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

多维概率分布的基本性质通常包括与越来越多的变量中的光滑泛函的全局行为相关的几何和分析特征。本文主要研究高维空间中不同类型概率测度的各种集中现象。乘积测度类、凸体上的均匀分布类或几何凹测度类是具有许多挑战性问题的很好例子。作为分布的主要参数的维数的作用的研究被放置在研究的中心。随机过程的许多重要性质假设样本轨迹的可能行为，并且通常涉及各种泛函的分布。获得基本信息的过程需要研究多维分布，并导致深层次的数学问题，这也是有趣的本身从角度来看，自然发展的数学科学。本研究的重点是研究随机过程的全局性质，以及它们如何与分析，几何和统计等不同对象相关联。

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.