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

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

doi:10.1214/11-AIHP448

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 661-687.

Résumé
Abstract

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 $ℕ$ -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.

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 $ℕ$ -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.

MR Zbl

DOI : 10.1214/11-AIHP448

Classification : 60J80, 60E10
Mots-clés : superprocesses,

N

-measure, spine decomposition, additive martingale, derivative martingale, travelling waves

@article{AIHPB_2012__48_3_661_0,
     author = {Kyprianou, A. E. and Liu, R.-L. and Murillo-Salas, A. and Ren, Y.-X.},
     title = {Supercritical super-brownian motion with a general branching mechanism and travelling waves},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {661--687},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {3},
     year = {2012},
     doi = {10.1214/11-AIHP448},
     mrnumber = {2976558},
     zbl = {1267.60094},
     language = {en},
     url = {https://www.numdam.org/articles/10.1214/11-AIHP448/}
}

TY  - JOUR
AU  - Kyprianou, A. E.
AU  - Liu, R.-L.
AU  - Murillo-Salas, A.
AU  - Ren, Y.-X.
TI  - Supercritical super-brownian motion with a general branching mechanism and travelling waves
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2012
SP  - 661
EP  - 687
VL  - 48
IS  - 3
PB  - Gauthier-Villars
UR  - https://www.numdam.org/articles/10.1214/11-AIHP448/
DO  - 10.1214/11-AIHP448
LA  - en
ID  - AIHPB_2012__48_3_661_0
ER  -

%0 Journal Article
%A Kyprianou, A. E.
%A Liu, R.-L.
%A Murillo-Salas, A.
%A Ren, Y.-X.
%T Supercritical super-brownian motion with a general branching mechanism and travelling waves
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2012
%P 661-687
%V 48
%N 3
%I Gauthier-Villars
%U https://www.numdam.org/articles/10.1214/11-AIHP448/
%R 10.1214/11-AIHP448
%G en
%F AIHPB_2012__48_3_661_0

Kyprianou, A. E.; Liu, R.-L.; Murillo-Salas, A.; Ren, Y.-X. Supercritical super-brownian motion with a general branching mechanism and travelling waves. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 661-687. doi : 10.1214/11-AIHP448. https://www.numdam.org/articles/10.1214/11-AIHP448/

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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

[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 | Zbl

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

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

[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

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

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

[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

[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

[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

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

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

[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

[37] 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 (1995) 1369-1393. | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

Cité par Sources :