Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T08:41:15.745Z Has data issue: false hasContentIssue false
  • English
  • Français

Mutually interacting superprocesses with migration

Part of: Special processes Stochastic analysis Markov processes

Published online by Cambridge University Press:  11 July 2022

Lina Ji Open the ORCID record for Lina Ji [Opens in a new window] ,
Huili Liu Open the ORCID record for Huili Liu [Opens in a new window]  and
Jie Xiong
Lina Ji*
Affiliation:
Shenzhen MSU-BIT University
Huili Liu*
Affiliation:
Hebei Normal University
Jie Xiong*
Affiliation:
Southern University of Science and Technology
*
*Postal address: Faculty of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, Shenzhen, Guangdong, 518172, China.
**Postal address: School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei, 050024, China. Email address: [email protected]
***Postal address: Department of Mathematics & National Center for Applied Mathematics (Shenzhen), Southern University of Science and Technology, Shenzhen, Guangdong, 518055, China.
Get access Rights & Permissions [Opens in a new window]

Abstract

A system of mutually interacting superprocesses with migration is constructed as the limit of a sequence of branching particle systems arising from population models. The uniqueness in law of the superprocesses is established using the pathwise uniqueness of a system of stochastic partial differential equations, which is satisfied by the corresponding system of distribution function-valued processes.

Keywords

Distribution function-valuedMartingale problemsuperprocessstochastic partial differential equationwell-posedness

MSC classification

Primary: 60J68: Superprocesses
Secondary: 60H15: Stochastic partial differential equations 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 904 - 929
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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

Dawson, D. A. (1975). Stochastic evolution equations and related measure processes. J. Multivariate Anal. 5, 1–52.10.1016/0047-259X(75)90054-8CrossRefGoogle Scholar
Dawson, D. A. (1993). Measure-Valued Markov Processes (Lecture Notes Math. 1541). Springer, Berlin.Google Scholar
Dawson, D. A. and Li, Z. (2012). Stochastic equations, flows and measure-valued processes. Ann. Prob. 40, 813–857.10.1214/10-AOP629CrossRefGoogle Scholar
Dawson, D. A. and Perkins, E. (1998). Long-time behavior and coexistence in a mutually catalytic branching model. Ann. Prob. 26, 1088–1138.10.1214/aop/1022855746CrossRefGoogle Scholar
Dawson, D. A., Etheridge, A., Fleischmann, K., Mytnik, L., Perkins, E. and Xiong, J. (2002). Mutually catalytic branching in the plane: finite measure states. Ann. Prob. 30, 1681–1762.10.1214/aop/1039548370CrossRefGoogle Scholar
Dawson, D. A., Etheridge, A., Fleischmann, K., Mytnik, L., Perkins, E. and Xiong, J. (2002). Mutually catalytic branching in the plane: infinite measure states. Electron. J. Prob. 7, 1–61.10.1214/EJP.v7-114CrossRefGoogle Scholar
Etheridge, A. (2000). An Introduction to Superprocesses (University Lecture Series 20). American Mathematical Society.Google Scholar
He, H., Li, Z. and Yang, X. (2014). Stochastic equations of super-Lévy processes with general branching mechanism. Stoch. Process. Appl. 124, 1519–1565.10.1016/j.spa.2013.12.007CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn (Grundlehren der Mathematischen Wissenschaften 288). Springer, Berlin.Google Scholar
Karoui, N. E. and Méléard, S. (1990). Martingale measures and stochastic calculus. Prob. Theory Related Fields 84, 83–101.10.1007/BF01288560CrossRefGoogle Scholar
Li, Z. (1992). Measure-valued branching processes with immigration. Stoch. Process. Appl. 43, 249–264.10.1016/0304-4149(92)90061-TCrossRefGoogle Scholar
Li, Z. (1996). Immigration structures associated with Dawson–Watanabe superprocesses. Stoch. Process. Appl. 62, 73–86.10.1016/0304-4149(95)00087-9CrossRefGoogle Scholar
Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer, Heidelberg.10.1007/978-3-642-15004-3CrossRefGoogle Scholar
Li, Z. and Shiga, T. (1995). Measure-valued branching diffusions: immigrations, excursions and limit theorems. J. Math. Kyoto Univ. 35, 233–274.Google Scholar
Li, Z., Xiong, J. and Zhang, M. (2010). Ergodic theory for a superprocess over a stochastic flow. Stoch. Process. Appl. 120, 1563–1588.10.1016/j.spa.2010.03.012CrossRefGoogle Scholar
Méléard, S. (1996). Asymptotic behaviour of some interacting particle systems; McKean–Vlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations (Lecture Notes Math. 1627), eds D. Talay et al., pp. 42–95. Springer, Berlin and Heidelberg.Google Scholar
Méléard, S. and Roelly, S. (1992). Interacting measure branching processes: some bounds for the support. Stoch. Stoch. Reports 44, 103–121.10.1080/17442509308833843CrossRefGoogle Scholar
Mytnik, L. (1998). Uniqueness for a mutually catalytic branching model. Prob. Theory Related Fields 112, 245–253.10.1007/s004400050189CrossRefGoogle Scholar
Mytnik, L. and Xiong, J. (2015). Well-posedness of the martingale problem for superprocess with interaction. Illinois J. Math. 59, 485–497.10.1215/ijm/1462450710CrossRefGoogle Scholar
Watanabe, S. (1968). A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167.Google Scholar
Xiong, J. (2008). An Introduction to Stochastic Filtering Theory (Oxford Graduate Texts in Mathematics 18). Oxford University Press.Google Scholar
Xiong, J. (2013). Super-Brownian motion as the unique strong solution to an SPDE. Ann. Prob. 41, 1030–1054.10.1214/12-AOP789CrossRefGoogle Scholar
Xiong, J. and Yang, X. (2016). Superprocesses with interaction and immigration. Stoch. Process. Appl. 126, 3377–3401.10.1016/j.spa.2016.04.032CrossRefGoogle Scholar