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

相互作用的布朗桥和随机交错的大型对称系统及其尺度限制

• 批准号: 2813903
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2813903
• 关键词: Large; symmetrised; systems; interacting; Brownian

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

总的主题是相互作用的粒子系统及其临界现象。新颖之处是结合联合收割机大偏差分析相互作用的布朗桥与最近发展起来的随机交织理论。另一个新奇是相互作用的布朗桥和随机交织的概率散射长度。第三个新奇是研究对称化下的扩散标度极限。在这里，目的是要证明的猜想，极限，实际上，所谓的薛定谔扩散的例子。这个想法是研究各种概率方法相互作用玻色子系统和玻色爱因斯坦凝聚的发病作为一个关键现象。相变表现在所谓的随机交错的开始或有限时间范围随机循环上的概率质量的损失。该项目主要是概率性的，使用大偏差技术，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。该项目分析了最吸引人和最具挑战性的模型之一：在正温度下相互作用的玻色子。自1990年代末的冷原子实验和两次诺贝尔奖以来，数学研究已经开始致力于证明所谓的玻色-爱因斯坦凝聚临界现象，例如液态氦在低温下的超流性。该项目使用量子相互作用系统的概率方法-所谓的Feynman-Kac公式将量子问题改写为概率论中的经典问题。概述：要使用的主要技术是大偏差分析，非线性扩散和浓度不等式的随机分析的变种。特别是，该项目研究路径措施和当地时间沿着与新的时空同构定理，该项目涉及以下步骤：研究活动分为三种不同的方法。在第一个中，对称化的效果对大系统的粒子，分别随机过程，研究使用的空间依赖性的排列和循环结构的组合。其目的是表明，唯一的最小化的大偏差率函数和标度限制的例子，所谓的薛定谔扩散。在第二种方法中，人们研究了相互作用的Gross-Pitaevskii标度;见下面的（II）。挑战是找到一个合适的概率表示的散射长度。此外，这种表示可以允许证明Gross-Pitaevskii变分公式。我们研究大型对称系统的布朗运动和交错的第三种方法。在这里，我们的目标是一个完整的本地时间分析使用新的同构定理的时空扩散，见（III）。(I)检验大N极限（布朗桥数）与经验路径测度的大时间范围极限。其目的是证明唯一的极小是薛定谔扩散，其对测度的密度分别由稀释系统的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)在Gross-Pitaevskii标度极限中，陷阱势中相互作用的布朗运动的大N极限（布朗桥数）与大时间极限耦合的检验。这两个悬而未决的问题，一方面，限制在零温度限制的基态描述，另一方面，散射长度的作用。散射长度通过变分问题（PDE）出现在分析中。该项目将不得不开发和采用概率版本。(III)布朗运动系统的对称化过程是一个双随机机制，涉及绘制随机排列，然后采样$N$随机粒子位置。在过去，不同的团体分析了这些