Large symmetrised systems of interacting Brownian bridges and random interlacements and their scaling limits

总的主题是相互作用的粒子系统及其临界现象。其新颖之处在于将相互作用的布朗桥的大偏差分析与最近发展的随机交错理论相结合。另一个新奇之处是相互作用的布朗桥和随机交错的概率散射长度。第三个创新是研究对称化下的扩散标度极限。这里的目的是证明这样一个猜想，即极限实际上是所谓的薛定谔扩散的例子。我们的想法是研究各种概率方法来相互作用的玻色子系统和玻色-爱因斯坦凝聚的开始作为一个关键现象。这种相变表现在所谓的随机交错的开始，或有限时间视界随机环上的概率质量的损失。该项目主要是概率分析，使用了大偏差技术、非线性扩散的随机分析、多尺度分析和相互作用的粒子系统。这个项目分析了最吸引人和最具挑战性的模型之一：正温度下的相互作用玻色子。自从20世纪90年代末的冷原子实验和两次诺贝尔奖以来，数学研究的目标就是证明所谓的玻色-爱因斯坦凝聚--即临界现象，比如液氦在低温下的超流性。该项目使用了量子相互作用系统的概率方法--所谓的费曼-卡克公式将量子问题重写为概率论中的经典问题。主要技术有大偏差分析的变种、非线性扩散的随机分析和浓度不等。特别是，该项目研究了路径度量和本地时间以及新的时空同构定理，该项目涉及以下步骤：研究活动分为三种不同的方法。在第一种方法中，利用排列的空间依赖性和循环结构的组合，研究了对称化对大粒子系统的影响，分别是随机过程。其目的是证明大偏差率函数的唯一极小值和标度极限是所谓的薛定谔扩散的例子。在第二种方法中，研究了这种相互作用的格罗斯-皮塔夫斯基尺度；见下文(2)。挑战在于找到一个合适的散射长度的概率表示。此外，这种表示还可以证明Gross-Pitaevskii变分公式。我们用第三种方法研究了布朗运动和交错的大型对称系统。在这里，我们的目标是使用新的时空扩散同构定理来进行完整的局部时间分析，见(Iii)。(I)考察经验路径度量的大N极限(布朗桥的数目)和大时间视界极限。其目的是证明唯一的极小化是薛定谔扩散，它的对测度具有一个密度，该密度分别由稀薄系统的Gross-Pitaevskii函数与桥密度函数的乘积给出，也称为Wasserstein扩散输运项。(Ii)考察了Gross-Pitaevskii标度极限中的大N极限(布朗桥数)和相互作用的布朗运动的大时间极限。这两个悬而未决的问题一方面涉及零温极限中对基态描述的极限，另一方面涉及散射长度的作用。散射长度在分析中是通过一个变分问题(PDE)出现的。该项目将不得不开发和使用一个概率版本。(Iii)布朗运动系统的对称化过程是一个双随机机制，它涉及到绘制随机排列，然后采样$N$随机粒子位置。在过去，不同的组织分析过这些

The overall theme is interacting particle systems and their critical phenomena. The novelty is to combine large deviation analysis for interacting Brownian bridges with the recently developed theory of random interlacements. Another novelty is the probabilistic scattering length for interacting Brownian bridges and random interlacements. A third novelty is to investigate the diffusive scaling limits under symmetrisation. Here, the aim is to prove the conjecture that the limits are, in fact, examples of the so-called Schrödinger diffusion. The idea is to study various probabilistic approaches to interacting Boson systems and the onset of the Bose-Einstein condensation as a critical phenomenon. The phase transition manifests in the onset of so-called random interlacements or the loss of probability mass on finite time horizon random loops. The project is primarily probabilistic, using large deviation techniques, stochastic analysis of nonlinear diffusions, multi-scale analysis and interacting particle systems. This project analyses one of the most fascinating and challenging models: interacting Bosons at positive temperatures. Since the experiments on cold atoms in the late 1990s and two Nobel Prizes, mathematical research has started aiming to prove the so-called Bose-Einstein condensation - critical phenomena like, for example, the superfluidity of liquid Helium at low temperatures. The project uses probabilistic methods for the quantum interacting systems - the so-called Feynman-Kac formula rewrites the quantum problem as a classical problem in probability theory. Outline: Main techniques to be used are variants of large deviation analysis, stochastic analysis for nonlinear diffusions and concentration inequalities. In particular, the project studies path measures and local times along with novel space-time isomorphism theorems, and the project involves the following steps: The research activity splits into three different approaches. In the first one, the effect symmetrisation has on large systems of particles, respectively stochastic processes, is studied using a combination of the spatial dependences of the permutations and the cycle structures. The aim is to show that the unique minimiser of the large deviation rate functions and scaling limits are examples of the so-called Schrödinger diffusion. In the second approach, one studies the Gross-Pitaevskii scaling of the interaction; see (II) below. The challenge is to find a suitable probabilistic representation of the scattering length. Furthermore, this representation may allow proving the Gross-Pitaevskii variational formula. We study large symmetrised systems of Brownian motions and interlacements in a third approach. Here, we aim for a complete local time analysis using novel isomorphism theorems for space-time diffusions, see (III).(I) Examination of the large N limit (number of Brownian bridges) coupled with the large time horizon limit for empirical path measures. The aim is to prove that the unique minimiser is Schrödinger diffusions whose pair measure has a density given by the product of the Gross-Pitaevskii functions for dilute systems respectively by the bridge density function, also known as the Wasserstein diffusion transport term. (II) Examination of the large N limit (number of Brownian bridges) coupled with the large time limit for interacting Brownian motions in trap potential in the Gross-Pitaevskii scaling limit. The two open questions concern, on the one hand, the limit to the ground state description in the zero-temperature limit and, on the other hand, the role of the scattering length. The scattering length appears in analysis via a variational problem (PDE). The project will have to develop and employ a probabilistic version. (III) The symmetrisation procedure for Brownian motion systems is a two-random mechanism involving drawing random permutations and then sampling $N$ random particle positions. In the past, different groups have analysed these