Occupation Time Fluctuations of Weakly Degenerate Branching Systems

Abstract

We establish limit theorems for re-scaled occupation time fluctuations of a sequence of branching particle systems in $\R^d$ with anisotropic space motion and weakly degenerate splitting ability. In the case of large dimensions, our limit processes lead to a new class of operator-scaling Gaussian random fields with non-stationary increments. In the intermediate and critical dimensions, the limit processes have spatial structures analogous to (but more complicated than) those arising from the critical branching particle system without degeneration considered by Bojdecki et al.{\it Stochastic Process. Appl. 2006}. Due to the weakly degenerate branching ability, temporal structures of the limit processes in all three cases are different from those obtained by Bojdecki et al. .{\it Stochastic Process. Appl. 2006}.Comment: 27page

Fluctuation limits of strongly degenerate branching systems

“…We find that the limit processes in any positive time interval are time-independent measure-valued Wiener processes, which always have the form Cλξ, where ξ is a standard normal random variable, λ is the Lebesgue measure in R d and C is a non-random constant. By comparison with the corresponding results in Li and Xiao [14], the current limit processes are simpler (please see Remark 2.1 in Section 2 for more details). To save the space of this paper, we leave the study on the case of large dimensions elsewhere because the potential limit processes deserve further investigation.…”

“…It is easy to see that when δ = 0, this model is similar to a classical (d, α, β)-branching particle system with β = 1 except that the moving mechanism is the anisotropic stable Lévy process ξ rather than a symmetric α-stable Lévy process. Li and Xiao [14] called this model as a (d, α, δ, γ)-degenerate branching particle system, where α := (α 1 , . .…”

“…Motivated by the work on occupation time fluctuations of classical branching particle systems and the work on the construction of anisotropic random fields (see, for example, [1]), Li and Xiao [14] explicitly studied the functional limits of the occupation time fluctuations of the models. Observe that a fixed (d, α, δ, γ)-branching particle system with δ > 0 will go to local extinction as time elapses because of the sub-critical branching laws at positive ages.…”

“…Observe that a fixed (d, α, δ, γ)-branching particle system with δ > 0 will go to local extinction as time elapses because of the sub-critical branching laws at positive ages. They [14] borrowed the idea of nearly critical branching processes (see [12,13,18]) and considered a sequence of (d, α, δ n , γ)-models with δ n → 0 as n → ∞. More precisely, let N n (s) denote the empirical measure of the (d, α, δ n , γ)-degenerate branching particle system at time s, i.e., N n (s)(A) is the number of particles in the set A ⊂ R d at time s. They studied the limit of a sequence of scaled occupation time fluctuations, X n (t) = 1 F n nt 0 (N n (s) − f n (s)λ)ds, (1.1) where F n is a scaling constant and f n (s) :=f n (s)e −δns :…”

“…We focus on the cases of critical and intermediate dimensions in this paper. The main methods used in this situation is the same as that in Li and Xiao [14], which was formulated and developed by Bojdecki et al in their serial papers (see [2][3][4][5]), except some complexities and differences arising from the strong degeneration. We find that the limit processes in any positive time interval are time-independent measure-valued Wiener processes, which always have the form Cλξ, where ξ is a standard normal random variable, λ is the Lebesgue measure in R d and C is a non-random constant.…”