Resumen: We study the long time behavior for the distribution of a critical birth and death diffusion process, motivated by population dynamics in changing environment (cf. a recent paper by Calvez, Henry, Méléard, Tran). The birth rates are bounded but death rates are unbounded. Our analysis is based on the spectral properties of the associated Feynman Kac semigroup. We require a standard spectral gap property for this semigroup with a dominant eigenfunction vanishing at infinity. Some examples of diffusions, diffusions with jump, pure jump dynamics are given for which it is true. We consider situations where the underlying diffusion process doesn’t come down rapidly from infinity but the compactness properties follow from the divergence of the death rate at infinity.
We prove the convergence in law of the branching diffusion process suitably normalized and conditioned to non-extinction. We also prove the existence of the $Q$-process. The main tool is the convergence of suitably normalized moments of the process, which follows from recursive relations for these moments.