Investigation of Interacting Particle Systems by Stochastic Analysis Methods

相互作用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。其中，它们可以用来模拟股票市场的资本分配、流体动力学、排队网络的稳定性和交通流等现象。该项目的主要目标是获得对相互作用粒子系统的更好的数学理解，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、随机杨氏表和对称函数。这种详细的数学分析将提供新的信息，说明相互作用粒子系统作为数学模型在上述领域中的适用性，以及它们的局限性。此外，由于相互作用粒子系统的数学是在概率论，偏微分方程和表示理论的界面上，拟议的研究有望揭示这些数学领域之间的联系。这种联系是非常有趣的，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打算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（或者更一般地说，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类似物。然而，状态空间的连续性允许人们将随机分析方法应用于这样的系统，并且由于描述相互作用粒子系统宏观行为的（随机）偏微分方程的连续性而自然。特别是，PI打算研究kardar - paris - zhang普适类中的系统，该普适类描述了随机表面生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简单不相容过程和O'Connell-Yor半离散聚合物。在这种情况下，人们对主要粒子的渐近行为和它们之间的间距特别感兴趣，对它们的研究是拟议研究的主要部分。

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.