Branching random motions, nonlinear hyperbolic systems and travellind waves
Faculty of Economics, Rosario University,
Cl. 14, No. 4-69, Bogotá, Colombia;
Revised: 20 August 2005
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.
Mathematics Subject Classification: 35L60 / 60J25 / 60J80 / 60J85
Key words: Branching random motion / travelling wave / Feynman-Kac connection / non-linear hyperbolic system / McKean solution.
© EDP Sciences, SMAI, 2006