Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-16T07:28:33.011Z Has data issue: false hasContentIssue false
  • English
  • Français

An Application of the Backbone Decomposition to Supercritical Super-Brownian Motion with a Barrier

Part of: Representations of solutions Markov processes

Published online by Cambridge University Press:  04 February 2016

A. E. Kyprianou ,
A. Murillo-Salas  and
J. L. Pérez
A. E. Kyprianou*
Affiliation:
University of Bath
A. Murillo-Salas*
Affiliation:
Universidad de Guanajuato
J. L. Pérez*
Affiliation:
Mexico Autonomous Institute of Technology
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
∗∗ Postal address: Departamento de Matemáticas, Universidad de Guanajuato, Jalisco s/n, Mineral de Valenciana, Guanajuato, Gto. CP 36240, México.
∗∗∗ Postal address: Department of Statistics, Mexico Autonomous Institute of Technology, Rio Hondo 1, Tizapan 1 San Angel, Mexico DF 01000, Mexico.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.

Keywords

Super-Brownian motionbackbone decompositionkilled super-Brownian motion

MSC classification

Primary: 60J68: Superprocesses 35C07: Traveling wave solutions
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 671 - 684
Copyright
© Applied Probability Trust 

References

Addario-Berry, L. and Broutin, N. (2011). Total progeny in killed branching random walk. Prob. Theory Relat. Fields 151, 265–295.Google Scholar
Aïdékon, E., Hu, Y. and Zindy, O. (2011). The precise tail behavior of the total progeny of a killed branching random walk. Preprint. Available at http://arxiv.org/abs/1102.5536v1.Google Scholar
Berestycki, J., Kyprianou, A. E. and Murillo-Salas, A. (2011). The prolific backbone decomposition for supercritical superdiffusions. Stoch. Process. Appl. 121, 1315–1331.CrossRefGoogle Scholar
Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Prob. 19, 1157–1194.Google Scholar
Dynkin, E. B. (1991). A probabilistic approach to one class of nonlinear differential equations. Prob. Theory Relat. Fields 89, 89–115.CrossRefGoogle Scholar
Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Prob. 21, 1185–1262.Google Scholar
Dynkin, E. B. (2001). Branching exit Markov systems and superprocess. Ann. Prob. 29, 1833–1858.Google Scholar
Dynkin, E. B. (2002). Diffusions, Superdiffusions and Partial Differential Equations. American Mathematical Society, Providence, RI.Google Scholar
Dynkin, E. B. and Kuznetsov, S. E. (2004). {\BBN}-measures for branching exit Markov systems and their applications to differential equations. Prob. Theory Relat. Fields 130, 135–150.Google Scholar
El Karoui, N. and Roelly, S. (1991). Propriétés de martingales, explosion et représentation de Lévy–Khintchine d'une classe de processus de branchement à valeurs mesures. Stoch. Process. Appl. 38, 239–266.CrossRefGoogle Scholar
Engländer, J. and Pinsky, R. G. (1999). On the construction and support properties of measure-valued diffusions on D {R}{d} with spatially dependent branching. Ann. Prob. 27, 684–730.Google Scholar
Evans, S. N. and O'Connell, N. (1994). Weighted occupation time for branching particle systems and a representation for the supercritical superprocess. Canad. Math. Bull. 37, 187–196.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64, 337–361.Google Scholar
Harris, J. W., Harris, S. C. and Kyprianou, A. E. (2006). Further probabilistic analysis of the Fisher–Kolmogorov–Pretrovskii–Piscounov equation: one sided travelling-waves. Ann. Inst. H. Poincaré Prob. Statist. 42, 125–145.CrossRefGoogle Scholar
Kametaka, Y. (1976). On the nonlinear diffusion equation of Kolmogorov–Petrovskii–Piskunov type. Osaka J. Math. 13, 11–66.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Kyprianou, A., 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.Google Scholar
Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.Google Scholar
Maillard, P. (2012). The number of absorbed individuals in branching Brownian motion with a barrier. To appear in Ann. Inst. H. Poincaré Prob. Statist. Google Scholar
Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes (Prog. Prob. Statist. 15), eds Çinlar, E., Chung, K. L. and Getoor, R. K., Birkhaüser, Boston, MA, pp. 223–241.Google Scholar
Pinsky, R. G. (1995). 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. Commun. Partial Differential Equat. 20, 1369–1393.Google Scholar
Salisbury, T. and Verzani, J. (1999). On the conditioned exit measures of super Brownian motion. Prob. Theory Relat. Fields 115, 237–285.Google Scholar
Salisbury, T. S. and Verzani, J. (2000). Non-degenerate conditionings of the exit measure of super Brownian motion. Stoch. Process. Appl. 87, 25–52.Google Scholar
Sheu, Y.-C. (1997). Lifetime and compactness of range for super-Brownian motion with a general branching mechanism. Stoch. Process. Appl. 70, 129–141.Google Scholar
Uchiyama, K. (1978). The behavior of solutions of some non-linear diffusion equations for large time. J. Math. Kyoto Univ. 18, 453–508.Google Scholar
Watanabe, K. (1968). A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167.Google Scholar
Yang, T. and Ren, Y.-X. (2011). Limit theorem for derivative martingale at criticality w.r.t. branching Brownian motion. Statist. Prob. Lett. 81, 195–200.Google Scholar