We consider a modified 2-type Markov branching processes incorporating with both immigration and instantaneous resurrection. We prove that if the sum of the immigration rates is finite then no such structure can exist. The existence criterion is presented for the case that the sum of the resurrection rates is infinite. We construct some equivalent criteria that are far easier to verify. The recurrence and ergodicity criterion of the researching process are analysed in detail. The explicit expression for the equilibrium distribution is also presented.

中文翻译：

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具有迁移和瞬时复活的2型马尔可夫分支过程的一些性质

我们考虑一种结合了移民和瞬时复活的修改后的 2 型马尔可夫分支过程。我们证明，如果移民率的总和是有限的，那么就不可能存在这样的结构。针对复活率之和为无穷大的情况提出存在准则。我们构建了一些更容易验证的等效标准。详细分析了研究过程的递归性和遍历性准则。还给出了平衡分布的显式表达式。