Persistence of a critical super-2 process

Abstract

It is shown that the critical two-level (2, d, 1, 1)-superprocess is persistent in dimensions d greater than 4. This complements the extinction result of Wu (1994) and implies that the critical dimension is 4.

A note on two-level superprocesses

“…In this paper, as a mass reproduction, we will consider twolevel branching, with a critical reproduction on the individual as well as the family level; the mass flow is that of Brownian motion, leading to so-called two-level super-Brownian motion. Those two-level superprocesses have been studied by many authors including Dawson, Gorostiza, Hochberg, Wakobinger and Wu ([2], [6], [10], [11]). Wu [11] confirmed Dawson's conjecture about extinction in lower dimensions d ≤ 4; whereas Gorostiza et al [6] proved the persistence property in higher dimensions d > 4 and thus established that d = 4 is the critical dimension.…”

“…Those two-level superprocesses have been studied by many authors including Dawson, Gorostiza, Hochberg, Wakobinger and Wu ([2], [6], [10], [11]). Wu [11] confirmed Dawson's conjecture about extinction in lower dimensions d ≤ 4; whereas Gorostiza et al [6] proved the persistence property in higher dimensions d > 4 and thus established that d = 4 is the critical dimension. For one-level superprocesses, this important property was proved by Dawson in the famous paper [1] in which the critical dimension is d = 2.…”

“…As pointed out in [6], the two-level superprocess X can be thought of as a 'supersuperprocess'. We will consider the occupation-time process Y ≡ {Y(t), t ≥ 0} of the two-level superprocess X, where…”

“…Let X(0) = R ∞ , where R ∞ is the canonical equilibrium measure of the one-level superprocess nontrivial equilibrium state X(∞) for d ≥ 3 (see [3] and [6]). Consider the centered process…”

Degrees of Transience and Recurrence and Hierarchical Random Walks

“…Multilevel branching systems were introduced by Dawson and Hochberg [11] and they have been studied by several authors [8,9,12,14,19,21,22,34]. In addition to the individual particle branching there is an independent branching of families of related particles (2-level branching), and this idea can be extended to higher levels of branching.…”