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

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

Number variance for hierarchical random walks and related fluctuations

We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the c-random walks. Such walks are used, e.g., in population genetics. The number variance problem consists in investigating if the variance of the number of "particles" N_n(L) lying in the ball of radius L at a given time n remains bounded, or even better, converges to a finite limit, as $L\to \infty$. We give a necessary and sufficient condition and discuss its relationship to transience/recurrence property of the walk. Next we consider normalized fluctuations of N_n(L) around the mean as $n\to \infty$ and L is increased in an appropriate way. We prove convergence of finite dimensional distributions to a Gaussian process whose properties are discussed. As the c-random walks mimic symmetric stable processes on R, we compare our results to those obtained by Hambly and Jones (2007,2009), where the number variance problem for an infinite system of symmetric stable processes on R was studied. Since the hierarchical group is an ultrametric space, corresponding results for symmetric stable processes and hierarchical random walks may be analogous or quite different, as has been observed in other contexts. An example of a difference in the present context is that for the stable processes a fluctuation limit process is a centered Gaussian process which is not Markovian and has long range dependent stationary increments, but the counterpart for hierarchical random walks is Markovian, and in a special case it has independent increments

Sub-fractional Brownian motion and its relation to occupation times

We study a long-range dependence Gaussian process which we call “sub-fractional Brownian motion” (sub-fBm), because it is intermediate between Brownian motion (Bm) and fractional Brownian motion (fBm) in the sense that it has properties analogous to those of fBm, but the increments on non-overlapping intervals are more weakly correlated and their covariance decays polynomially at a higher rate. Sub-fBm has a parameter h E (0, 2), we show how it arises from occupation time fluctuations of branching particle systems for h >= 1 and we exhibit the long memory effect of the initial condition.Long-range dependence; Fractional Brownian motion; Sub-fractional Brownian motion; Occupation time fluctuations; Branching systems.

A long range dependence stable process and an infinite variance branching system

We prove a functional limit theorem for the rescaled occupation time fluctuations of a $(d,\alpha,\beta)$-branching particle system [particles moving in $\mathbb {R}^d$ according to a symmetric $\alpha$-stable L\'{e}vy process, branching law in the domain of attraction of a $(1+\beta)$-stable law, $0<\beta<1$, uniform Poisson initial state] in the case of intermediate dimensions, $\alpha/\beta<d<\alpha(1+\beta)/\beta$. The limit is a process of the form $K\lambda\xi$, where $K$ is a constant, $\lambda$ is the Lebesgue measure on $\mathbb {R}^d$, and $\xi=(\xi_t)_{t\geq0}$ is a $(1+\beta)$-stable process which has long range dependence. For $\alpha<2$, there are two long range dependence regimes, one for $\beta>d/(d+\alpha)$, which coincides with the case of finite variance branching $(\beta=1)$, and another one for $\beta\leq d/(d+\alpha)$, where the long range dependence depends on the value of $\beta$. The long range dependence is characterized by a dependence exponent $\kappa$ which describes the asymptotic behavior of the codifference of increments of $\xi$ on intervals far apart, and which is $d/\alpha$ for the first case (and for $\alpha=2$) and $(1+\beta-d/(d+\alpha))d/\alpha$ for the second one. The convergence proofs use techniques of $\mathcal{S}'(\mathbb {R}^d)$-valued processes.Comment: Published at http://dx.doi.org/10.1214/009117906000000737 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org