Fluctuation limits of site-dependent branching systems in critical and large dimensions

Notes on occupation time fluctuations of binary branching particle systems

“…As the first step of this paper, we will prove this result with more details. Note that the same conclusions hold in the cases ofᾱ ≥ 2 (see Theorem 2.2 in Li [9]).…”

“…(see Remark 2.1 (2) in Li [9]). As the first step of this paper, we will prove this result with more details.…”

“…While this assumption can not technically change the arguments in this paper, we believe that it will be helpful in some situations such as investigation on the problem whether/when/how the anisotropy caused by the micro particles affects the large-scaled macro phenomena. By the notation in Li [9], we refer to such a model as a (d, α, σ(x))-branching particle system. Let N (s) denote the random counting measure of a (d, α, σ(x))-branching particle system at time s, i.e.…”

Riemann–liouville Processes Arising From Branching Particle Systems

“…where E x (f (N )) := E(f (N )|N (0) = ǫ x ) and ǫ x denotes the unit measure concentrated at x ∈ R. From Section 3 in [14], we get that where…”

“…Motivated by the aforementioned works, the main purpose of this paper is to report a similar but new result. We show that under certain conditions, RL processes arise from the limits of occupation time fluctuations of the so-called site-dependent branching particle systems (see Li [14] for details). This result is not obvious at a first sight.…”