Supercritical superprocesses: Proper normalization and non-degenerate strong limit

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（8）
• 页码范围: 1519-1552
• 页数: 34页
• 语种: 英文

摘要

Suppose that X ={Xt,t ≥ 0;Pμ} is a supercritical superprocess in a locally compact separable metric space E.Let φ0 be a positive eigenfunction corresponding to the first eigenvalue λ0 of the generator of the mean semigroup of X.Then Mt:=e-λ0t<φ0,Xt> is a positive martingale.Let M∞ be the limit of Mt.It is known (see Liu et al.(2009)) that M∞ is non-degenerate if and only if the L log L condition is satisfied.In this paper we are mainly interested in the case when the L log L condition is not satisfied.We prove that,under some conditions,there exist a positive function γt on [0,∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E,limt→∞ γt<φ0,Xt> =W,a.s.-PμWe also give the almost sure limit of γt<f,Xt> for a class of general test functions f.