Document ESAIM: P&S, April 2006, Vol. 10, pp. 236-257
DOI: 10.1051/ps:2006009
Branching random motions, nonlinear hyperbolic systems and travellind waves
Nikita RatanovFaculty of Economics, Rosario University, Cl. 14, No. 4-69, Bogotá, Colombia; nratanov@urosario.edu.co
(Received April 25, 2005. Revised August 20, 2005. / Published online: 3 May 2006)
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.
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