Supercritical superprocesses: Proper normalization and non-degenerate strong limit

• Published in: Science China. Mathematics Vol. 62; no. 8; pp. 1519 - 1552
• Main Authors: Ren, Yan-Xia, Song, Renming, Zhang, Rui
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.08.2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Eigenvalues; Eigenvectors; Martingales; Mathematics; Mathematics and Statistics; Metric space; Random variables; Seneta-Heyde norming; superprocesses; 60J68; non-degenerate strong limit; martingales; 60F15
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-018-9402-4

Suppose that X = { X t , t ⩾ 0;ℙ μ } is a supercritical superprocess in a locally compact separable metric space E . Let φ 0 be a positive eigenfunction corresponding to the first eigenvalue λ 0 of the generator of the mean semigroup of X . Then M t := e − λ 0 t 〈 ϕ 0 , X t 〉 is a positive martingale. Let M ∞ be the limit of M t . 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 γ t on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E , M t := e - λ 0 t 〈 φ 0 , X t 〉 We also give the almost sure limit of γ t 〈 f, X t 〉 for a class of general test functions f .