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

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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
Keywords: branching random motion, travelling wave, Feynman-Kac connection, non-linear hyperbolic system, McKean solution 
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     author = {Ratanov, Nikita},
     title = {Branching random motions, nonlinear hyperbolic systems and travelling waves},
     journal = {ESAIM: Probability and Statistics},
     pages = {236--257},
     publisher = {EDP-Sciences},
     volume = {10},
     year = {2006},
     doi = {10.1051/ps:2006009},
     mrnumber = {2219342},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2006009/}
}
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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

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