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The conditional Feynman-Kac functional is used to derive the Laplace transforms of conditional maximum distributions of processes related to third- and fourth-order equations. These distributions are then obtained explicitly and are expressed in terms of stable laws and the fundamental solutions of these higher-order equations. Interestingly, it is shown that in the third-order case, a genuine non-negative real-valued probability distribution is obtained. (C) 2000 Elsevier Science B.V. All rights reserved

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We consider a dynamic multilevel population or information system. At each level individuals or information units undergo a Galton–Watson-type branching process in which they can be replicated or removed. In addition, a collection of individuals or information units at a given level constitutes an information unit at the next higher level. Each collection of units also undergoes a Galton–Watson branching process, either dying or replicating. In this paper, we represent this multilevel branching model as a measure-valued stochastic process, study its moment structure, identify the limiting continuous-state approximation and analyse the long-time behavior in both non-critical and critical cases. For example, we obtain an asymptotic expression for the extinction probability for the total population mass process and an analogue of Yaglom's conditioned limit theorem in the critical case.