### 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) |
---|---|

Pages (from-to) | 607-634 |

Number of pages | 28 |

Journal | Communications In Mathematical Physics |

Volume | 167 |

Issue number | 3 |

DOIs | |

State | Published - Feb 1 1995 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Communications In Mathematical Physics*, vol. 167, no. 3, pp. 607-634. https://doi.org/10.1007/BF02101538

**The Markov branching random walk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov) equations.** / Kelbert, M. Ya; Suhov, Yu M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Markov branching random walk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov) equations

AU - Kelbert, M. Ya

AU - Suhov, Yu M.

PY - 1995/2/1

Y1 - 1995/2/1

N2 - A general model of a branching random walk in R1 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, Xn, k is the position of the kth particle in the nth generation, Nn is the number of particles in the nth generation (regardless of their type). It turns out that the distribution of X0 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.

AB - A general model of a branching random walk in R1 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, Xn, k is the position of the kth particle in the nth generation, Nn is the number of particles in the nth generation (regardless of their type). It turns out that the distribution of X0 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.

UR - http://www.scopus.com/inward/record.url?scp=21844503971&partnerID=8YFLogxK

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U2 - 10.1007/BF02101538

DO - 10.1007/BF02101538

M3 - Article

AN - SCOPUS:21844503971

VL - 167

SP - 607

EP - 634

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -