Singular diffusion limit of a tagged particle in zero range processes with Sinai-type random environment

TL;DR

This paper establishes a singular diffusion limit for a tagged particle in zero range processes within a Sinai-type random environment, revealing the limiting behavior as the environment regularization vanishes and connecting it to a singular PDE.

Contribution

It introduces a two-step approach to derive the diffusion limit, involving environment regularization and homogenization, and characterizes the limit via a singular PDE and White noise.

Findings

The tagged particle's limit is a diffusion driven by spatial White noise.

The regularized environment converges to a singular PDE for the density.

The particle's limiting behavior is described by a stochastic differential equation with singular coefficients.

Abstract

We derive a singular diffusion limit for the position of a tagged particle in zero range interacting particle processes on a one dimensional torus with a Sinai-type random environment via two steps. In the first step, a regularization is introduced by averaging the random environment over an $\varepsilon N$-neighborhood. With respect to such an environment, the microscopic drift of the tagged particle is in form $\frac{1}{N}W_\varepsilon'$, where $W_\varepsilon'$ is a regularized White noise. Scaling diffusively, we find the nonequilibrium limit of the tagged particle $x^\varepsilon_t$ is the unique weak solution of $d x_t^{\varepsilon} = 2\frac{\Phi(\rho^{\varepsilon}(t, x_t^{\varepsilon}))}{\rho^{\varepsilon}(t, x_t^\varepsilon)} \,W_{\varepsilon}'(x_t^\varepsilon) + \sqrt{\frac{\Phi(\rho^{\varepsilon}(t, x_t^\varepsilon))}{\rho^{\varepsilon}(t, x_t^\varepsilon)}} \,dB_t$, in terms of the hydrodynamic mass density $\rho^\varepsilon$ recently identified and homogenized interaction rate $\Phi$. In the second step, we show that $x^\varepsilon$, as $\varepsilon$ vanishes, converges in law to the diffusion $x^0$ described informally by $d x_t^0 = 2\frac{\Phi(\rho^{0}(t, x_t^{0}))}{\rho^{0}(t, x_t^0)} \,W'(x_t^0) + \sqrt{\frac{\Phi(\rho^{0}(t, x_t^0))}{\rho^{0}(t, x_t^0)}} \,dB_t$, where $W'$ is a spatial White noise and $\rho^0$ is the para-controlled limit of $\rho^\varepsilon$ also recently identified, solving the singular PDE $ \partial_t \rho^0 = \frac{1}{2}\Delta \Phi(\rho^0) - 2\nabla \big(W' \Phi(\rho^0)\big)$.