Independent random cascades on Galton-Watson trees

Consider an independent random cascade acting on the positive Borel measures defined on the boundary of a Galton-Watson tree. Assuming an offspring distribution with finite moments of all orders, J. Peyriere computed the fine scale structure of an independent random cascade on Galton-Watson trees. In this paper we use developments in the cascade theory to relax and clarify the moment assumptions on the offspring distribution. Moreover a larger class of initial measures is covered and, as a result, it is shown that it is the Holder exponent of the initial measure which is the critical parameter in the Peyriere theory. We start by defining the space of Galton-Watson trees and introducing some related terminology. 1. GALTON-WATSON TREES Let Tr be the space of labelled tree graphs rooted at 4. An element r of ?7 is a set of finite sequences of positive integers (v1, v2,... , Vn) E r such that: (i) X5E r is coded as the empty sequence. (ii) If (Vl,...,Vk) ET, then (vl,...,vi) ETV 1 1. The countable dense subset To of finite labelled tree graphs rooted at 0 makes Tr a Polish space. An important class of probability distributions on the Borel sigma field B(T) of Tr for this paper is the Galton-Watson distribution with single progenitor and offspring distribution Pk, k = 0, ,.. ., for which the probability Received by the editors May 14, 1998 and, in revised form, October 8, 1998. 2000 Mathematics Subject Classification. Primary 60G57, 60G30, 60G42; Secondary 60K35. @)2000 American Mathematical Society