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

M. Ya Kelbert, Yu M. Suhov

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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.

Original languageEnglish (US)
Pages (from-to)607-634
Number of pages28
JournalCommunications In Mathematical Physics
Issue number3
StatePublished - Feb 1995

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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