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Branching random motions,nonlinear hyperbolic systems and travellind waves

Published online by Cambridge University Press:  03 May 2006

Nikita Ratanov
Nikita Ratanov*
Affiliation:
Faculty of Economics, Rosario University, Cl. 14, No. 4-69, Bogotá, Colombia; [email protected]
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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.

Keywords

Branching random motiontravelling waveFeynman-Kac connectionnon-linear hyperbolic systemMcKean solution.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 236 - 257
Copyright
© EDP Sciences, SMAI, 2006

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References

K.B. Athreya and P.E. Ney, Branching processes. Dover Publ. Inc. Mineola, NY (2004).
Beghin, L., Nieddu, L. and Orsingher, E., Probabilistic analysis of the telegrapher's process with drift by mean of relativistic transformations. J. Appl. Math. Stoch. Anal. 14 (2001) 11–25. CrossRef
Bramson, M., Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978) 531–581. CrossRef
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (1983) iv+190.
Chauvin, B. and Rouault, A., Supercritical branching Brownian motion and K-P-P equation in the critical speed-are. Math. Nachr. 19 (1990) 41–59. CrossRef
Di Crescenzo, A. and Pellerey, F., On prices' evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry 18 (2002) 171–184. CrossRef
Dunbar, S.R., A branching random evolution and a nonlinear hyperbolic equation. SIAM J. Appl. Math. 48 (1988) 1510–1526. CrossRef
Dunbar, S.R. and Othmer, H.G., On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks, in Nonlinear oscillations in biology and chemistry, (Salt Lake City, Utah, 1985). Lect. Notes Biomath. 66 (1986) 274–289. CrossRef
Fisher, R.A., The advance of advantageous genes. Ann. Eugenics 7 (1937) 335–369.
Fort, J. and Mendez, V., Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment. Rep. Prog. Phys. 65 (2002) 895–954. CrossRef
Goldstein, S., On diffusion by discontinuous movements and on the telegraph equation. Quart. J. Mech. Apl. Math. 4 (1951) 129–156. CrossRef
K.P. Hadeler, Nonlinear propagation in reaction transport systems. Differential equations with applications to biology, Halifax, NS, 1997, Fields Inst. Commun. 21 Amer. Math. Soc., Providence, RI (1999) 251–257.
Hadeler, K.P., Reaction transport systems in biological modelling, In Mathematics inspiring by biology. Lect. Notes in Math. 1714 (1999) 95–150. CrossRef
Hadeler, K.P., Hillen, T. and Lutscher, F., The Langevin or Kramer approach to biological modelling. Math. Mod. Meth. Appl. Sci. 14 (2004) 1561–1583. CrossRef
Hillen, T. and Othmer, H.G., The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61 (2000) 751–775. H.G. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations. SIAM J. Appl. Math. 62, (2002) 1222–1250.
Horsthemke, W., Spatial instabilities in reaction random walks with direction-independent kinetics. Phys. Rev. E 60 (1999) 2651–2663. CrossRef
Horsthemke, W., Fisher waves in reaction random walks. Phys. Lett. A 263 (1999) 285–292. CrossRef
Joseph, D.D. and Preziosi, L., Heat waves. Rev. Mod. Phys. 61 (1989) 41–73. CrossRef
Joseph, D.D. and Preziosi, L., Addendum to the paper “Heat waves”. Rev. Mod. Phys. 62 (1990) 375–391. CrossRef
M. Kac, Probability and related topics in physical sciences. Interscience, London (1959).
Kac, M., Stochastic, A model related to the telegraph equation. Rocky Mountain J. Math. 4 (1974) 497–509. CrossRef
Kolmogorov, A., Petrovskii, I. and Piskunov, N., Étude de l'équation de la diffusion avec croissance de la quantité de la matière et son application à un problème biologique. Bull. Math. 1 (1937) 1–25.
Lyne, O.D., Travelling waves for a certain first-order coupled PDE system. Electronic J. Prob. 5 (2000) 1–40. CrossRef
Lyne, O.D. and Williams, D., Weak solutions for a simple hyperbolic system. Electronic J. Prob. 6 (2001) 1–21. CrossRef
H.P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. XXVIII (1975) 323–331.
H.P. McKean, Correction to above. Comm. Pure Appl. Math. XXIX (1976) 553–554.
S. Mizohata, The theory of partial differential equations. Cambridge University Press, New York (1973) xii+490.
Mendez, V. and Camacho, J., Dynamics and thermodynamics of delayed population growth. Phys. Rev. E 55 (1997) 6476–6482. CrossRef
Mendez, V. and Compte, A., Wavefronts in bistable hyperbolic reaction-diffusion systems. Physica A 260 (1998) 90–98. CrossRef
M. Nagasawa, Schrödinger equations and diffusion theory. Monographs in Mathematics, Birkhäuser Verlag, Basel 86 (1993) pp. x+319.
Orsingher, E., Probability law, flow function, maximum distribution of wave governed random motions and their connections with Kirchoff's laws. Stoch. Proc. Appl. 34 (1990) 49–66. CrossRef
Othmer, H.G., Dunbar, S.R. and Alt, W., Models of dispersal in biological systems. J. Math. Biol. 26 (1988) 263–298. CrossRef
M. Pinsky, Lectures on random evolution. World Scient. Publ. Co., River Edge, NY (1991).
Ratanov, N.E., Telegraph processes with reflecting and absorbing barriers in inhomogeneous media. Theor. Math. Phys. 112 (1997) 857–865. CrossRef
Ratanov, N., Reaction-advection random motions in inhomogeneous media. Physica D 189 (2004) 130–140. CrossRef
Ratanov, N., Pricing options under telegraph processes. Rev. Econ. Ros. 8 (2005) 131–150.
A.I. Volpert, V.A. Volpert and Vl.A. Volpert, Travelling wave solutions of parabolic systems. Translated from the Russian manuscript by James F. Heyda. Translations of Mathematical Monographs. 140 Amer. Math. Soc. Providence, RI, (1994) pp. xii+448.
G.H. Weiss, Aspects and applications of the random walk. North-Holland, Amsterdam (1994).
Weiss, G.H., Some applications of persistent random walks and the telegrapher's equation. Physica A 311 (2002) 381–410. CrossRef