Yaglom theorem for critical branching random walk on ${\mathbb{Z}}^d$

We study the critical branching random walk on ${\mathbb{Z}}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in ${\mathbb{Z}}^d$. We are interested in the occupation time in $K$ and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\mathrm{\parallel }x{\mathrm{\parallel }}^{4-d}$ in dimensions $d\le 3$, of order $\log \mathrm{\parallel }x\mathrm{\parallel }$ in dimension $d=4$, and of order 1 in dimensions $d\ge 5$. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265).