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光滑泛函在越来越多的变量中的整体行为有关的几何和解析特征。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。乘积度量、凸体上的均匀分布或对数凹的度量都是一些具有挑战性的问题的很好的例子。将维度作为分布的主要参数的作用放在研究的中心位置。随机过程的许多重要性质假设了样本轨迹的可能行为，并经常涉及各种泛函的分布。要获得有关这一过程的基本信息，就需要研究多维分布，并产生从数学科学自然发展的角度来看本身也很有趣的深层数学问题。本研究从分析、几何和统计学的角度研究了随机过程的全局性质以及它们与不同对象的关系。

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.