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

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 $\epsilon N$-neighborhood. With respect to such an environment, the microscopic drift of the tagged particle is in form $\frac{1}{N}{W}_{\epsilon }^{\prime }$, where $W_{\epsilon }^{\prime }$ is a regularized White noise. Scaling diffusively, we find the nonequilibrium limit of the tagged particle $x_t^{\epsilon }$ is the unique weak solution of $d{x}_t^{\epsilon }=2\frac{\hat{I}¦(\ddot{I}{}^{\epsilon }(t,x_t^{\epsilon }))}{\ddot{I}{}^{\epsilon }(t,x_t^{\epsilon })}{W}_{\epsilon }^{\prime }(x_t^{\epsilon })+\sqrt{\frac{\hat{I}¦(\ddot{I}{}^{\epsilon }(t,x_t^{\epsilon }))}{\ddot{I}{}^{\epsilon }(t,x_t^{\epsilon })}}d{B}_t$, in terms of the hydrodynamic mass density $\ddot{I}{}^{\epsilon }$ recently identified and homogenized interaction rate $\hat{I}¦$. In the second step, we show that $x^{\epsilon }$, as $\epsilon$ vanishes, converges in law to the diffusion $x^0$ described informally by $d{x}_t^0=2\frac{\hat{I}¦(\ddot{I}{}^0(t,x_t^0))}{\ddot{I}{}^0(t,x_t^0)}{W}^{\prime }(x_t^0)+\sqrt{\frac{\hat{I}¦(\ddot{I}{}^0(t,x_t^0))}{\ddot{I}{}^0(t,x_t^0)}}d{B}_t$, where $W^{\prime }$ is a spatial White noise and $\ddot{I}{}^0$ is the para-controlled limit of $\ddot{I}{}^{\epsilon }$ also recently identified, solving the singular PDE ${\partial }_t\ddot{I}{}^0=\frac{1}{2}\hat{I}”\hat{I}¦(\ddot{I}{}^0)-2\nabla (W^{\prime }\hat{I}¦(\ddot{I}{}^0))$.

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