Investigation of Interacting Particle Systems by Stochastic Analysis Methods

相互作用的粒子系统起源于数学物理，但现在被用于应用数学的许多领域。此外，它们还可以用来模拟股票市场中的资本分配、流体动力学、排队网络的稳定性和交通流量等现象。这个项目的主要目标是更好地从数学上理解相互作用的粒子系统，并将它们与其他数学对象联系起来，包括偏微分方程、随机矩阵、随机聚合物、随机Young表和对称函数。这种详细的数学分析将提供关于相互作用的粒子系统作为上述领域内的数学模型的适用性以及它们的局限性的新信息。此外，由于相互作用的粒子系统的数学处于概率论、偏微分方程式和表示论的交界处，拟议的研究有望揭示这些数学领域之间的联系。这种联系非常有趣，因为它们允许人们在未来的研究中结合所有这些不同领域的工具。更具体地说，PI旨在研究具有连续状态空间的相互作用粒子系统，其中粒子动力学可以通过相互作用的布朗运动(或更一般地，扩散运动)来描述。通常，这样的系统是更经典的离散相互作用粒子系统的连续模拟。然而，状态空间的连续性允许人们将随机分析的方法应用于这样的系统，并且由于描述相互作用的粒子系统的宏观行为的(随机)偏微分方程组的连续性质，这是自然的。特别是，PI打算研究Kardar-Parisi-Zhang普适性类中的系统，该普适性类描述了随机表面增长背后的微观动力学。这类体系的例子包括布朗完全不对称的简单排斥过程和O‘Connell-Yor半离散聚合物。在这种背景下，人们对领先粒子的渐近行为和它们之间的间距特别感兴趣，而对它们的研究是拟议中的研究的主要部分。

Interacting particle systems originate from mathematical physics, but are now used in a number of areas in applied mathematics. Among others, they can be used to model phenomena such as capital distribution in equity markets, fluid dynamics, stability of queueing networks, and traffic flow. The main goal of this project is to obtain a better mathematical understanding of interacting particle systems and to relate them to other mathematical objects, including partial differential equations, random matrices, random polymers, random Young tableaux, and symmetric functions. Such detailed mathematical analysis would provide new information on the applicability of interacting particle systems as mathematical models within the fields described above, as well as on their limitations. In addition, with the mathematics of interacting particle systems being on the interface of probability theory, partial differential equations, and representation theory, the proposed research is expected to unveil connections between these areas of mathematics. Such connections are of great interest as they allow one to combine tools from all these different areas in future research.More specifically, the PI intends to study interacting particle systems with continuous state spaces in which the particle dynamics can be described by interacting Brownian (or, more generally, diffusive) motions. Typically, such systems are the continuous analogues of the more classical discrete interacting particle systems. However, the continuity of the state space allows one to apply methods of stochastic analysis to such systems and is natural due to the continuous nature of the (stochastic) partial differential equations describing the macroscopic behavior of interacting particle systems. In particular, the PI intends to study systems in the Kardar-Parisi-Zhang universality class, which describes the microscopic dynamics behind the growth of random surfaces. Examples of such systems include, among others, the Brownian totally asymmetric simple exclusion process and the O'Connell-Yor semi-discrete polymer. In this context one is particularly interested in the asymptotic behavior of the leading particles and the spacings between them, and their investigation is a major part of the proposed research.