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Stationary fluctuation for the occupation time of the multi-species stirring process

Xiaofeng Xue

TL;DR

The paper proves a fluctuation theorem for the occupation time of a multi-species stirring process on the lattice starting from a stationary distribution. Using a resolvent strategy and a graphical coupling with an auxiliary process, it decomposes the occupation-time fluctuations into a martingale part and a remainder, then analyzes the limit in each dimension. The main result is a dimension-dependent Gaussian limit: for $d\ge 3$ and $d=2$ the limit is a Gaussian process formed by applying $\mathcal{A}^{1/2}$ to independent Brownian motions, while for $d=1$ the limit involves a fractional Brownian motion with Hurst parameter $3/4$; in all cases, different brands interact via negative correlations, and for $d\ge 2$ the limit is independent of $k$, with a special dependence in one dimension. The graphical representation and exchanging-path construction are central to controlling correlations and proving tightness across the three regime cases.

Abstract

In this paper, we prove a fluctuation theorem for the occupation time of the multi-species stirring process on a lattice starting from a stationary distribution. Our result shows that the occupation times of different species interact with each other at the level of equilibrium fluctuation. The proof of our result utilizes the resolvent strategy introduced in \cite{Kipnis1987}. A coupling relationship between the multi-species stirring process and an auxiliary process and a graphical representation of the auxiliary process play the key roles in the proof.

Stationary fluctuation for the occupation time of the multi-species stirring process

TL;DR

The paper proves a fluctuation theorem for the occupation time of a multi-species stirring process on the lattice starting from a stationary distribution. Using a resolvent strategy and a graphical coupling with an auxiliary process, it decomposes the occupation-time fluctuations into a martingale part and a remainder, then analyzes the limit in each dimension. The main result is a dimension-dependent Gaussian limit: for and the limit is a Gaussian process formed by applying to independent Brownian motions, while for the limit involves a fractional Brownian motion with Hurst parameter ; in all cases, different brands interact via negative correlations, and for the limit is independent of , with a special dependence in one dimension. The graphical representation and exchanging-path construction are central to controlling correlations and proving tightness across the three regime cases.

Abstract

In this paper, we prove a fluctuation theorem for the occupation time of the multi-species stirring process on a lattice starting from a stationary distribution. Our result shows that the occupation times of different species interact with each other at the level of equilibrium fluctuation. The proof of our result utilizes the resolvent strategy introduced in \cite{Kipnis1987}. A coupling relationship between the multi-species stirring process and an auxiliary process and a graphical representation of the auxiliary process play the key roles in the proof.

Paper Structure

This paper contains 6 sections, 11 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Graphical representation
  4. The proof of Theorem \ref{['theorem 2.1 main theorem']}: $d=1$ case
  5. The proof of Theorem \ref{['theorem 2.1 main theorem']}: $d=2$ case
  6. The proof of Theorem \ref{['theorem 2.1 main theorem']}: $d\geq 3$ case

Key Result

Theorem 2.1

Let $\eta_0$ be distributed with $\nu_{\vec{p}}$, then $V^N$ converges weakly, with respect to the uniform topology of $C\left([0, T], \mathbb{R}^l\right)$, to $V$ as $N\rightarrow+\infty$, where where $\{B_t^1\}_{t\geq 0}, \ldots, \{B_t^l\}_{t\geq 0}$ are independent copies of $\{B_t\}_{t\geq 0}$ and $\{\zeta_t^1\}_{t\geq 0}, \ldots, \{\zeta_t^l\}_{t\geq 0}$ are independent copies of $\{\zeta_t\

Theorems & Definitions (11)

  • Theorem 2.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 6.1
  • ...and 1 more