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

M. Ya Kelbert, Yu M. Suhov

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

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
Volume167
Issue number3
DOIs
StatePublished - Feb 1 1995

Fingerprint

Branching Random Walk
Random Systems
Reaction-diffusion
random walk
Minimal Solution
pulse detonation engines
Nonlinear Parabolic Equations
random variables
Traveling Wave Solutions
traveling waves
Stochastic Equations
Branching
Monotone
Continuum
Random variable
Linearly
continuums
Necessary Conditions
Sufficient Conditions

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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

In: Communications In Mathematical Physics, Vol. 167, No. 3, 01.02.1995, p. 607-634.

Research output: Contribution to journalArticle

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