# Long-time behavior for subcritical measure-valued branching processes with immigration

## TL;DR

This paper investigates the long-term behavior of subcritical measure-valued branching processes with immigration, establishing the existence, uniqueness, and exponential convergence to an invariant measure under certain conditions.

## Contribution

It proves the existence and uniqueness of an invariant measure and demonstrates exponential convergence in Wasserstein and Laplace transform-based distances.

## Key findings

- Existence and uniqueness of invariant measure
- Exponential convergence to invariant measure
- Application to super-Lévy processes and lattice particle systems

## Abstract

In this work we study the long-time behavior for subcritical measure-valued branching processes with immigration on the space of tempered measures. Under some reasonable assumptions on the spatial motion, the branching and immigration mechanisms, we prove the existence and uniqueness of an invariant measure for the corresponding Markov transition semigroup. Moreover, we show that it converges with exponential rate to the unique invariant measure in the Wasserstein distance as well as in a distance defined in terms of Laplace transforms. Finally, we consider an application of our results to super-L\'evy processes as well as branching particle systems on the lattice with noncompact spins.