Abstract
A general model of a branching random walk inR 1 is considered, with several types of particles, where the branching occurs with probabilities determined by the type of a parent particle. Each new particle starts moving from the place where it was born, independently of other particles. The distribution of the displacement of a particle, before it splits, depends on its type. A necessary and sufficient condition is given for the random variable
to be finite. Here,X n, k is the position of thek th particle in then th generation,N n is the number of particles in then th generation (regardless of their type). It turns out that the distribution ofX 0 gives a minimal solution to a natural system of stochastic equations which has a linearly ordered continuum of other solutions. The last fact is used for proving the existence of a monotone travelling-wave solution to systems of coupled non-linear parabolic PDE's.
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Communicated by Ya.G. Sinai
This research was supported in part by the Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 5545, USA, and the EC Grant “Human Capital and Mobility”, No 16296 (Contract No CHRX-CT 93-0411).
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Kelbert, M.Y., Suhov, Y.M. The Markov branching random walk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov) equations. Commun.Math. Phys. 167, 607–634 (1995). https://doi.org/10.1007/BF02101538
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DOI: https://doi.org/10.1007/BF02101538