test-statistics as many of the random functionals considered in applications are known statistics that have been studied thoroughly.<\/p>\nPablo L\u00f3pez-Rivera, Universit\u00e9 Paris Cit\u00e9
\n<\/em><\/strong><\/h5>\nPreservation of functional inequalities under log-Lipschitz perturbations<\/strong><\/p>\nGiven a probability measure satisfying some functional inequalities (Poincar\u00e9, log-Sobolev, etc.), it is natural to wonder if these remain valid for a perturbation of the measure. In particular, if there exists a globally Lipschitz map pushing forward the source measure towards its perturbation, then it is easy to transport certain functional inequalities. For example, Caffarelli\u2019s contraction theorem states that the optimal transport map between the Gaussian measure and a log-concave perturbation is 1-Lipschitz.<\/p>\n
In this talk I will show how such a map exists if we consider log-Lipschitz perturbations of a measure on a Riemannian manifold, via the interpolation given by the Langevin diffusion associated to the source measure (aka Kim-Milman\u2019s heat flow transport map), assuming as well control on the curvature of the manifold at first and second order in the sense of Bakry-\u00c9mery-Ricci. No prerequisites in geometry are needed.<\/p>\n
<\/p>\n
Gonzalo Panizo, IMCA<\/em>
\n<\/strong><\/h5>\nRandom Interlacements of directed paths<\/strong><\/p>\nWe present the model of Random Interlacements and a variation of it where the simple symmetric random walk is replaced by a directed random walk. We talk about the existence and percolation problems of this model and its relationship with the so-called escape problems.<\/p>\n
Jaime San Mart\u00edn, Universidad de Chile<\/em><\/strong><\/h5>\nSome New and Classical results for Schr\u00f6dinger operators in dimension 1<\/strong><\/p>\nWe will see classical and new results for Schr\u00f6dinger operators in dimension 1, in the context of Branching Diffusions. We will give asymptotic for the mean number of\u00a0 particles surviving to time t and its limiting conditional Laplace transform.<\/p>\n
<\/p>\n