TY - JOUR

T1 - Supercritical superprocesses

T2 - Proper normalization and non-degenerate strong limit

AU - Ren, Yan Xia

AU - Song, Renming

AU - Zhang, Rui

N1 - Publisher Copyright:
© 2019, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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, Mt:=e-λ0t〈φ0,Xt〉 We also give the almost sure limit of γt〈f, Xt〉 for a class of general test functions f.

AB - 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, Mt:=e-λ0t〈φ0,Xt〉 We also give the almost sure limit of γt〈f, Xt〉 for a class of general test functions f.

KW - 60F15

KW - 60J68

KW - Seneta-Heyde norming

KW - martingales

KW - non-degenerate strong limit

KW - superprocesses

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U2 - 10.1007/s11425-018-9402-4

DO - 10.1007/s11425-018-9402-4

M3 - Article

AN - SCOPUS:85068909376

SN - 1674-7283

VL - 62

SP - 1519

EP - 1552

JO - Science China Mathematics

JF - Science China Mathematics

IS - 8

ER -