Travelling wave solutions to the K-P-P equation : alternatives to Simon Harris' probabilistic analysis
Annales de l'I.H.P. Probabilités et statistiques (2004)
- Volume: 40, Issue: 1, page 53-72
- ISSN: 0246-0203
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topKyprianou, A. E.. "Travelling wave solutions to the K-P-P equation : alternatives to Simon Harris' probabilistic analysis." Annales de l'I.H.P. Probabilités et statistiques 40.1 (2004): 53-72. <http://eudml.org/doc/77799>.
@article{Kyprianou2004,
author = {Kyprianou, A. E.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {branching Brownian motion; KPP equation; travelling wave solutions; size biased measures; Bessel-3 processes; Brownian motion; spine decompositions},
language = {eng},
number = {1},
pages = {53-72},
publisher = {Elsevier},
title = {Travelling wave solutions to the K-P-P equation : alternatives to Simon Harris' probabilistic analysis},
url = {http://eudml.org/doc/77799},
volume = {40},
year = {2004},
}
TY - JOUR
AU - Kyprianou, A. E.
TI - Travelling wave solutions to the K-P-P equation : alternatives to Simon Harris' probabilistic analysis
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2004
PB - Elsevier
VL - 40
IS - 1
SP - 53
EP - 72
LA - eng
KW - branching Brownian motion; KPP equation; travelling wave solutions; size biased measures; Bessel-3 processes; Brownian motion; spine decompositions
UR - http://eudml.org/doc/77799
ER -
References
top- [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. Zbl0325.35050MR427837
- [2] K. Athreya, P. Ney, Branching Processes, Springer-Verlag, Berlin, 1972. Zbl0259.60002MR373040
- [3] K. Athreya, Change of measures for Markov chains and the LlogL theorem for branching processes, Bernoulli6 (1999) 323-338. Zbl0969.60076MR1748724
- [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. Zbl0757.60067MR1232850
- [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. Zbl0973.60098MR1765165
- [6] J.D. Biggins, Martingale convergence in the branching random walk, J. Appl. Probab.14 (1977) 25-37. Zbl0356.60053MR433619
- [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. Zbl0770.60077MR1193068
- [8] J.D. Biggins, Uniform convergence of martingales in the branching random walk, Ann. Probab.20 (1992) 137-151. Zbl0748.60080MR1143415
- [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. Zbl0864.60070
- [10] J.D. Biggins, A.E. Kyprianou, Measure change in multi-type branching, Advances of Applied Probability, in press. Zbl1056.60082
- [11] M. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math.31 (1978) 531-581. Zbl0361.60052MR494541
- [12] M. Bramson, Convergence of solutions to the Kolmogorov nonlinear diffusion equation to travelling waves, Mem. Amer. Math. Soc.44 (1983) 1-190. Zbl0517.60083MR705746
- [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. Zbl0824.60070
- [14] B. Chauvin, Multiplicative martingales and stopping lines for branching Brownian motion, Ann. Probab.30 (1991) 1195-1205. Zbl0738.60079MR1112412
- [15] B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields80 (1988) 299-314. Zbl0653.60077MR968823
- [16] B. Chauvin, A. Rouault, Supercritical branching Brownian motion and KPP equation in the critical speed area, Math. Nachr.149 (1990) 41-59. Zbl0724.60091MR1124793
- [17] B. Chauvin, A. Rouault, A. Wakolbinger, Growing conditioned trees, Stochastic Process. Appl.39 (1991) 117-130. Zbl0747.60077MR1135089
- [18] R. Durrett, Probability: Theory and Examples, Duxbury, Belmont, CA, 1996. Zbl1202.60001MR1609153
- [19] E.B. Dynkin, Superprocesses and partial differential equations, Ann. Probab.21 (1993) 1185-1262. Zbl0806.60066MR1235414
- [20] J. Engländer, A.E. Kyprianou, Local extinction versus local exponential growth for spatial branching processes, Ann. Probab., 2002, submitted for publication. Zbl1056.60083MR2040776
- [21] A. Etheridge, An Introduction to Superprocesses, Univ. Lecture Ser., Amer. Math. Soc, Providence, RI, 2000. Zbl0971.60053MR1779100
- [22] S. Evans, Two representations of a superprocess, Proc. Roy. Soc. Edinburgh Sect. A125 (1993) 959-971. Zbl0784.60052MR1249698
- [23] R.A. Fisher, The advance of advantageous genes, Ann. Eugenics7 (1937) 355-369. Zbl63.1111.04JFM63.1111.04
- [24] S.C. Harris, D. Williams, Large-deviations and martingales for a typed branching diffusion: I, Astérisque236 (1996) 133-154. Zbl0857.60088MR1417979
- [25] S.C. Harris, Convergence of a Gibbs–Boltzmann random measure for a typed branching diffusion, in: Séminaire de Probabilités, vol. XXXIV, 2000. Zbl0985.60053
- [26] S.C. Harris, Travelling-waves for the F-K-P-P equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A129 (1999) 503-517. Zbl0946.35040MR1693633
- [27] P. Jagers, General branching processes as Markov fields, Stochastic Process. Appl.32 (1989) 193-212. Zbl0678.92009MR1014449
- [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. Zbl0864.60069
- [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. Zbl0820.60068
- [30] J.F.C. Kingman, The first birth problem for an age-dependent branching processes, Ann. Probab.3 (1975) 790-801. Zbl0325.60079MR400438
- [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. Zbl0018.32106
- [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. Zbl0868.60068
- [33] J.-F. LeGall, Spatial Branching Processes, Random Snakes and Partial Differential Equations, Lectures Math., Birkhäuser, Basel, 1999. Zbl0938.60003MR1714707
- [34] O.D. Lyne, Travelling waves for a certain first-order coupled PDE system, Electron. J. Probab.5 (2000), Paper 14. Zbl0954.35105MR1781026
- [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. Zbl0897.60086MR1601749
- [36] R. Lyons, R. Pemantle, Y. Peres, Conceptual proofs of LlogL criteria for mean behaviour of branching processes, Ann. Probab.23 (1995) 1125-1138. Zbl0840.60077MR1349164
- [37] H.P. McKean, Excursions of a non-singular diffusion, Z. Wahr. werv. Geb.1 (1963) 230-239. Zbl0117.35903MR162282
- [38] H.P. McKean, Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov, Comm. Pure Appl. Math.29 (1975) 323-331. Zbl0316.35053
- [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. Zbl0652.60089MR1046418
- [40] P. Olofsson, The xlogx condition for general branching processes, J. Appl. Probab.35 (1998) 537-554. Zbl0926.60063MR1659492
- [41] Pitman, One dimensional Brownian motion and three dimensional Bessel processes, Adv. Appl. Probab.7 (1975) 511-526. Zbl0332.60055MR375485
- [42] T. Shiga, S. Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahr. Verw. Geb.27 (1973) 37-46. Zbl0327.60047MR368192
- [43] A.V. Skorohod, Branching diffusion processes, Theory Probab. Appl.9 (1964) 492-497. Zbl0264.60058MR168030
- [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. Zbl0408.35053MR509494
- [45] D. Williams, Path decomposition and continuity of local times for one-dimensional diffusions, Proc. London Math. Soc.28 (1974) 737-768. Zbl0326.60093MR350881
Citations in EuDML Documents
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- J. W. Harris, S. C. Harris, A. E. Kyprianou, Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation : one sided travelling-waves
- Pascal Maillard, The number of absorbed individuals in branching brownian motion with a barrier
- Anne-Laure Basdevant, On the equivalence of some eternal additive coalescents
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