Wasserstein-type distances of two-type continuous-state branching processes in L\'{e}vy random environments

TL;DR

This paper establishes exponential ergodicity in Wasserstein and total variance distances for two-type continuous-state branching processes in Lévy environments, providing explicit parameter expressions and employing coupling and superprocess techniques.

Contribution

It introduces new ergodicity results for multi-type branching processes in Lévy environments, with explicit parameter characterization and novel methodological approaches.

Findings

Proved exponential ergodicity in Wasserstein distance.

Derived explicit parameters for the exponential rate.

Established ergodicity in total variance distance.

Abstract

Under natural conditions, we proved the exponential ergodicity in Wasserstein distance of two-type continuous-state branching processes in L\'evy random environments with immigration. Furthermore, we expressed accurately the parameters of the exponent. The coupling method and the conditioned branching property play an important role in the approach. Using the tool of superprocesses, the ergodicity in total variance distance is also proved.