Branching random motions, nonlinear hyperbolic systems and travellind waves

Nikita Ratanov

ESAIM: Probability and Statistics (2006)

  • Volume: 10, page 236-257
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Ratanov, Nikita. "Branching random motions, nonlinear hyperbolic systems and travellind waves." ESAIM: Probability and Statistics 10 (2006): 236-257. <http://eudml.org/doc/249754>.

@article{Ratanov2006,
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. },
author = {Ratanov, Nikita},
journal = {ESAIM: Probability and Statistics},
keywords = {Branching random motion; travelling wave; Feynman-Kac connection; non-linear hyperbolic system; McKean solution.; branching random motion; McKean solution; Kolmogorov type backward equation; Kolmogorov-Petrovskii-Piskunov equation},
language = {eng},
month = {5},
pages = {236-257},
publisher = {EDP Sciences},
title = {Branching random motions, nonlinear hyperbolic systems and travellind waves},
url = {http://eudml.org/doc/249754},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Ratanov, Nikita
TI - Branching random motions, nonlinear hyperbolic systems and travellind waves
JO - ESAIM: Probability and Statistics
DA - 2006/5//
PB - EDP Sciences
VL - 10
SP - 236
EP - 257
AB - 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.
LA - eng
KW - Branching random motion; travelling wave; Feynman-Kac connection; non-linear hyperbolic system; McKean solution.; branching random motion; McKean solution; Kolmogorov type backward equation; Kolmogorov-Petrovskii-Piskunov equation
UR - http://eudml.org/doc/249754
ER -

References

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  1. K.B. Athreya and P.E. Ney, Branching processes. Dover Publ. Inc. Mineola, NY (2004).  
  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.  
  3. M. Bramson, Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math.31 (1978) 531–581.  
  4. M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc.44 (1983) iv+190.  
  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.  
  6. A. Di Crescenzo and F. Pellerey, On prices' evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry18 (2002) 171–184.  
  7. S.R. Dunbar, A branching random evolution and a nonlinear hyperbolic equation. SIAM J. Appl. Math.48 (1988) 1510–1526.  
  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.  
  9. R.A. Fisher, The advance of advantageous genes. Ann. Eugenics7 (1937) 335–369.  
  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.  
  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.  
  13. K.P. Hadeler, Reaction transport systems in biological modelling, In Mathematics inspiring by biology. Lect. Notes in Math.1714 (1999) 95–150.  
  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.  
  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.  
  16. W. Horsthemke, Spatial instabilities in reaction random walks with direction-independent kinetics. Phys. Rev. E60 (1999) 2651–2663.  
  17. W. Horsthemke, Fisher waves in reaction random walks. Phys. Lett. A263 (1999) 285–292.  
  18. D.D. Joseph and L. Preziosi, Heat waves. Rev. Mod. Phys.61 (1989) 41–73.  
  19. D.D. Joseph and L. Preziosi, Addendum to the paper “Heat waves”. Rev. Mod. Phys.62 (1990) 375–391.  
  20. M. Kac, Probability and related topics in physical sciences. Interscience, London (1959).  
  21. M. Kac, A Stochastic model related to the telegraph equation. Rocky Mountain J. Math.4 (1974) 497–509.  
  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.  
  23. O.D. Lyne, Travelling waves for a certain first-order coupled PDE system. Electronic J. Prob.5 (2000) 1–40.  
  24. O.D. Lyne and D. Williams, Weak solutions for a simple hyperbolic system. Electronic J. Prob.6 (2001) 1–21.  
  25. H.P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math.XXVIII (1975) 323–331.  
  26. H.P. McKean, Correction to above. Comm. Pure Appl. Math.XXIX (1976) 553–554.  
  27. S. Mizohata, The theory of partial differential equations. Cambridge University Press, New York (1973) xii+490.  
  28. V. Mendez and J. Camacho, Dynamics and thermodynamics of delayed population growth. Phys. Rev. E55 (1997) 6476–6482.  
  29. V. Mendez and A. Compte, Wavefronts in bistable hyperbolic reaction-diffusion systems. Physica A260 (1998) 90–98.  
  30. M. Nagasawa, Schrödinger equations and diffusion theory. Monographs in Mathematics, Birkhäuser Verlag, Basel 86 (1993) pp. x+319.  
  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.  
  32. H.G. Othmer, S.R. Dunbar and W. Alt, Models of dispersal in biological systems. J. Math. Biol.26 (1988) 263–298.  
  33. M. Pinsky, Lectures on random evolution. World Scient. Publ. Co., River Edge, NY (1991).  
  34. N.E. Ratanov, Telegraph processes with reflecting and absorbing barriers in inhomogeneous media. Theor. Math. Phys.112 (1997) 857–865.  
  35. N. Ratanov, Reaction-advection random motions in inhomogeneous media. Physica D189 (2004) 130–140.  
  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. 140Amer. Math. Soc. Providence, RI, (1994) pp. xii+448.  
  38. G.H. Weiss, Aspects and applications of the random walk. North-Holland, Amsterdam (1994).  
  39. G.H. Weiss, Some applications of persistent random walks and the telegrapher's equation. Physica A311 (2002) 381–410.  

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