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Local properties for 1-dimensional critical branching Lévy process

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Listed:
  • Hou, Haojie
  • Ren, Yan-Xia
  • Song, Renming

Abstract

Consider a one dimensional critical branching Lévy process ((Zt)t≥0,Px). Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction of some α-stable distribution with α∈(1,2), and that the underlying Lévy process (ξt)t≥0 is non-lattice and has finite 2+δ∗ moment for some δ∗>0. We first prove that t1α−11−Etyexp−1t1α−1−12∫h(x)Zt(dx)−1t1α−1∫gxtZt(dx)converges as t→∞ for any non-negative bounded Lipschitz function g and any non-negative directly Riemann integrable function h of compact support. Then for any y∈R and bounded Borel set A of positive Lebesgue measure with its boundary having zero Lebesgue measure, under a higher moment condition on ξ, we find the decay rate of the probability Pty(Zt(A)>0). As an application, we prove some convergence results for Zt under the conditional law Pty(⋅|Zt(A)>0).

Suggested Citation

  • Hou, Haojie & Ren, Yan-Xia & Song, Renming, 2026. "Local properties for 1-dimensional critical branching Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:spapps:v:192:y:2026:i:c:s0304414925002789
    DOI: 10.1016/j.spa.2025.104834
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    References listed on IDEAS

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    1. Ren, Yan-Xia & Song, Renming & Zhang, Rui, 2021. "The extremal process of super-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 1-34.
    2. Jean-François Le Gall, 1996. "A probabilistic approach to the trace at the boundary for solutions of a semilinear parabolic partial differential equation," International Journal of Stochastic Analysis, Hindawi, vol. 9, pages 1-16, January.
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