Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-06-02T00:53:52.145Z Has data issue: false hasContentIssue false
  • English
  • Français

Skeletal stochastic differential equations for superprocesses

Part of: Stochastic processes Markov processes Stochastic analysis

Published online by Cambridge University Press:  23 November 2020

Dorottya Fekete ,
Joaquin Fontbona  and
Andreas E. Kyprianou
Dorottya Fekete*
Affiliation:
University of Exeter
Joaquin Fontbona*
Affiliation:
Universidad de Chile
Andreas E. Kyprianou*
Affiliation:
University of Bath
*
*Postal address: College of Engineering, Mathematics and Physical Sciences, Harrison Building, University of Exeter, North Park Road, Exeter EX4 4QF, UK. Email address: d.fekete@exeter.ac.uk
**Postal address: Center for Mathematical Modeling, DIM CMM, UMI 2807 UChile-CNRS, Universidad de Chile, Beauchef 851, Edificio Norte – Piso 7, Santiago, Chile. Email address: fontbona@dim.uchile.cl
***Postal address: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. Email address: a.kyprianou@bath.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).

In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.

Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

Keywords

SuperprocessesSDEsskeletal decomposition

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1111 - 1134
Check for updates
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abraham, R. and Delmas, J.-F. (2012). A continuum-tree-valued Markov process. Ann. Prob. 40, 1167–1211.10.1214/11-AOP644CrossRefGoogle Scholar
Berestycki, J., Kyprianou, A. and Murillo-Salas, A. (2011). The prolific backbone for supercritical superprocesses. Stoch. Proc. Appl. 121, 1315–1331.10.1016/j.spa.2011.02.004CrossRefGoogle Scholar
Bertoin, J., Fontbona, J. and Martínez, S. (2008). On prolific individuals in a supercritical continuous-state branching process. J. Appl. Prob. 45, 714–726.10.1239/jap/1222441825CrossRefGoogle Scholar
Duquesne, T. and Le Gall, J.-F. (2002). Random Trees, Lévy Processes, and Spatial Branching Processes (Astérisque 281). Société mathématique de France.Google Scholar
Duquesne, T. and Winkel, M. (2007). Growth of Lévy trees. Prob. Theory Relat. Fields 139, 313–371.10.1007/s00440-007-0064-3CrossRefGoogle Scholar
Dynkin, E. B. (1991). A probabilistic approach to one class of nonlinear differential equations. Prob. Theory Relat. Fields 89, 89–115.10.1007/BF01225827CrossRefGoogle Scholar
Dynkin, E. B. (1994). An Introduction to Branching Measure-Valued Processes (CRM Monograph Series 6). American Mathematical Society, Providence, RI.10.1090/crmm/006CrossRefGoogle Scholar
Dynkin, E. B. (2002). Diffusions, Superprocesses and Partial Differential Equations (Colloquium Publications 50). American Mathematical Society, Providence, RI.Google Scholar
Dynkin, E. B. and Kuznetsov, S. E. (2004). $\mathbb{N}$-measures for branching exit Markov systems and their applications to differential equations. Prob. Theory Relat. Fields 130, 135–150.Google Scholar
Eckhoff, M., Kyprianou, A. E. and Winkel, M. (2015). Spines, skeletons and the strong law of large numbers for superdiffusions. Ann. Prob. 43, 2545–2610.10.1214/14-AOP944CrossRefGoogle Scholar
Engländer, J. and Pinsky, R. G. (1999). On the construction and support properties of measure-valued diffusions on ${D} \subseteq \mathbb{R}^d$ with spatially dependent branching. Ann. Prob. 27, 684–730.10.1214/aop/1022677383CrossRefGoogle Scholar
Etheridge, A. M. (2000). An Introduction to Superprocesses (University Lecture Series 20). American Mathematical Society, Providence, RI.Google Scholar
Etheridge, A. M. and Williams, D. R. E. (2003). A decomposition of the ($1 + \beta$)-superprocess conditioned on survival. Proc. Roy. Soc. Edinburgh Sect. A Math. 133, 829.10.1017/S0308210500002699CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics). John Wiley, New York.10.1002/9780470316658CrossRefGoogle 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, 187–196.10.4153/CMB-1994-028-3CrossRefGoogle Scholar
Fekete, D., Fontbona, J. and Kyprianou, A. E. (2019). Skeletal stochastic differential equations for continuous-state branching process. J. Appl. Prob. 56, 1122–1150.10.1017/jpr.2019.67CrossRefGoogle Scholar
Fekete, D., Palau, S., Pardo, J. C. and Pérez, J. L. (2018). Backbone decomposition of multitype superprocesses. Available at arXiv:1803.09620.Google Scholar
Fittipaldi, M. and Fontbona, J. (2012). On SDE associated with continuous-state branching processes conditioned to never be extinct. Electron. Commun. Prob. 17, 49, 1–13.10.1214/ECP.v17-1972CrossRefGoogle Scholar
Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64, 337–361.10.1007/BF02882426CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes (North-Holland Math. Library). North-Holland.Google Scholar
Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales (Lecture Notes Math. 714). Springer.10.1007/BFb0064907CrossRefGoogle Scholar
Kyprianou, A. and Ren, Y.-X. (2012). Backbone decomposition for continuous-state branching processes with immigration. Statist. Prob. Lett. 82, 139–144.10.1016/j.spl.2011.09.013CrossRefGoogle Scholar
Kyprianou, A., Pérez, J. L. and Ren, Y.-X. (2014). The backbone decomposition for spatially dependent supercritical superprocesses. In Séminaire de Probabilités XLVI (Lecture Notes Math. 2123), pp. 33–59. Springer International Publishing, Switzerland.10.1007/978-3-319-11970-0_2CrossRefGoogle Scholar
Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Prob. 12, 14, 420–446.10.1214/EJP.v12-402CrossRefGoogle Scholar
Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24, 45–163.10.1080/15326340802437728CrossRefGoogle Scholar
Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations (Lectures in Mathematics, ETH Zürich). Birkhäuser, Basel.Google Scholar
Li, Z. (2011). Measure Valued Branching Markov Processes. Springer, Berlin and Heidelberg.10.1007/978-3-642-15004-3CrossRefGoogle Scholar
Murillo-Salas, A. and Pérez, J.-L. (2015). The Backbone Decomposition for Superprocesses with Non-local Branching. In XI Symposium on Probability and Stochastic Processes (Progress in Probability 69), pp. 199–216. Springer International Publishing, Cham.Google Scholar
øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications (Hochschultext/Universitext). Springer.Google Scholar
Pinsky, R. (2008). Positive Harmonic Functions and Diffusion (Cambridge Studies in Advanced Mathematics). Cambridge University Press.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations: A New Approach (Applications of Mathematics (New York) 21). Springer, Berlin.10.1007/978-3-662-02619-9CrossRefGoogle Scholar
Roelly-Coppoletta, S. and Rouault, A. (1989). Processus de Dawson–Watanabe conditionné par le futur lointain. C.R. Acad. Sci. Paris Sér. I Math. 309, 867–872.Google Scholar
Salisbury, T. S. and Verzani, J. (1999). On the conditioned exit measures of super Brownian motion. Prob. Theory Relat. Fields 115, 237–285.10.1007/s004400050271CrossRefGoogle Scholar
Xiong, J. (2013). Three Classes of Nonlinear Stochastic Partial Differential Equations. World Scientific.10.1142/8728CrossRefGoogle Scholar