Branching random motions, nonlinear hyperbolic systems and travelling waves
ESAIM: Probability and Statistics, Tome 10 (2006), pp. 236-257.

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.

DOI : 10.1051/ps:2006009
Classification : 35L60, 60J25, 60J80, 60J85
Mots-clés : branching random motion, travelling wave, Feynman-Kac connection, non-linear hyperbolic system, McKean solution 
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Ratanov, Nikita. Branching random motions, nonlinear hyperbolic systems and travelling waves. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 236-257. doi : 10.1051/ps:2006009. http://www.numdam.org/articles/10.1051/ps:2006009/

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