Criterion on the limits of superprocesses
The author considers superprocesses with an underlying process which is a diffusion on \(\mathbb{R}^d\) satisfying certain natural conditions, and a branching function of the form \[ \Phi(z)= bz+cz^2+ \int^\infty_0 (e^{-zu}-1+zu) n(du), \] where \(c\geq 0\), \(b\in\mathbb{R}\) and \(n\) is a measure on \((0,\infty)\) such that \(\int^\infty_0 u\wedge u^2n(du)<\infty\). The objective of the paper is to study the behavior of such superdiffusions as time \(t\to\infty\), mainly the usual problems of extinction, persistence and equilibrium states in the case of infinite measure values. These problems have been studied extensively for branching functions of the form \(\Phi(z)=c z^{1+\beta}\), \(c>0\), \(0<\beta\leq 1\) [see, e.g. the reviewer and \textit{A. Wakolbinger}, Ann. Probab. 19, No. 1, 266-288 (1991; Zbl 0732.60093)], \textit{M. Bramson}, \textit{J. T. Cox} and \textit{A. Greven}, ibid. 25, No. 1, 56-70 (1997; Zbl 0882.60091) and \textit{A. Klenke}, ibid. 26, 1057-1087 (1998)] using several methods (e.g. Campbell measures and backward trees). The author employs a different method. Assuming that the Lebesgue measure \(m\) is invariant for the underlying diffusion, and starting the superdiffusion from \(m\), it is shown that the superdiffusion has a nontrivial equilibrium state (resp. has only the zero equilibrium state) if \(b<0\) (resp. \(b>0)\). If \(b=0\) and in addition \(\int^\infty_0u^2n(du)<\infty\), then extinction (resp. persistence) holds when \(d<3\) (resp. \(d\geq 3)\).
scientific article; zbMATH DE number 1207972
scientific article