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Loewner theory and non-commutative probability theory

Loewner理论和非交换概率论

基本信息

批准号：
401281084
负责人：
Dr. Sebastian Schleißinger
金额：
--
依托单位：
Lehrstuhl für Mathematik IV (Komplexe Analysis)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/401281084?language=en
关键词：
Loewner theory non commutative probability

项目摘要

This project focuses on exploring connections between geometric function theory, in particular Loewner theory, and non-commutative probability theory.In 1923, C. Loewner has introduced certain evolution equations for conformal mappings, which soon became an important tool in geometric function theory. They are used to model two-dimensional growth processes or to tackle extremal problems for univalent functions. Since the introduction of the Schramm-Loewner Evolution (SLE) by O. Schramm in 2000, Loewner theory has become an active research field with interdisciplinary topics from complex analysis, probability theory, statistical mechanics, and conformal field theory.Non-commutative probability theory provides frameworks for abstract probability spaces consisting of random variables which do not commute in general. This is motivated by quantum mechanics, where observables can be regarded as non-commutative random variables. Many goals in this field ask to transfer notions and theorems from classical probability theory to this non-commutative setting. For instance, there is a theory of quantum stochastic processes and quantum stochastic differential equations, which in turn is useful for providing mathematical models for certain quantum systems. The notion of independence plays a central role in classical probability theory and it has been shown that, in a certain sense, there are five ways of defining it in non-commutative probability theory. This leads to tensor, free, Boolean, monotone and anti-monotone probability theory.All five notions lead to certain convolutions of holomorphic mappings, and at this point, complex analysis enters the theory. So far, however, the methods used for this purpose are rather elementary. I noticed that there is a deeper relation between complex analysis and monotone probability theory: both, most studied Loewner equations (the ``radial'' and the "chordal" equation) can be regarded as the Lévy-Khintchine representation of quantum stochastic processes with monotonically independent increments. (And a time reversion of these equations corresponds to the anti-monotone analogues.)This interpretation of Loewner's differential equation leads to several interesting questions and in this project, I plan to investigate the connection between the two theories systematically. Solutions to the problems I describe would be of interest in non-commutative probability theory on the one hand, but they would also enrich complex analysis on the other hand, as they would add a new, probabilistic perspective to Loewner theory.For instance, Loewner theory and univalent mappings have been studied also in higher dimensions. So far, applications to other disciplines have not been found yet, in contrast to the one-dimensional case. However, a multivariate generalization of monotone independence naturally leads to an evolution equation for univalent mappings in several variables.

这个项目的重点是探索几何函数理论，特别是Loewner理论，和非交换概率论之间的联系。Loewner为保形映射引入了某些演化方程，它很快成为几何函数论中的一个重要工具。它们被用来模拟二维增长过程或解决单叶函数的极值问题。自O. Schramm于2000年提出的Loewner理论是复分析、概率论、统计力学和共形场论等多学科交叉的研究领域，非对易概率论为抽象的概率空间提供了框架，这些概率空间由一般不对易的随机变量组成。这是由量子力学激发的，在量子力学中，可观测量可以被视为非交换随机变量。这个领域的许多目标要求将概念和定理从经典概率论转移到这个非交换的环境中。例如，有量子随机过程和量子随机微分方程的理论，这反过来又有助于为某些量子系统提供数学模型。独立性的概念在经典概率论中起着核心作用，并且已经表明，在某种意义上，在非交换概率论中有五种定义它的方法。这导致了张量、自由、布尔、单调和反单调概率论。所有这五个概念都导致了全纯映射的某些卷积，在这一点上，复分析进入了理论。然而，到目前为止，用于此目的的方法相当初级。我注意到复分析和单调概率论之间有更深层次的关系：大多数研究的Loewner方程（“径向”和“弦”方程）都可以被视为具有单调独立增量的量子随机过程的Lévy-Khintchine表示。(And这些方程的时间反演对应于反单调类似物。Loewner微分方程的这种解释导致了几个有趣的问题，在这个项目中，我计划系统地研究这两个理论之间的联系。我所描述的问题的解决方案一方面会引起非交换概率论的兴趣，但另一方面也会丰富复分析，因为它们会为Loewner理论增加一个新的概率视角。例如，Loewner理论和单叶映射也在高维中被研究。到目前为止，应用到其他学科尚未发现，在一维的情况下。然而，单调独立性的多元推广自然会导致单叶映射在多个变量的发展方程。