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.
Mots clés : branching random motion, travelling wave, Feynman-Kac connection, non-linear hyperbolic system, McKean solution
@article{PS_2006__10__236_0, 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://www.numdam.org/articles/10.1051/ps:2006009/} }
TY - JOUR AU - Ratanov, Nikita TI - Branching random motions, nonlinear hyperbolic systems and travelling waves JO - ESAIM: Probability and Statistics PY - 2006 SP - 236 EP - 257 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006009/ DO - 10.1051/ps:2006009 LA - en ID - PS_2006__10__236_0 ER -
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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