Particle systems with quasi-homogeneous initial states and their occupation time fluctuations

TL;DR

This paper extends the analysis of occupation time fluctuations in particle systems to a broader class of initial measures, revealing that sub-fractional Brownian motion appears more generally in the limits than previously recognized.

Contribution

It introduces a unified approach to study occupation time fluctuations for quasi-homogeneous initial states, broadening the understanding of limit processes in particle systems.

Findings

Sub-fractional Brownian motion appears in non-branching systems in low dimensions.

Fractional Brownian motion is linked to systems with quasi-homogeneous initial measures.

The new approach generalizes previous results to a wider class of initial conditions.

Abstract

Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In previous papers, in the homogeneous Poisson case, for the branching system in "low" dimensions, the limit was characterized by a long-range dependent Gaussian process called sub-fractional Brownian motion (sub-fBm), and this effect was attributed to the branching because it had appeared only in that case. An unexpected finding in this paper is that sub-fBm is more prevalent than previously thought. Namely, it is a natural ingredient of the limit process in the non-branching case (for "low" dimension), as well. On the other hand, fractional Brownian motion is not only related to systems in equilibrium (e.g., non-branching system with initial homogeneous Poisson measure), but it also appears here for a wider class of initial measures of quasi-homogeneous type.