Central limit theorems for additive functionals of long-range zero-range processes

TL;DR

This paper extends the central limit theorem for additive functionals of zero-range processes to long-range interactions, showing that the limit processes are often driven by fractional Brownian motions with specific Hurst parameters.

Contribution

It introduces a long-range version of the CLT for zero-range processes, revealing fractional Brownian motion as the limit process in several cases.

Findings

Limit processes are driven by fractional Brownian motions with Hurst parameters in (1/2, 3/4].

A local CLT for long-range random walks is established.

The paper proves a relaxation to equilibrium for long-range zero-range processes.

Abstract

In this paper, we extend the central limit theorem of the additive functional of the nearest-neighbor zero-range process given in \cite{Quastel2002} to the long-range case. Our main results show that in several cases the limit processes are driven by fractional Brownian motions with Hurst parameters in $(1/2, 3/4]$. A local central limit theorem of the long-range random walk and a relaxation to equilibrium theorem of the long-range zero-range process play the key roles in the proofs of our main results.