Long-Time Behavior for Subcritical Measure-Valued Branching Processes with Immigration

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