Marcel Hudiani: Singular diffusion limit of a tagged particle in a zero range process with Sinai-type random environment

We prove a non-equilibrium functional central limit theorem for the position of a tagged particle in a one dimensional zero range process random environment. A regularization by averaging over an epsilon neighborhood of a Sinai random environment is introduced to the zero range model. As a result, a derivative-like term of the Brownian motion appears in the microscopic drift of the tagged particle. Under the diffusive scaling, such a term survives in the limit so that the resulting tagged particle macroscopic dynamics is described by a stochastic differential equation with a random drift term.

Thanks to the results by Funaki, Hoshino, Sethuraman, Xie, such a diffusion has a limit if expressed as a time-changed Brownian Motion. Thus, we have obtained an interpretation of particle dynamics in a random environment. This is a generalization of the work by Jara-Landim-Sethuraman and a continuation of the work by Landim-Pacheco-Sethuraman-Xue.