Abstract
The $(d,\alpha,\beta,\gamma)$-branching particle system consists of particles moving in $\mathbb{R}^d$ according to a symmetric $\alpha$-stable L\'evy process $(0<\alpha\leq 2)$, splitting with a critical $(1+\beta)$-branching law $(0<\beta\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $\mu_\gamma(dx)=dx/(1+|x|^\gamma), \gamma\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_T\mu_\gamma$, occupation time fluctuation limits for the system as $T\to\infty$ have been obtained in two special cases: Lebesgue measure ($\gamma=0$, the homogeneous case), and finite measures $(\gamma > d)$. In some cases $H_T\equiv 1$ and in others $H_T\to\infty$ as $T\to\infty$ (high density systems). The limit processes are quite different for Lebesgue and for finite measures. Therefore the question arises of what kinds of limits can be obtained for Poisson intensity measures that are intermediate between Lebesgue measure and finite measures. In this paper the measures $\mu_\gamma, \gamma\in (0,d]$, are used for investigating this question. Occupation time fluctuation limits are obtained which interpolate in some way between the two previous extreme cases. The limit processes depend on different arrangements of the parameters $d,\alpha,\beta,\gamma$. There are two thresholds for the dimension $d$. The first one, $d=\alpha/\beta+\gamma$, determines the need for high density or not in order to obtain non-trivial limits, and its relation with a.s. local extinction of the system is discussed. The second one, $d=[\alpha(2+\beta)-\gamma\vee \alpha]/\beta$\ (if $\gamma < d$), interpolates between the two extreme cases, and it is a critical dimension which separates different qualitative behaviors of the limit processes, in particular long-range dependence in ``low'' dimensions, and independent increments in ``high'' dimensions. In low dimensions the temporal part of the limit process is a new self-similar stable process which has two different long-range dependence regimes depending on relationships among the parameters. Related results for the corresponding $(d,\alpha,\beta,\gamma)$-superprocess are also given.
Highlights
Occupation time fluctuation limits have been proved for the so-called (d, α, β)-branching particle systems in Rd with initial Poisson states in two special cases, namely, if the Poisson intensity measure is either Lebesgue measure, denoted by λ, or a finite measure [BGT1], [BGT2], [BGT3], [BGT4], [BGT6]
The question arises of what happens with Poisson intensity measures that are intermediate between Lebesgue measure and finite measures
That is the main motivation for the present paper, and our aim is to obtain limit processes that interpolate in some way between those of the two special cases
Summary
Occupation time fluctuation limits have been proved for the so-called (d, α, β)-branching particle systems in Rd with initial Poisson states in two special cases, namely, if the Poisson intensity measure is either Lebesgue measure, denoted by λ, or a finite measure [BGT1], [BGT2], [BGT3], [BGT4], [BGT6]. For μ = λ, N0 is homogeneous Poisson This case is special (and technically simpler) because λ is invariant for the α-stable process (which implies in particular that ENt = λ for all t), and there is the following persistence/extinction dichotomy [GW], which heuristically explains the need for high density in some cases in order to obtain non-trivial occupation time fluctuation limits, and anticipates the situation we will encounter in this paper:. High density is indispensable to obtain a non-trivial limit if either d ≤ α/β + γ or α < γ ≤ d In the latter case the total occupation time of any bounded Borel set by the process N is finite a.s.
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