Mathematical Sciences: Probability Theory with Minimum Structures

这项工作的主题是“最小结构的概率”，特别是研究一般条件下随机过程的上界和下界。连接的边线应用于巴拿赫空间，组合学，难题的概率分析，随机傅立叶级数当然还有更经典的概率问题，比如随机过程的轨迹和高斯过程。其核心理念是优先考虑需要很少结构的工具，而不是更具体的参数。产品空间中的等距概念，给了概率独立性的概念一个新的理解，将继续是一个中心工具。这些想法的真正范围直到最近才被认识到，未来的进展和应用的潜力不应被低估。这项研究的主要目的是发展抽象的概率工具，特别是与经典的等径测量思想的关系。研究者意识到，等渗法的经典概念可以在比以前认为的更普遍的情况下变得有意义。这产生了新的不等式族，解决了抽象概率中的许多问题，并对概率作为工具而不是研究对象的领域产生了影响，特别是运筹学和理论计算机科学。这些想法的潜力，无论是在理论层面还是在应用层面，都刚刚开始被理解，并且看起来非常有希望。

The main theme of this work is "probabilities with minimal structures", and in particular the research of upper and lower bounds for stochastic processes under general conditions. Connected sidelines are applications to Banach spaces, to combinatorics, to probabilistic analysis of hard problems, to random Fourier series and of course to more classical probabilistic questions such as trajectories of stochastic processes and gaussian processes. The central philosophy is to give priority to tools that require very little structure over more specific arguments. The ideas of isoperimetry in product spaces, that have given a new understanding of the notion of probabilistic independence will continue to be a central tool. The true scope of these ideas has been realized only recently, and the potential for future progress and applications should not be underestimated. The main thrust of this research is to pursue the development of abstract tools of probability in particular in relationship with the classical idea of isoperimetry. It was realised by the investigator that the classical idea of isoperimetry can be made meaningful in a much more general setting than was previously believed. This gave rise to new families of inequalities that have solved numerous problems in abstract probability, and had impact in areas where probability is a tool rather than an object of study in itself, and in particular operation research and theoretical computer science. The potential of these idea, both at the theoretical level and the level of applications starts only to be understood, and appears very promising.