Isoperimetry, Concentration of Measure and Related Sobolev-Type Inequalities in High Dimensional Probability Theory

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Essential properties of multidimensional probability distributions often include geometric and analytic characteristics related to global behavior of smooth functionals in a growing number of variables. This research focuses on various concentration phenomena for different classes of probability measures in spaces of high dimension. The classes of product measures, uniform distributions over convex bodies, or logarithmically concave measures are good examples with a number of challenging problems. The study of the role of the dimension as the main parameter of a distribution is placed in the center of the research. Many important properties of stochastic processes postulate possible behavior of sample trajectories and often refer to distributions of various functionals. Obtaining essential information on the process requires the study of multidimensional distributions and leads to deep mathematical problems which are also interesting in themselves from the point view of the natural development of mathematical sciences. This research focuses on the study of global properties of stochastic processes and on how they relate to different objects from analysis, geometry and statistics.