1. Introduction.
Consider a Markov process Y taking values in a Polish space,
M M
E. For M€ IN, K(M) particles start at xx (u) , . . . ,xR,.- . (J) E and
follow independent copies of Y on the time interval [0,1/M). At t
= 1/M each particle dies or splits into two, each with probability
1/2, independently of each other- The new individuals then follow
independent copies of Y on [L/M,2/M), and this pattern of
alternating spatial motions and branching mechanisms continues
indefinitely. In this way we construct a random "Y-tree" of
branching particles. Define a random measure on (E,&) (& is the
Borel cr-field) by
X^(u)(A) = M"1(no. of particles in A at time t), Ae&, t^O.
is a process taking values in the space M„(E) of finite
measures on E with the topology of weak convergence. Watanabe
(1968) (see Ethier-Kurtz (1986, Ch. 9) for the required tightness)
showed that if Y is a Feller process and E is locally compact,
then X^ —- mQ (in Mp(E)) implies XM ^-X on D( [0,«) ,Mp(E) ) (weak
convergence of laws), where X is a M^fE)-valued diffusion starting
r
at mn. Let Q denote the limiting law on the appropriate space
Research of both authors partially supported by the Natural
Sciences and Engineering Research Council of Canada.
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