## Abstract

A general model of a branching random walk in R^{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 {Mathematical expression} to be finite. Here, X_{n, k} is the position of the k^{th} particle in the n^{th} generation, N_{n} is the number of particles in the n^{th} generation (regardless of their type). It turns out that the distribution of X^{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.

Original language | English (US) |
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Pages (from-to) | 607-634 |

Number of pages | 28 |

Journal | Communications In Mathematical Physics |

Volume | 167 |

Issue number | 3 |

DOIs | |

State | Published - Feb 1995 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics