Article,
Growing conditioned trees
Stochastic Process. Appl., 39 (1): 117--130 (1991)
DOI: 10.1016/0304-4149(91)90036-C
Abstract
MR: Consider a Markov branching particle system in $R^d$Rd. Let $\Phi$Φ denote the random genealogical tree generated up to time $t$t, and $Z_t$Zt the random population at time $t$t. The object of interest in this paper is the conditional distribution of $\Phi$Φ given that $Z_t$Zt populates a given site. More precisely, denoting by $P$P the distribution of $\Phi$Φ, the aim is to compute the disintegration of the joint measure $P(d\Phi)Z_t(dy)$P(dΦ)Zt(dy) with respect to its second marginal (assuming that $Z_t$Zt a.s. has no multiple points). The problem is solved in two different ways. One uses "tree methods'' developed by Neveu and Chauvin. The other uses Kallenberg's backward technique. Both methods require growing "Palm trees'' for the individual occupying a given site. These "Palm trees'' are Palm-type distributions on the genealogical trees. The results are obtained first under the assumption of spatially homogeneous branching, and then extended to the inhomogeneous case.
