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Branching random motions, nonlinear hyperbolic systems and travelling waves

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Ratanov, Nikita

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2006

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Abstract
A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independent of random motion, and intensities of reverses are defined by a particle’s current direction. A solution of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) has a so-called McKean representation via such processes. Commonly this system possesses travelling-wave solutions. The convergence of solutions with Heaviside terminal data to the travelling waves is discussed. The paper realizes the McKean’s program for the Kolmogorov-Petrovskii-Piskunov equation in this case. The Feynman-Kac formula plays a key role.
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Branching random motion , Travelling wave , Feynman-Kac connection , Non-linear hyperbolic system , Mckean solution
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