Asymptotic expansion for branching killed Brownian motion with drift

Let $Z_t^{(0,\mathrm{\infty })}$ be the point process formed by the positions of all particles alive at time $t$ in a branching Brownian motion with drift and killed upon reaching 0. We study the asymptotic expansions of $Z_t^{(0,\mathrm{\infty })}(A)$ for $A=(a,b)$ and $A=(a,\mathrm{\infty })$ under the assumption that ${\sum }_{k=1}^{\mathrm{\infty }}k(\log k{)}^{1+\lambda }{p}_k<\mathrm{\infty }$ for large $\lambda$ in the regime of $\theta \in [0,\sqrt{2})$. These results extend and sharpen the results of Louidor and Saglietti [J. Stat. Phys, 2020] and Kesten [Stochastic Process. Appl., 1978].