Asymptotic behaviour of measure-valued critical branching processes

Author:
Alison M. Etheridge

Journal:
Proc. Amer. Math. Soc. **118** (1993), 1251-1261

MSC:
Primary 60J80; Secondary 60G57, 60J60

DOI:
https://doi.org/10.1090/S0002-9939-1993-1100650-X

MathSciNet review:
1100650

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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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1100650-X

Keywords:
Measure-valued diffusions,
critical branching process,
diffusion approximation,
asymptotic distribution

Article copyright:
© Copyright 1993
American Mathematical Society