2017/08/15 by Yan-Xia Ren, Renming Song, Ren, Yan-Xia +3 · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1708.04422
arxiv created 2018/10/18 · arxiv updated 2018/10/19
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, J. Appl. Probab. 46 (2009), 479--496) that M_∞ is non-degenerate iff the Llog L condition is satisfied. In this paper we are mainly interested in the case when the Llog L condition is not satisfied. We prove that, under some conditions, there exist 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,\qquada.s.-ℙμ. We also give the almost sure limit of γt⟨ f,Xt⟩ for a class of general test functions f.