Continuous Time Mixed State Branching Processes and Stochastic Equations

A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes. The process can also be obtained by the pathwise unique solution to a stochastic equation system. From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup. Meanwhile, we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.

Automated summary

In this study, a continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes. The distribution of local jumps is derived from the stochastic equation system, and exponential ergodicity in Wasserstein-type distances of the transition semigroup is established. Immigration structures associated with the process are also investigated, along with the existence of the stationary distribution of the process with immigration.