LIMITING BEHAVIOR FOR AGE AND POSITIONDEPENDENT BRANCHING PROCESSES.
Technical summary rept.,
WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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In this paper a model is studied for the population transition probabilities for a branching process composed of particles diffusing in a finite interval. The model is in general non-Markovian assuming the branching transformation probabilities for a particle depend on its age and position. The process is described by the random number N sub t x of particles in the interval I at time t that are generated by a single particle initially at the point x in I. By considering N sub t x as a regenerative process with respect to the random age and position of the initial particle when it is transformed, a functional equation is developed for the generating function for N sub t x. This functional equation is the basis for the study of the population probabilities PN sub t x n, n 0,1,2, ... , as function of x in I and t in 0, infinity. The principal results develop the behavior of N sub t x for large t and the dependence of such behavior on x. A convergence is established in distribution for those processes for which N sub t x can increase without bound with positive probability as t approaches infinity. Author