Local extinction versus local exponential growth for spatial branching processes
Local extinction versus local exponential growth for spatial branching processes
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DOI：
10.1214/aop/1078415829
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发表时间：
2004
影响因子：
2.3
通讯作者：
J. Engländer;A. Kyprianou
中科院分区：
文献类型：
--
作者：
J. Engländer;A. Kyprianou
设X是D <$Rd上的算子Lu+β（u2−u）对应的分支扩散[其中β（x）≥0且β <$0是上有界的]或D <$Rd上的算子Lu+βu−αu2对应的超过程（其中α>0且β是上有界的但对其符号没有限制）。设λc表示D上算子L+β的广义主特征值.我们证明了以下二分法：或者λc≤0，X表现出局部绝灭，或者λc>0，D的紧集的质量以速率λc指数增长.对于superdiffusions，这完成了本地灭绝标准获得平斯基[安。24（1996）237- 267]和Englander和Turaev [Ann. Probab. 30（2002）683- 722]。在上述两篇论文中的证明是基于偏微分方程技术，但我们提供的证明是概率概念。在大多数情况下，它们是基于“spine”分解或“不朽粒子表示”，沿着鞅收敛和大数定律。此外，它们是通用的，因为它们适用于两种类型的过程。
Let X be either the branching diffusion corresponding to the operator Lu+β(u2−u) on D⊆ Rd [where β(x)≥0 and β≡0 is bounded from above] or the superprocess corresponding to the operator Lu+βu−αu2 on D⊆ Rd (with α>0 and β is bounded from above but no restriction on its sign). Let λc denote the generalized principal eigenvalue for the operator L+β on D. We prove the following dichotomy: either λc≤0 and X exhibits local extinction or λc>0 and there is exponential growth of mass on compacts of D with rate λc. For superdiffusions, this completes the local extinction criterion obtained by Pinsky [Ann. Probab. 24 (1996) 237--267] and a recent result on the local growth of mass under a spectral assumption given by Englander and Turaev [Ann. Probab. 30 (2002) 683--722]. The proofs in the above two papers are based on PDE techniques, however the proofs we offer are probabilistically conceptual. For the most part they are based on "spine'' decompositions or "immortal particle representations'' along with martingale convergence and the law of large numbers. Further they are generic in the sense that they work for both types of processes.