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

Tomasz Bojdecki,Luis Gorostiza,Anna Talarczyk

doi:10.1214/ecp.v15-1547

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.

Highlights

In a series of papers [2, 3, 4, 5, 6] we studied particle systems in d starting from a configuration determined by a random point measure ν, and independently moving according to a standard α-stable Lévy process (0 < α ≤ 2)
The object of interest is the limit of the time-rescaled and normalized occupation time fluctuation process X T defined by XT (t) = FT
The results show that within the class the fluctuations caused by the branching are so large that X T “forgets” the randomness of the initial state of the system

Summary

Introduction

In a series of papers [2, 3, 4, 5, 6] (and others) we studied particle systems in d starting from a configuration determined by a random point measure ν, and independently moving according to a standard α-stable Lévy process (0 < α ≤ 2). It turns out that the only case where new limits appear is the non-branching case with (d = )1 < α They have the form Kλξ, where K is a constant, λ is the Lebesgue measure, and ξ is the sum of two independent processes, one of them is a sub-fractional Brownian motion (see (2.3)), and the other one is a new (centered continuous long-range dependent) Gaussian process (see (2.4) and Remark 2.3(c)). For a deterministic ν this process reduces to a sub-fractional Brownian motion, and in the homogeneous Poisson case, as well as for any ν with Eθ0 = Var θ0, it yields a fractional Brownian motion (see Remark 2.3(a)(b) for more comments) This result seems surprising since in all earlier papers sub-fractional Brownian motion was related only to branching systems, and was attributed to the branching, but in the present framework this process is more “natural” than fractional Brownian motion. That paper, which is a full-length version of the present one, contains more references and an example where convergence in C([0, τ], ( )) is proved (Proposition 2.6)

Results

Auxiliary facts related to the stable density

Scheme of proofs

T t Ts