Asymptotic behaviour of measure-valued critical branching processes

Abstract

Abstract. Measure-valued branching processes can be characterized in terms of the Laplace transform of their transition densities and this gives rise to a second order nonlinear p.d.e.-the evolution equation of the process. We write the solution to this evolution equation as a series, each of whose coefficients is expressed in terms of the linear semigroup corresponding to the spatial part of the measure-valued process. From this we obtain a simple proof that if the spatial part of the process is a recurrent (resp., transient) Markov process on a standard Borel space and the initial value of the process is an invariant measure of this spatial process, then the process has no (resp., has a unique) nontrivial limiting distribution.

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

“…Tempered measures have been first introduced for the study of critical branching Markov processes without immigration (ψ = 0) where the long-time behavior of the Markov process started at the Lebesgue measure was investigated, see Iscoe [17]. Further developments in this direction have been obtained in [3,4,6,9,22] and [23].…”

Non‐parametric Estimation of the Death Rate in Branching Diffusions

“…(16) follows by conditioning on the first branching event in the subfamily under consideration, cf. also Etheridge (1993). See Hö pfner & Lö cherbach (1999) for details concerning (a) and (b) and Hö pfner & Lö cherbach (2000, rem.…”

“…Spatially branching diffusions are fairly well understood from a probabilistic point of view, see e.g. Etheridge (1993), Wakolbinger (1995) and the references therein. However, statistical inference has not reached a unified form yet and only several particular cases of the general model of branching diffusions have been covered.…”