Stable central limit theorems for super Ornstein-Uhlenbeck processes

Abstract

In this paper, we study the asymptotic behavior of a supercritical (ξ, ψ)superprocess (X t ) t≥0 whose underlying spatial motion ξ is an Ornstein-Uhlenbeck process on R d with generator L = 1 2 σ 2 ∆ − bx · ∇ where σ, b > 0; and whose branching mechanism ψ satisfies Grey's condition and some perturbation condition which guarantees that, when z → 0, ψ(z) = −αz + ηz 1+β (1 + o(1)) with α > 0, η > 0 and β ∈ (0, 1). Some law of large numbers and (1 + β)-stable central limit theorems are established for (X t (f )) t≥0 , where the function f is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding the branching rate being relatively small, large or critical at a balanced value.2010 Mathematics Subject Classification. 60J68, 60F05.

Almost sure, L1- and L2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration

“…Very recently, Ren et al [18] derived stable central limit theorems for some kind of projections of (measure-valued) super Ornstein-Uhlenbeck processes having a branching mechanism which is close to a function of the form −a 1 z + a 2 z 2 + a 3 z 1+α , z 0, with a 1 > 0, a 2 0, a 3 > 0 and α ∈ (0, 1) in some sense (see the Assumption 2 in Ren et al [18]).…”

Equivalence of mean-field avalanches and branching diffusions: from the Brownian force model to the super-Brownian motion

“…More limit theorems can be found in [104] and relations to random trees in [105]. Finally generalizations for Ornstein-Uhlenbeck superprocesses have been studied [106].…”

On the derivative martingale in a branching random walk

“…The latter article discusses, among others, limit theorems for some models of statistical mechanics. A large selection of rate of convergence results for more complicated branching processes, including branching diffusions and superprocesses, can be traced via the references given in [34]. Theorem 2.4 is a counterpart of Proposition 2.1 in [28] obtained for the derivative martingale which corresponds to a branching Brownian motion.…”