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.
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