Makiko Sasada (University of Tokyo, Japan)
Title: Topological structures and the role of symmetry in the hydrodynamic limit
Abstract: Hydrodynamic limit provides a rigorous mathematical method to derive the time evolution of macroscopic parameters, from a microscopic stochastic large scale interacting system. Recently, we introduce a general framework in order to systematically investigate hydrodynamics limits of various models in a unified fashion. In particular, we introduced a new cohomology theory called the uniformly cohomology to investigate the underlying topological structure of the interacting system. Our theory gives a new interpretation of the macroscopic parameters, the role played by the group action on the microscopic system, and the origin of the diffusion matrix associated to the macroscopic hydrodynamic equation. Furthermore, we rigorously formulate and prove for a relatively general class of models Varadhan’s decomposition of closed forms, which plays a key role in the proof of hydrodynamic limits of nongradient models. Our result is applicable for many important models including generalized exclusion processes, multi-species exclusion processes, exclusion processes on crystal lattices and so on. Based on joint papers with Kenichi Bannai and Yukio Kametani (https://arxiv.org/abs/2009.04699, https://arxiv.org/abs/2111.08934).