Clustering and invariant measures for spatial branching models with infinite variance

• 成果类型: Article
• 署名作者: Klenke, A
• 署名单位: University of Erlangen Nuremberg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1998
• 页码: 1057-1087
• 关键词: particle-systems; persistence; EQUATIONS; limit; time

摘要

We consider two spatial branching models on R-d: branching Brownian motion with a branching law in the domain of normal attraction of a (1 + beta) stable law, 0 < beta less than or equal to 1, and the corresponding high density limit measure valued diffusion. The longtime behavior of both models depends highly on beta and d. We show that for d less than or equal to 2/beta the only invariant measure is delta(0), the unit mass on the empty configuration. Furthermore, we give a precise condition for convergence toward delta(0). For d > 2/beta it is known that there exists a family (nu(theta), theta is an element of [0, infinity)) of nontrivial invariant measures. We show that every invariant measure is a convex combination of the ya. Both results have been known before only under an additional finite mean assumption. For the critical dimension d = 2/beta we show that both models display the phenomenon of diffusive clustering. This means that clusters grow spatially on a random scale. We give a precise description of the clusters via multiple scale analysis. Our methods rely mainly on studying sub- and supersolutions of the reaction diffusion equation partial derivative u/partial derivative t - 1/2 Delta u + u(1+beta) = 0.