Local properties for $1$-dimensional critical branching L\'{e}vy process

TL;DR

This paper investigates the asymptotic properties of a one-dimensional critical branching Lévy process, establishing convergence results and decay rates under specific moment and distribution conditions.

Contribution

It provides new limit theorems and decay rate analyses for the process, extending understanding of its local properties under various offspring and Lévy process conditions.

Findings

Convergence of a scaled expectation involving the process as time tends to infinity.

Decay rate of the probability that the process hits a set, under higher moment conditions.

Convergence results for the process conditioned on hitting a set.

Abstract

Consider a one dimensional critical branching L\'{e}vy process $((Z_t)_{t\geq 0}, \mathbb {P}_x)$. Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction to some $\alpha$-stable distribution with $\alpha\in (1, 2)$, and that the underlying L\'{e}vy process $(\xi_t)_{t\geq 0}$ is non-lattice and has finite $2+\delta^*$ moment for some $\delta^*>0$. We first prove that $$t^{\frac{1}{\alpha-1}}\left(1- \mathbb{E}_{\sqrt{t}y}\left(\exp\left\{-\frac{1}{t^{\frac{1}{\alpha-1}-\frac{1}{2}}}\int h(x) Z_t(\mathrm{d}x) -\frac{1}{t^{\frac{1}{\alpha-1}}} \int g\left(\frac{x}{\sqrt{t}}\right)Z_t(\mathrm{d}x)\right\}\right)\right)$$ converges as $t\to\infty$ for any non-negative bounded Lipschtitz function $g$ and any non-negative directly Riemann integrable function $h$ of compact support. Then for any $y\in \R$ and bounded Borel set of positive Lebesgue measure with its boundary having zero Lebesgue measure, under a higher moment condition on $\xi$, we find the decay rate of the probability $\mathbb {P}_{\sqrt{t}y}(Z_t(A)>0)$. As an application, we prove some convergence results for $Z_t$ under the conditional law $\mathbb {P}_{\sqrt{t}y}(\cdot| Z_t(A)>0).$