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

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-Lévy processes as well as branching particle systems on the lattice with noncompact spins.

References in corpus (6)

• Super-Brownian motion as the unique strong solution to an SPDE
• Stochastic equation and exponential ergodicity in Wasserstein distances for affine processes
• Affine Jump-Diffusions: Stochastic Stability and Limit Theorems
• Existence of limiting distribution for affine processes
• Stochastic averaging principle for spatial Markov evolutions in the continuum
• Exponential ergodicity for stochastic equations of nonnegative processes with jumps