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

Xue Xiaofeng

TL;DR

The paper extends the central limit theorem for additive functionals of nearest-neighbor zero-range processes to long-range interactions on $\mathbb{Z}^d$ with kernel $\|y-x\|_2^{-(d+\alpha)}$. By leveraging a local central limit theorem for the long-range random walk and a relaxation-to-equilibrium principle, it derives dimension- and parameter-dependent CLTs in which the limit processes are Brownian motion or fractional Brownian motion with Hurst parameters in $(\tfrac12, \tfrac34]$, depending on $(d,\alpha)$. The analysis employs a martingale decomposition (Kipnis–Varadhan) and a Poisson-flow approach to control the additive functionals, with explicit normalizations $\Lambda_{d,\alpha}(N)$ and variance factors $\sigma_\gamma(V)$ that encode the effect of long-range jumps and the equilibrium fluctuations. The results unify and extend prior nearest-neighbor results (Quastel 2002) and align with related findings for long-range interacting systems, highlighting phase-transition-like changes in the scaling and limiting behavior as $\alpha$ varies. Overall, the work provides a comprehensive framework for understanding temporal fluctuations in long-range zero-range dynamics and their connections to fractional Brownian limits.

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.

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

TL;DR

The paper extends the central limit theorem for additive functionals of nearest-neighbor zero-range processes to long-range interactions on with kernel . By leveraging a local central limit theorem for the long-range random walk and a relaxation-to-equilibrium principle, it derives dimension- and parameter-dependent CLTs in which the limit processes are Brownian motion or fractional Brownian motion with Hurst parameters in , depending on . The analysis employs a martingale decomposition (Kipnis–Varadhan) and a Poisson-flow approach to control the additive functionals, with explicit normalizations and variance factors that encode the effect of long-range jumps and the equilibrium fluctuations. The results unify and extend prior nearest-neighbor results (Quastel 2002) and align with related findings for long-range interacting systems, highlighting phase-transition-like changes in the scaling and limiting behavior as varies. Overall, the work provides a comprehensive framework for understanding temporal fluctuations in long-range zero-range dynamics and their connections to fractional Brownian limits.

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 . 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.
Paper Structure (8 sections, 16 theorems, 248 equations)

This paper contains 8 sections, 16 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Proof of Theorem \ref{['theorem 2.1 main theorem d=1']}
  5. The case of $\alpha<1$
  6. The case of $\alpha=1$
  7. The case of $\alpha>1$
  8. Proof of Theorem \ref{['theorem 2.2 main theorem d=2']} in the case $\alpha\geq 2$

Key Result

Theorem 2.1

Let $T>0$, $\gamma>0$, local $V: \mathbb{N}^{\mathbb{Z}^1}\rightarrow \mathbb{R}$ with polynomial bound satisfy that $\overline{V}(\gamma)=0, \overline{V}^\prime(\gamma)\neq 0$ and $\{\eta_t\}_{t\geq 0}$ on $\mathbb{Z}^1$ with parameter $\alpha>0$ start from $\nu_\gamma$. If $\alpha\neq 1$, then converges weakly, with respect to the uniform topology, to $\left\{Y_t^{1, \alpha}\right\}_{0\leq t\le

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • ...and 7 more