Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation : one sided travelling-waves
Harris, J. W. ; Harris, S. C. ; Kyprianou, A. E.
Annales de l'I.H.P. Probabilités et statistiques, Tome 42 (2006), p. 125-145 / Harvested from Numdam
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@article{AIHPB_2006__42_1_125_0,
     author = {Harris, J. W. and Harris, S. C. and Kyprianou, A. E.},
     title = {Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation : one sided travelling-waves},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {42},
     year = {2006},
     pages = {125-145},
     doi = {10.1016/j.anihpb.2005.02.005},
     zbl = {1093.60059},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2006__42_1_125_0}
}

Harris, J. W.; Harris, S. C.; Kyprianou, A. E. Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation : one sided travelling-waves. Annales de l'I.H.P. Probabilités et statistiques, Tome 42 (2006) pp. 125-145. doi : 10.1016/j.anihpb.2005.02.005. http://gdmltest.u-ga.fr/item/AIHPB_2006__42_1_125_0/

[1] D.G. Aronson, H.F. Weinberger, Nonlinear diffusions in population genetics, combustion and nerve propagation, in: Goldstein J. (Ed.), Partial Differential Equations and Related Topics, Lecture Notes in Math., vol. 446, Springer-Verlag, Berlin, 1975. | MR 427837 | Zbl 0325.35050

[2] K.B. Athreya, Change of measures for Markov chains and the L log L theorem for branching processes, Bernoulli 6 (2) (2000) 323-338. | MR 1748724 | Zbl 0969.60076

[3] J.D. Biggins, A.E. Kyprianou, Measure change in multitype branching, Adv. Appl. Probab. 36 (2) (2004) 544-581. | MR 2058149 | Zbl 1056.60082

[4] A.N. Borodin, P. Salminen, Handbook of Brownian Motion-Facts and Formulae, Probability and its Applications, Birkhäuser, Basel, 1996. | Zbl 0859.60001

[5] M.D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (5) (1978) 531-581. | MR 494541 | Zbl 0361.60052

[6] M.D. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc. 44 (285) (1983), iv+190. | MR 705746 | Zbl 0517.60083

[7] A. Champneys, S.C. Harris, J. Toland, J. Warren, D. Williams, Algebra, analysis and probability for a coupled system of reaction-diffusion equations, Philos. Trans. Roy. Soc. London 350 (1995) 69-112. | MR 1325205 | Zbl 0824.60070

[8] B. Chauvin, Product martingales and stopping lines for branching Brownian motion, Ann. Probab. 19 (3) (1991) 1195-1205. | MR 1112412 | Zbl 0738.60079

[9] B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299-314. | MR 968823 | Zbl 0653.60077

[10] B. Chauvin, A. Rouault, Supercritical branching Brownian motion and K-P-P equation in the critical speed-area, Math. Nachr. 149 (1990) 41-59. | MR 1124793 | Zbl 0724.60091

[11] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York, 1955. | MR 69338 | Zbl 0064.33002

[12] E.B. Dynkin, Superprocesses and partial differential equations, Ann. Probab. 21 (3) (1993) 1185-1262. | MR 1235414 | Zbl 0806.60066

[13] J. Engländer, A.E. Kyprianou, Local extinction versus local exponential growth for spatial branching processes, Ann. Probab. 32 (1A) (2004) 78-99. | MR 2040776 | Zbl 1056.60083

[14] R.A. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937) 355-369. | JFM 63.1111.04 | MR 2079

[15] M. Freidlin, Functional Integration and Partial Differential Equations, Ann. Math. Stud., vol. 109, Princeton University Press, Princeton, NJ, 1985. | MR 833742 | Zbl 0568.60057

[16] Y. Git, J.W. Harris, S.C. Harris, Exponential growth-rates and large deviations for a typed branching diffusion, University of Bath preprint, 2004. | Zbl 1131.60077

[17] R. Hardy, S.C. Harris, A new formulation of the spine approach, University of Bath preprint, 2004.

[18] S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503-517. | MR 1693633 | Zbl 0946.35040

[19] S.C. Harris, D. Williams, Large deviations and martingales for a typed branching diffusion. I, Astérisque 236 (1996) 133-154, Hommage à P.A. Meyer et J. Neveu. | MR 1417979 | Zbl 0857.60088

[20] N. Ikeda, M. Nagasawa, S. Watanabe, Branching Markov processes. I, J. Math. Kyoto Univ. 8 (1968) 233-278. | MR 232439 | Zbl 0233.60068

[21] N. Ikeda, M. Nagasawa, S. Watanabe, Branching Markov processes. II, J. Math. Kyoto Univ. 8 (1968) 365-410. | MR 238401 | Zbl 0233.60069

[22] N. Ikeda, M. Nagasawa, S. Watanabe, Branching Markov processes. III, J. Math. Kyoto Univ. 9 (1969) 95-160. | MR 246376 | Zbl 0233.60070

[23] Y. Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka J. Math. 13 (1) (1976) 11-66. | MR 422875 | Zbl 0344.35050

[24] H. Kesten, Branching Brownian motion with absorption, Stochastic Process. Appl. 7 (1) (1978) 9-47. | MR 494543 | Zbl 0383.60077

[25] A.N. Kolmogorov, I. Petrovski, N. Piscounov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problem biologique, Mosc. Univ. Bull. Math. 1 (1937) 1-25, Translated and reprinted in P. Pelce, Dynamics of Curved Fronts, Academic, San Diego, 1988. | Zbl 0018.32106

[26] D. Kuhlbusch, On weighted branching processes in random environment, Stochastic Process. Appl. 109 (1) (2004) 113-144. | MR 2024846 | Zbl 1075.60111

[27] A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincaré Probab. Statist. 40 (1) (2004) 53-72. | Numdam | MR 2037473 | Zbl 1042.60057

[28] A.E. Kyprianou, A. Rahimzadeh Sani, Martingale convergence and the functional equation in the multi-type branching random walk, Bernoulli 7 (4) (2001) 593-604. | MR 1849369 | Zbl 1017.60090

[29] R. Lyons, R. Pemantle, Y. Peres, Conceptual proofs of L log L criteria for mean behavior of branching processes, Ann. Probab. 23 (3) (1995) 1125-1138. | MR 1349164 | Zbl 0840.60077

[30] R. Lyons, A simple path to Biggins' martingale convergence for branching random walk, in: Athreya K.B., Jagers P. (Eds.), Classical and Modern Branching Processes, Minneapolis, MN, 1994, IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 217-221. | MR 1601749 | Zbl 0897.60086

[31] H.P. Mckean, Application of Brownian motion to the equation of Kolmogorov-Petrovskiĭ-Piskunov, Comm. Pure Appl. Math. 28 (3) (1975) 323-331. | MR 400428 | Zbl 0316.35053

[32] H.P. Mckean, A correction to: “Application of Brownian motion to the equation of Kolmogorov-Petrovskiĭ-Piskunov” (Comm. Pure Appl. Math. 28 (3) (1975) 323-331), Comm. Pure Appl. Math. 29 (5) (1976) 553-554. | Zbl 0354.35051

[33] J. Neveu, Multiplicative martingales for spatial branching processes, in: Seminar on Stochastic Processes, Princeton, NJ, 1987, Progr. Probab. Statist., vol. 15, Birkhäuser Boston, Boston, MA, 1988, pp. 223-242. | MR 1046418 | Zbl 0652.60089

[34] P. Olofsson, The x log x condition for general branching processes, J. Appl. Probab. 35 (3) (1998) 537-544. | MR 1659492 | Zbl 0926.60063

[35] R.G. Pinsky, K-P-P-type asymptotics for nonlinear diffusion in a large ball with infinite boundary data and on R d with infinite initial data outside a large ball, Comm. Partial Differential Equations 20 (7-8) (1995) 1369-1393. | MR 1335755 | Zbl 0831.35078

[36] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (3) (1978) 453-508. | MR 509494 | Zbl 0408.35053

[37] S. Watanabe, On the branching process for Brownian particles with an absorbing boundary, J. Math. Kyoto Univ. 4 (1965) 385-398. | MR 178505 | Zbl 0134.34401