The Properties of Quadratic Weighted Markov Branching Processes with Immigration and Instantaneous Resurrection

We consider quadratic weighted branching processes with immigration and instantaneous resurrection. Under the condition that the resurrection rate can be summed, we show that the target process does not exist. The existence and uniqueness criterion for the process are obtained under the assumption that the sum of the resurrection is infinite. Also, the equivalent conditions for the existence criterion are given for easy verification. It is proved that there exist infinitely many of Q processes. Among them, there exists a unique honest process and the corresponding construction method is then investigated. We prove that this honest process is always ergodic and the second-order differential equation of the equilibrium distribution is established.