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

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 : https://doi.org/10.1051/ps:2006009
Classification : 35L60,  60J25,  60J80,  60J85
Mots clés : branching random motion, travelling wave, Feynman-Kac connection, non-linear hyperbolic system, McKean solution
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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. http://www.numdam.org/articles/10.1051/ps:2006009/

[1] K.B. Athreya and P.E. Ney, Branching processes. Dover Publ. Inc. Mineola, NY (2004). | MR 2047480 | Zbl 1070.60001

[2] L. Beghin, L. Nieddu and E. Orsingher, Probabilistic analysis of the telegrapher's process with drift by mean of relativistic transformations. J. Appl. Math. Stoch. Anal. 14 (2001) 11-25. | Zbl 0986.60096

[3] M. Bramson, Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 (1978) 531-581. | Zbl 0361.60052

[4] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (1983) iv+190. | MR 705746 | Zbl 0517.60083

[5] B. Chauvin and A. Rouault, Supercritical branching Brownian motion and K-P-P equation in the critical speed-are. Math. Nachr. 19 (1990) 41-59. | Zbl 0724.60091

[6] A. Di Crescenzo and F. Pellerey, On prices' evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry 18 (2002) 171-184. | Zbl 1011.91041

[7] S.R. Dunbar, A branching random evolution and a nonlinear hyperbolic equation. SIAM J. Appl. Math. 48 (1988) 1510-1526. | Zbl 0664.60082

[8] S.R. Dunbar and H.G. Othmer, 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. | Zbl 0592.92003

[9] R.A. Fisher, The advance of advantageous genes. Ann. Eugenics 7 (1937) 335-369. | JFM 63.1111.04

[10] J. Fort and V. Mendez, Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment. Rep. Prog. Phys. 65 (2002) 895-954.

[11] S. Goldstein, On diffusion by discontinuous movements and on the telegraph equation. Quart. J. Mech. Apl. Math. 4 (1951) 129-156. | Zbl 0045.08102

[12] 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. | Zbl 0918.35066

[13] K.P. Hadeler, Reaction transport systems in biological modelling, In Mathematics inspiring by biology. Lect. Notes in Math. 1714 (1999) 95-150. | Zbl 1002.92506

[14] K.P. Hadeler, T. Hillen and F. Lutscher, The Langevin or Kramer approach to biological modelling. Math. Mod. Meth. Appl. Sci. 14 (2004) 1561-1583. | Zbl 1057.92012

[15] T. Hillen and H.G. Othmer, 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. | Zbl 1103.35098

[16] W. Horsthemke, Spatial instabilities in reaction random walks with direction-independent kinetics. Phys. Rev. E 60 (1999) 2651-2663.

[17] W. Horsthemke, Fisher waves in reaction random walks. Phys. Lett. A 263 (1999) 285-292. | Zbl 0940.82030

[18] D.D. Joseph and L. Preziosi, Heat waves. Rev. Mod. Phys. 61 (1989) 41-73. | Zbl 1129.80300

[19] D.D. Joseph and L. Preziosi, Addendum to the paper “Heat waves”. Rev. Mod. Phys. 62 (1990) 375-391. | Zbl 1129.80300

[20] M. Kac, Probability and related topics in physical sciences. Interscience, London (1959). | MR 102849 | Zbl 0087.33003

[21] M. Kac, A Stochastic model related to the telegraph equation. Rocky Mountain J. Math. 4 (1974) 497-509. | Zbl 0314.60052

[22] A. Kolmogorov, I. Petrovskii and N. Piskunov, 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. | Zbl 0018.32106

[23] O.D. Lyne, Travelling waves for a certain first-order coupled PDE system. Electronic J. Prob. 5 (2000) 1-40. | Zbl 0954.35105

[24] O.D. Lyne and D. Williams, Weak solutions for a simple hyperbolic system. Electronic J. Prob. 6 (2001) 1-21. | Zbl 0984.35101

[25] H.P. Mckean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. XXVIII (1975) 323-331. | Zbl 0316.35053

[26] H.P. Mckean, Correction to above. Comm. Pure Appl. Math. XXIX (1976) 553-554. | Zbl 0354.35051

[27] S. Mizohata, The theory of partial differential equations. Cambridge University Press, New York (1973) xii+490. | MR 599580 | Zbl 0263.35001

[28] V. Mendez and J. Camacho, Dynamics and thermodynamics of delayed population growth. Phys. Rev. E 55 (1997) 6476-6482.

[29] V. Mendez and A. Compte, Wavefronts in bistable hyperbolic reaction-diffusion systems. Physica A 260 (1998) 90-98.

[30] M. Nagasawa, Schrödinger equations and diffusion theory. Monographs in Mathematics, Birkhäuser Verlag, Basel 86 (1993) pp. x+319. | MR 1227100 | Zbl 0780.60003

[31] E. Orsingher, 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. | Zbl 0693.60070

[32] H.G. Othmer, S.R. Dunbar and W. Alt, Models of dispersal in biological systems. J. Math. Biol. 26 (1988) 263-298. | Zbl 0713.92018

[33] M. Pinsky, Lectures on random evolution. World Scient. Publ. Co., River Edge, NY (1991). | MR 1143780 | Zbl 0925.60139

[34] N.E. Ratanov, Telegraph processes with reflecting and absorbing barriers in inhomogeneous media. Theor. Math. Phys. 112 (1997) 857-865. | Zbl 0978.82508

[35] N. Ratanov, Reaction-advection random motions in inhomogeneous media. Physica D 189 (2004) 130-140. | Zbl 1052.35115

[36] N. Ratanov, Pricing options under telegraph processes. Rev. Econ. Ros. 8 (2005) 131-150.

[37] 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. | MR 1297766 | Zbl 1001.35060

[38] G.H. Weiss, Aspects and applications of the random walk. North-Holland, Amsterdam (1994). | MR 1280031 | Zbl 0925.60079

[39] G.H. Weiss, Some applications of persistent random walks and the telegrapher's equation. Physica A 311 (2002) 381-410. | Zbl 0996.35040

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