Investigation of Interacting Particle Systems by Stochastic Analysis Methods

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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.