A super-stable motion with infinite mean branching

Abstract

A class of finite measure-valued càdlàg superprocesses X with Neveu’s (1992) continuous-state branching mechanism is constructed. To this end, we start from certain supercritical (α,d,β)-superprocesses X(β) with symmetric α-stable motion and (1+β)-branching and prove convergence on path space as β ↓ 0. The log-Laplace equation related to X has the locally non-Lipschitz function ulog u as non-linear term (instead of u1+β in the case of X(β)). It can nevertheless be shown to be well-posed. X has infinite expectation, is immortal in all finite times, propagates mass instantaneously everywhere in space, and has locally countably infinite biodiversity.