Continuous Time Mixed State Branching Processes and Stochastic Equations

TL;DR

This paper introduces a new class of continuous-time mixed state branching processes derived from Galton-Watson processes, analyzes their stochastic equations, and explores their ergodic and stationary properties.

Contribution

It constructs mixed state branching processes as scaling limits and provides a detailed analysis of their stochastic equations and ergodic behavior.

Findings

Derived the distribution of local jumps.

Proved exponential ergodicity in Wasserstein distances.

Established existence of stationary distribution with immigration.

Abstract

A continuous time mixed state branching process is constructed as the scaling limits 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 the exponential ergodicity in Wasserstein-type distances of the transition semigroup is given. Meanwhile, we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.