Prerequisite
It is desirable that the audience had attended my course in the last semester. But, at the beginning of the course, I will try to summarize the methods and results which I explained in the last semester.
Abstract
This is a continuation of my course given in the last semester (Oct 2022-Jan 2023). In the last semester, we started with a quick introduction to modern probability theory and stochastic analysis including strong law of large numbers, central limit theorem, continuous time martingales, Brownian motion, Poisson point processes. Then, we briefly discussed the construction and equilibrium states of interacting particle systems such as exclusion process (Kawasaki dynamics), zero-range process, Glauber dynamics. The core of the course was the study of hydrodynamic large space-time scaling limits of interacting particle systems. We explained two methods: the entropy method (GPV method) and the relative entropy method (due to H.T. Yau). Important notions and tools are local ergodicity, one block estimate, two blocks estimate, local equilibrium states, entropy inequality, large deviation principle. We also discussed linear and nonlinear (KPZ) fluctuations via Boltzmann-Gibbs principle. We touched non-gradient models. Finally, we briefly gave some ideas to derive an interface motion from interacting particle systems. The course in this semester gives some applications and extensions of the methods explained in the last semester. More details are found in the syllabus.
Lecturer Intro.
Funaki Tadahisa was a professor at University of Tokyo and then at Waseda University in Japan. His research subject is probability theory mostly related to statistical physics, specifically interacting systems and stochastic PDEs, whose importance increases as several Fields medals are given to this area.
