Supercritical super-brownian motion with a general branching mechanism and travelling waves

Kyprianou, A. E. ; Liu, R.-L. ; Murillo-Salas, A. ; Ren, Y.-X.

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 661-687 / Harvested from Numdam

Résumé

Nous proposons une approche probabiliste au problème classique de l'existence, de l'unicité et du comportement asymptotique des solutions monotones de l'équation de propagation de front associée à l'équation parabolique du super-mouvement brownien de mécanisme de branchement général. Bien que largement inspiré par l'approche de Kyprianou (Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53-72) pour le mouvement brownien branchant, cet article ouvre plusieurs perspectives nouvelles. Notre analyse inclut le rôle de la normalisation de Seneta-Heyde qui, dans cette situation, s'inspire du travail classique de Grey (J. Appl. Probab. 11 (1974) 669-677). Nous donnons une explication trajectorielle de la décomposition en épine (la particule immortelle d’Evans), en utilisant la $\mathbb {N}$-mesure de Dynkin-Kuznetsov comme ingrédient clef. En outre, dans l’esprit des lignes d’arrêt de Neveu nous utilisons à plusieurs reprises les mesures de sortie de Dynkin. La nature générale du mécanisme de branchement rend l’analyse du problème plus délicate et nous proposons une dichotomie exacte basée sur un moment $X(\log X)^{2}$ pour la convergence presque-sûre de la martingale dérivée (pour la valeur critique de son paramètre) vers une limite non-triviale. Ceci diffère du cas du mouvement brownien branchant (Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53-72) et de la marche aléatoire branchante (Adv. in Appl. Probab. 36 (2004) 544-581), où un écart dans les hypothèses sur les moments apparaît entre les conditions nécessaires et suffisantes. Notre approche probabiliste permet de retrouver des résultats connus d'existence, d'unicité et de comportement asymptotique pour l'équation de propagation de front reliée au super-mouvement brownien.

Abstract

We offer a probabilistic treatment of the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the reasoning in Kyprianou (Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53-72) for branching Brownian motion, the current paper offers a number of new insights. Our analysis incorporates the role of Seneta-Heyde norming which, in the current setting, draws on classical work of Grey (J. Appl. Probab. 11 (1974) 669-677). We give a pathwise explanation of Evans’ immortal particle picture (the spine decomposition) which uses the Dynkin-Kuznetsov $\mathbb {N}$-measure as a key ingredient. Moreover, in the spirit of Neveu’s stopping lines we make repeated use of Dynkin’s exit measures. Additional complications arise from the general nature of the branching mechanism. As a consequence of the analysis we also offer an exact $X(\log X)^{2}$ moment dichotomy for the almost sure convergence of the so-called derivative martingale at its critical parameter to a non-trivial limit. This differs to the case of branching Brownian motion (Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53-72), and branching random walk (Adv. in Appl. Probab. 36 (2004) 544-581), where a moment ‘gap' appears in the necessary and sufficient conditions. Our probabilistic treatment allows us to replicate known existence, uniqueness and asymptotic results for the travelling wave equation, which is related to a super-Brownian motion.

Détail

Publié le : 2012-01-01

MR 2976558 | Zbl 1267.60094

DOI : https://doi.org/10.1214/11-AIHP448
Classification:  60J80,  60E10

Bibliographie

[1] D. G. Aronson and H. F. Weinberger. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30 (1978) 33-76. | MR 511740 | Zbl 0407.92014

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

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

[4] B. Chauvin. Multiplicative martingales and stopping lines for branching Brownian motion. Ann. Probab. 30 (1991) 1195-1205. | MR 1112412 | Zbl 0738.60079

[5] R. Durrett. Probability Theory and Examples, 2nd edition. Duxbury Press, 1996. | MR 1609153 | Zbl 1202.60002

[6] R. Durrett and C. Neuhauser. Particle systems and reaction-diffusion equations. Ann. Probab. 22 (1994) 289-333. | MR 1258879 | Zbl 0799.60093

[7] E. B. Dynkin. A probabilistic approach to one class of non-linear differential equations. Probab. Theory Related Fields 89 (1991) 89-115. | MR 1109476 | Zbl 0722.60062

[8] E. B. Dynkin. Branching particle systems and superprocesses. Ann. Probab. 19 (1991) 1157-1194. | MR 1112411 | Zbl 0732.60092

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

[10] E. B. Dynkin. Branching exit Markov systems and superprocesses. Ann. Probab. 29 (2001) 1833-1858. | MR 1880244 | Zbl 1014.60079

[11] E. B. Dynkin. Diffusions, Superdiffusions and Partial Differential Equations. Amer. Math. Soc., Providence, RI, 2002. | MR 1883198 | Zbl 0999.60003

[12] E. B. Dynkin and S. E. Kuznetsov. $\mathbb {N}$-measures for branching Markov exit systems and their applications to differential equations. Probab. Theory Related Fields 130 (2004) 135-150. | MR 2092876 | Zbl 1068.31002

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

[14] S. N. Evans. Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 959-971. | MR 1249698 | Zbl 0784.60052

[15] P. C. Fife and J. B. Mcleod. The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65 (1977) 335-361. | MR 442480 | Zbl 0361.35035

[16] R. A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics 7 (1937) 355-369. | JFM 63.1111.04

[17] P. J. Fitzsimmons. Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64 (1988) 337-361. | MR 995575 | Zbl 0673.60089

[18] Y. Git, J. W. Harris and S. C. Harris. Exponential growth rates in a typed branching diffusion. Ann. Appl. Probab. 17 (2007) 609-653. | MR 2308337 | Zbl 1131.60077

[19] D. R. Grey. Asymptotic behavior of continuous time, continuous state-space branching processes. J. Appl. Probab. 11 (1974) 669-677. | MR 408016 | Zbl 0301.60060

[20] R. Hardy and S. C. Harris. A spine approach to branching diffusions with applications to $L^{p}$-convergence of martingales. In Séminaire de Probabilités XLII 281-330. Berlin, 2009. | MR 2599214 | Zbl 1193.60100

[21] 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 1693633 | Zbl 0946.35040

[22] S. C. Harris and M. Roberts. Measure changes with extinction. Stat. Probab. Lett. 79 (2009) 1129-1133. | MR 2510780 | Zbl 1163.60309

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

[24] A. Kolmogorov, I. Petrovskii and N. Piskounov. Étude de l'équation de la diffusion avec croissance de la quantité de la matière at son application a un problèm biologique. Moscow Univ. Math. Bull. 1 (1937) 1-25. | Zbl 0018.32106

[25] A. E. Kyprianou. Travelling wave solution to the K-P-P equation: Alternatives to Simon Harris' probabilistic analysis Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53-72. | Numdam | MR 2037473 | Zbl 1042.60057

[26] A. E. Kyprianou. Asymptotic radial speed of the support of supercritical branching Brownian motion and super-Brownian motion in $\mathbb {R}^{d}$. Markov Process. Related Fields 11 (2005) 145-156. | Zbl 1076.60074

[27] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006. | Zbl pre06176054

[28] A. E. Kyprianou and A. Murillo-Salas. Super-Brownian motion: $L^{p}$-convergence of martingales through the pathwise spine decomposition. Preprint, 2011. Available at http://arxiv.org/abs/1106.2678. | Zbl 1279.60111

[29] K.-S. Lau. On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov. J. Differential Equations 59 (1985) 44-70. | Zbl 0584.35091

[30] J. F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1999. | Zbl 0938.60003

[31] R.-L. Liu, Y.-X. Ren and R. Song. $L\log L$ criterion for a class of superdiffusions. J. Appl. Probab. 46 (2009) 479-496. | Zbl 1175.60077

[32] R. Lyons. A simple path to Biggins' martingale convergence theorem. In Classical and Modern Branching Processes 217-222. K. B. Athreya and P. Jagers (Eds). Springer, New York, 1997. | Zbl 0897.60086

[33] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of $L\log L$ criteria for mean behaviour of branching processes. Ann. Probab. 23 (1995) 1125-1138. | Zbl 0840.60077

[34] P. Maillard. The number of absorbed individuals in branching Brownian motion with a barrier. Available at arXiv:1004.1426. | Numdam | Zbl 1281.60070

[35] H. P. Mckean. Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28 (1975) 323-331. | Zbl 0316.35053

[36] J. Neveu. Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes 1987 223-242. E. Çinlar, K. L. Chung and R. K. Getoor (Eds). Progress in Probability and Statistics 15. Birkhäuser, Boston, 1988. | Zbl 0652.60089

[37] R. G. Pinsky. K-P-P-type asymptotics for nonlinear diffusion in a large ball with infinite boundary data and on $\mathbf {R}^d$ with infinite initial data outside a large ball. Comm. Partial Differential Equations 20 (1995) 1369-1393. | Zbl 0831.35078

[38] Y.-X. Ren and H. Wang. On states of total weighted occupation times of a class of infinitely divisible superprocesses on a bounded domain. Potential Anal. 28 (2008) 105-137. | MR 2373101 | Zbl 1205.60151

[39] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin, 1980. | Zbl 0731.60002

[40] Y. C. Sheu. Lifetime and compactness of range for $\psi $-super-Brownian motion with a general branching mechanism. Stochastics Process. Appl. 70 (1997) 129-141. | MR 1472962 | Zbl 0911.60060

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

[42] A. I. Volpert, V. A. Volpert and V. A. Volpert. Traveling Wave Solutions of Parabolic Systems. Translations of Mathematical Monographs 140. Amer. Math. Soc., 1994. | MR 1297766 | Zbl 1001.35060

[43] S. Watanabe. A limit theorem of branching processes and continuous-state branching processes. J. Math. Kyoto Univ. 8 (1968) 141-167. | MR 237008 | Zbl 0159.46201