Supercritical superprocesses: Proper normalization and non-degenerate strong limit
Supercritical superprocesses: Proper normalization and non-degenerate strong limit
复制标题
超临界超过程：适当的归一化和非简并强极限
DOI：
10.1007/s11425-018-9402-4
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Zhang Rui
中科院分区：
文献类型：
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作者：
Ren Yan Xia;Song Renming;Zhang Rui
设X = {Xt，t <$0;<$μ}是局部紧可分度量空间E中的超临界超过程.设φ 0是X的平均半群的生成元的第一特征值λ 0所对应的正特征函数。这是一个正鞅。设M ∞是Mt的极限。已知（参见Liu et al.（2009））M ∞是非退化的当且仅当满足LlogL条件。在本文中，我们主要感兴趣的情况下，LlogL条件不满足。本文证明了在一定条件下，存在正函数γ t [0，∞）和非退化随机变量W，使得对任意有限非零Borel测度μonE，我们还给出了一类一般检验函数f的γt<f，Xt>的几乎必然极限.
Suppose thatX= {Xt,t⩾ 0;ℙμ} is a supercritical superprocess in a locally compact separable metric spaceE. Letφ0be a positive eigenfunction corresponding to the first eigenvalue λ0of the generator of the mean semigroup ofX. Thenis a positive martingale. LetM∞be the limit ofMt. It is known (see Liu et al. (2009)) thatM∞is non-degenerate if and only if theLlogLcondition is satisfied. In this paper we are mainly interested in the case when theLlogLcondition is not satisfied. We prove that, under some conditions, there exist a positive function γton [0, ∞) and a non-degenerate random variableWsuch that for any finite nonzero Borel measureμonE,We also give the almost sure limit of γt<f, Xt> for a class of general test functionsf.