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Branching random motions, nonlinear hyperbolic systems and traveling waves

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  • Nikita Ratanov

Abstract

A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independient of random motion, and intensities of reverses are defined by a particle's current direction. A soluton of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) have a so-called McKean representation via such processes. Commonly this system possesses traveling-wave solutions. The convergence of solutions with Heaviside terminal data to the travelling waves is discussed.This Paper realizes the McKean programme for the Kolmogorov-Petrovskii-Piskunov equation in this case. The Feynman-Kac formula plays a key role.

Suggested Citation

  • Nikita Ratanov, 2004. "Branching random motions, nonlinear hyperbolic systems and traveling waves," Borradores de Investigación 004331, Universidad del Rosario.
  • Handle: RePEc:col:000091:004331
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    File URL: http://repository.urosario.edu.co/bitstream/handle/10336/11126/4331.pdf
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    References listed on IDEAS

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    1. Weiss, George H, 2002. "Some applications of persistent random walks and the telegrapher's equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 311(3), pages 381-410.
    2. Méndez, Vicenç & Compte, Albert, 1998. "Wavefronts in bistable hyperbolic reaction–diffusion systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 260(1), pages 90-98.
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    Cited by:

    1. Bogachev, Leonid & Ratanov, Nikita, 2011. "Occupation time distributions for the telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1816-1844, August.

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