Hydrodynamic limit and the fluctuating hydrodynamics for large-scale interacting systems

In these decades, a great deal of works has been devoted to understand macroscopic phenomena, such as diffusion, aggregation or pattern formation, from the viewpoint of microscopic systems. Hydrodynamic limit, or fluctuating hydrodynamics, is a fundamental framework to explain the macroscopic behavior of physical quantities in mathematically rigorous ways from a system of the vast numbers of microscopic agents under random interactions, which system is called the large-scale interacting system. In this framework, our central aim is to derive partial differential equations (PDEs) which describe time evolution of some macroscopic quantities, starting from the large-scale interacting systems; hydrodynamic limit is a procedure to derive deterministic PDEs with help of the law of large numbers, whereas stochastic PDEs are derived under the scale of the central limit theorem by fluctuating hydrodynamics. In this talk, I would like to explain basic concepts of hydrodynamic limit and fluctuating hydrodynamics, through some simple models. In the first part, I will give a concise exposition on Markov processes as preliminaries and then state some results on scaling limits of simple exclusion processes as a pedagogical example. In the second part, I will talk about recent progress on universality which appears in fluctuating hydrodynamics. Especially, I would like to talk about the universality of the Kardar-Parisi-Zhang equation, and its mathematical background.