Ren, Yan-Xia; Yang, Ting Multitype branching Brownian motion and traveling waves. (English) Zbl 1303.60077 Adv. Appl. Probab. 46, No. 1, 217-240 (2014). Authors’ abstract: We study the parabolic system of equations which is closely related to a multitype branching Brownian motion. Particular attention is paid to the monotone traveling wave solutions of this system. Provided with some moment conditions, we show the existence, uniqueness, and asymptotic behaviors of such waves with speed greater than or equal to a critical value \(\underline{c}\) and nonexistence of such waves with speed smaller than \(\underline{c}\). Reviewer: P. R. Parthasarathy (Chennai) Cited in 2 Documents MSC: 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 35C07 Traveling wave solutions Keywords:multitype branching Brownian motion; spine approach; additive martingale; traveling wave solution × Cite Format Result Cite Review PDF Full Text: DOI Euclid References: [1] Athreya, K. B. (1968). Some results on multitype continuous time Markov branching processes. Ann. Math. Statist. 39 , 347-357. · Zbl 0169.49202 · doi:10.1214/aoms/1177698395 [2] Athreya, K. B. and Ney, P. E. (2004). Branching Processes . Dover, Mineola, NY. · Zbl 1070.60001 [3] Champneys, A. et al. (1995). Algebra, analysis and probability for a coupled system of reaction-diffusion equations. Philos. Trans. R. Soc. London Ser. A 350 , 69-112. · Zbl 0824.60070 · doi:10.1098/rsta.1995.0003 [4] Chauvin, B. and Rouault, A. (1988). KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Prob. Theory Relat. Fields 80 , 299-314. · Zbl 0653.60077 · doi:10.1007/BF00356108 [5] Durrett, R. (1996). Probability: Theory and Examples , 2nd edn. Duxbury Press, Belmont, CA. · Zbl 1202.60001 [6] Georgii, H.-O. and Baake, E. (2003). Supercritical multitype branching processes: the ancestral types of typical individuals. Adv. Appl. Prob. 35 , 1090-1110. · Zbl 1044.60080 · doi:10.1239/aap/1067436336 [7] Hardy, R. and Harris, S. C. (2009). A spine approach to branching diffusions with applications to \(L^p\)-convergence of martingales. In Séminaire de Probabilités XLII (Lecture Notes Math. 1979 ), Springer, Berlin, pp. 281-330. · Zbl 1193.60100 · doi:10.1007/978-3-642-01763-6_11 [8] Harris, S. C. (1999). Traveling-waves for the FKPP equation via probabilistic arguments. Proc. R. Soc. Edinburgh Sect. A 129 , 503-517. · Zbl 0946.35040 · doi:10.1017/S030821050002148X [9] Jagers, P. (1989). General branching processes as Markov fields. Stoch. Process. Appl. 32 , 183-212. · Zbl 0678.92009 · doi:10.1016/0304-4149(89)90075-6 [10] Kurtz, T., Lyons, R., Pemantle, R. and Peres, Y. (1997). A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes. In Classical and Modern Branching Processes , Springer, New York,pp. 181-185. · Zbl 0868.60068 · doi:10.1007/978-1-4612-1862-3_14 [11] Kyprianou, A. E. (2004). Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris’ probabilistic analysis. Ann. Inst. H. Poincaré Prob. Statist. 40 , 53-72. · Zbl 1042.60057 · doi:10.1016/S0246-0203(03)00055-4 [12] Kyprianou, A. E., Liu, R.-L., Murillo-Salas, A. and Ren, Y.-X. (2012). Supercritical super-Brownian motion with a general branching mechanism and travelling waves. Ann. Inst. H. Poincaré Prob. Statist. 48 , 661-687. · Zbl 1267.60094 · doi:10.1214/11-AIHP448 [13] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes , Springer, New York, pp. 217-221. · Zbl 0897.60086 · doi:10.1007/978-1-4612-1862-3_17 [14] McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Commun. Pure Appl. Math. 28 , 323-331. (Correction: 29 (1976), 553-554.) · Zbl 0316.35053 · doi:10.1002/cpa.3160280302 [15] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion . Springer, Berlin. · Zbl 0731.60002 [16] Seneta, E. (1973). Non-Negative Matrices. An Introduction to Theory and Applications . Halsted Press, New York. · Zbl 0278.15011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.