Travelling wave solutions to the K-P-P equation : alternatives to Simon Harris' probabilistic analysis

Kyprianou, A. E.

Annales de l'I.H.P. Probabilités et statistiques, Tome 40 (2004) no. 1, pp. 53-72.

[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/New York, 1975. | MR | Zbl

[2] K. Athreya, P. Ney, Branching Processes, Springer-Verlag, Berlin, 1972. | MR | Zbl

[3] K. Athreya, Change of measures for Markov chains and the LlogL theorem for branching processes, Bernoulli 6 (1999) 323-338. | MR | Zbl

[4] J. Bertoin, Splitting at the infimum and excursions in half-lines for random walks and Lévy processes, Stochastic Process. Appl. 42 (1993) 307-313. | MR | Zbl

[5] M. Bachmann, Limit theorems for the minimal position in a branching random walk with independent logconcave displacements, Adv. in Appl. Probab. 32 (2000) 159-176. | MR | Zbl

[6] J.D. Biggins, Martingale convergence in the branching random walk, J. Appl. Probab. 14 (1977) 25-37. | MR | Zbl

[7] J.D. Biggins, Uniform convergence of martingales in the one-dimensional branching random walk, in: Selected proceedings of the Sheffield Symposium on Applied Probability, 1989 , IMS Lecture Notes Monogr. Ser., vol. 18, 1991, pp. 159-173. | MR | Zbl

[8] J.D. Biggins, Uniform convergence of martingales in the branching random walk, Ann. Probab. 20 (1992) 137-151. | MR | Zbl

[9] J.D. Biggins, A.E. Kyprianou, Branching random walk: Seneta-Heyde norming, in: Chauvin B., Cohen S., Rouault A. (Eds.), Trees: Proceedings of a Workshop, Versailles, June 14-16, 1995 , Birkhäuser, Basel, 1996. | Zbl

[10] J.D. Biggins, A.E. Kyprianou, Measure change in multi-type branching, Advances of Applied Probability, in press. | Zbl

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

[12] M. Bramson, Convergence of solutions to the Kolmogorov nonlinear diffusion equation to travelling waves, Mem. Amer. Math. Soc. 44 (1983) 1-190. | MR | Zbl

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

[14] B. Chauvin, Multiplicative martingales and stopping lines for branching Brownian motion, Ann. Probab. 30 (1991) 1195-1205. | MR | Zbl

[15] 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 (1988) 299-314. | MR | Zbl

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

[17] B. Chauvin, A. Rouault, A. Wakolbinger, Growing conditioned trees, Stochastic Process. Appl. 39 (1991) 117-130. | MR | Zbl

[18] R. Durrett, Probability: Theory and Examples, Duxbury, Belmont, CA, 1996. | MR

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

[20] J. Engländer, A.E. Kyprianou, Local extinction versus local exponential growth for spatial branching processes, Ann. Probab., 2002, submitted for publication. | MR | Zbl

[21] A. Etheridge, An Introduction to Superprocesses, Univ. Lecture Ser., Amer. Math. Soc, Providence, RI, 2000. | MR | Zbl

[22] S. Evans, Two representations of a superprocess, Proc. Roy. Soc. Edinburgh Sect. A 125 (1993) 959-971. | MR | Zbl

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

[24] S.C. Harris, D. Williams, Large-deviations and martingales for a typed branching diffusion: I, Astérisque 236 (1996) 133-154. | Numdam | MR | Zbl

[25] S.C. Harris, Convergence of a Gibbs-Boltzmann random measure for a typed branching diffusion, in: Séminaire de Probabilités, vol. XXXIV, 2000. | Numdam | Zbl

[26] S.C. Harris, Travelling-waves for the F-K-P-P equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503-517. | MR | Zbl

[27] P. Jagers, General branching processes as Markov fields, Stochastic Process. Appl. 32 (1989) 193-212. | MR | Zbl

[28] F.I. Karpelevich, M.Ya. Kelbert, Yu.M. Suhov, The branching diffusion, stochastic equations and travelling wave solutions to the equation of Kolmogorov-Petrovskii-Piskounov, in: Boccara N., Goles E., Martinez S., Picco P. (Eds.), Cellular Automata and Cooperative Behaviour, Kluwer Academic, Dordrecht, 1993, pp. 343-366. | Zbl

[29] M.Ya. Kelbert, Yu.M. Suhov, The Markov branching random walk and systems of reaction-diffusion (Kolmogorov-Petrovskii-Piskunov) equations, Comm. Math. Phys. 167 (1995) 607-634. | Zbl

[30] J.F.C. Kingman, The first birth problem for an age-dependent branching processes, Ann. Probab. 3 (1975) 790-801. | MR | Zbl

[31] A. Kolmogorov, I. Petrovskii, N. Piskounov, Étude de l'équation de la diffusion avec croissance de la quantité de la matière et son application a un problèm biologique, in: Moscow Univ. Math. Bull., vol. 1, 1937, pp. 1-25. | Zbl

[32] T. Kurtz, R. Lyons, R. Pemantle, Y. Peres, A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes, in: Athreya K.B., Jagers P. (Eds.), Classical and Modern Branching Processes, Math. Appl., vol. 84, Springer-Verlag, New York, 1997, pp. 181-186. | Zbl

[33] J.-F. Legall, Spatial Branching Processes, Random Snakes and Partial Differential Equations, Lectures Math., Birkhäuser, Basel, 1999. | MR | Zbl

[34] O.D. Lyne, Travelling waves for a certain first-order coupled PDE system, Electron. J. Probab. 5 (2000), Paper 14. | MR | Zbl

[35] R. Lyons, A simple path to Biggins' martingale convergence theorem, in: Athreya K.B., Jagers P. (Eds.), Classical and Modern Branching Processes, Math. Appl., vol. 84, Springer-Verlag, New York, 1997, pp. 217-222. | MR | Zbl

[36] R. Lyons, R. Pemantle, Y. Peres, Conceptual proofs of LlogL criteria for mean behaviour of branching processes, Ann. Probab. 23 (1995) 1125-1138. | MR | Zbl

[37] H.P. Mckean, Excursions of a non-singular diffusion, Z. Wahr. werv. Geb. 1 (1963) 230-239. | MR | Zbl

[38] H.P. Mckean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 29 (1975) 323-331. | Zbl

[39] J. Neveu, Multiplicative martingales for spatial branching processes, in: Çinlar E., Chung K.L., Getoor R.K. (Eds.), Seminar on Stochastic Processes, 1987, Progr. Probab. and Statist., vol. 15, Birkhaüser, Boston, 1988, pp. 223-241. | MR | Zbl

[40] P. Olofsson, The xlogx condition for general branching processes, J. Appl. Probab. 35 (1998) 537-554. | MR | Zbl

[41] Pitman, One dimensional Brownian motion and three dimensional Bessel processes, Adv. Appl. Probab. 7 (1975) 511-526. | MR | Zbl

[42] T. Shiga, S. Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahr. Verw. Geb. 27 (1973) 37-46. | MR | Zbl

[43] A.V. Skorohod, Branching diffusion processes, Theory Probab. Appl. 9 (1964) 492-497. | MR | Zbl

[44] K. Uchiyama, The behaviour of solutions of some non-linear diffusion equations for large time. 1, J. Math. Kyoto Univ. 18 (1978) 453-508. | MR | Zbl

[45] D. Williams, Path decomposition and continuity of local times for one-dimensional diffusions, Proc. London Math. Soc. 28 (1974) 737-768. | MR | Zbl