Local properties for 1-dimensional critical branching Lévy process

Haojie Hou, Yan-Xia Ren and Renming Song

Stochastic Processes and their Applications, 2026, vol. 192, issue C

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).

Keywords: Critical branching Lévy process; Lévy process; Super-Brownian motion; Feynman–Kac formula (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2025.104834

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