Mathematical Sciences: Probability Theory with Minimum Structures

本论文的主题是“最小结构的概率”，特别是在一般条件下随机过程的上、下界的研究。 连接边线是应用到Banach空间，组合学，概率分析的困难问题，随机傅立叶级数，当然更经典的概率问题，如轨迹的随机过程和高斯过程。 核心理念是优先考虑那些需要很少结构的工具，而不是更具体的论点。 积空间中的等周思想，对概率独立性的概念有了新的理解，将继续成为一个核心工具。 直到最近才认识到这些想法的真正范围，不应低估未来进展和应用的潜力。 本研究的主旨是追求抽象的概率工具的发展，特别是与经典的等周思想的关系。研究者意识到，等周法的经典思想在比以前认为的更一般的环境中是有意义的。这引起了新的家庭的不平等，解决了许多问题，在抽象的概率，并产生影响的领域，概率是一种工具，而不是一个对象的研究本身，特别是操作研究和理论计算机科学。 这些想法的潜力，无论是在理论层面和应用层面开始才被理解，似乎非常有前途。

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.