Local properties for $1$-dimensional critical branching Lévy process

Consider a one dimensional critical branching Lévy process $((Z_t{)}_{t\ge 0},{\mathbb{P}}_x)$. 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 Lévy process $({\xi }_t{)}_{t\ge 0}$ is non-lattice and has finite $2+{\delta }^{\ast }$ moment for some ${\delta }^{\ast }>0$. We first prove that $$t^{\frac{1}{\alpha -1}}(1-{\mathbb{E}}_{\sqrt{t}y}(\exp \{-\frac{1}{t^{\frac{1}{\alpha -1}-\frac{1}{2}}}\int h(x)Z_t(\mathrm{d}x)-\frac{1}{t^{\frac{1}{\alpha -1}}}\int g\left(\frac{x}{\sqrt{t}}\right)Z_t(\mathrm{d}x)\}))$$ converges as $t\to \mathrm{\infty }$ for any non-negative bounded Lipschtitz function $g$ and any non-negative directly Riemann integrable function $h$ of compact support. Then for any $y\in \mathbb{R}$ 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 ${\mathbb{P}}_{\sqrt{t}y}(\cdot \mathrm{\mid }{Z}_t(A)>0)\mathrm{.}$