Branching random walk in Z^4 with branching at the origin only

For the critical branching random walk in $\mathbb{Z}^{4}$ with branching at the origin only we find the asymptotic behavior of the probability of the event that there are particles at the origin at moment $t\rightarrow \infty$ and prove a Yaglom type conditional limit theorem for the number of individuals at the origin given that there are particles at the origin.

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