Hostname: page-component-76d6cb85b7-pn7tm Total loading time: 0 Render date: 2026-07-19T03:18:18.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixed-state branching evolution for cell division models

Part of: Markov processes Stochastic analysis

Published online by Cambridge University Press:  29 September 2025

Shukai Chen ,
Lina Ji Open the ORCID record for Lina Ji [Opens in a new window]  and
Jie Xiong
Shukai Chen*
Affiliation:
Fujian Normal University
Lina Ji*
Affiliation:
Shenzhen MSU-BIT University
Jie Xiong*
Affiliation:
Southern University of Science and Technology
*
*Postal address: School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350007, China. Email: skchen@fjnu.edu.cn
**Postal address: MSU-BIT-SMBU Joint Research Center of Applied Mathematics, Shenzhen MSU-BIT University, Shenzhen 518172, China. Email: jiln@smbu.edu.cn
***Postal address: Department of Mathematics & National Center for Applied Mathematics (Shenzhen), Southern University of Science and Technology, Shenzhen 518055, China. Email: xiongj@sustech.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a scaling limit theorem for two-type Galton–Watson branching processes with interaction. The limit theorem gives rise to a class of mixed-state branching processes with interaction used to simulate evolution for cell division affected by parasites. Such processes can also be obtained by the pathwise-unique solution to a stochastic equation system. Moreover, we present sufficient conditions for extinction with probability 1 and the exponential ergodicity in the $L^1$-Wasserstein distance of such processes in some cases.

Keywords

Mixed-state branching processstochastic integral equationinteraction

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 60H15: Stochastic partial differential equations

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 63 , Issue 1 , March 2026 , pp. 135 - 156
Check for updates
Copyright
© The Author(s), 2025. 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.)

Article purchase

Temporarily unavailable

References

Aldous, D. (1978). Stopping times and tightness. Ann. Prob. 6, 335–340.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Bansaye, V. and Tran, V. C. (2011). Branching Feller diffusion for cell division with parasite infection. ALEA, Lat. Am. J. Prob. Math. Statist. 8, 95–127.Google Scholar
Berestycki, J., Kyprianou, A. E. and Murillo-Salas, A. (2011). The prolific backbone for supercritical superprocesses. Stoch. Process. Appl. 121, 1315–1331.CrossRefGoogle Scholar
Bertoin, J., Fontbona, J. and Martinez, S. (2008). On prolific individuals in a supercritical continuous-state branching process. J. Appl. Prob. 45, 714–726.CrossRefGoogle Scholar
Chen, M. (2005). Eigenvalues, Inequalities, and Ergodic Theory. Springer, New York.Google Scholar
Chen, S. and Li, Z. (2021). Continuous-time mixed-state branching processes and stochastic equations. Acta Math. Sci. 41, 1445–1473.CrossRefGoogle Scholar
Cranston, M. and Greven, A. (1995). Coupling and harmonic functions in the case of continuous-time Markov processes. Stoch. Process. Appl. 60, 261–286.CrossRefGoogle Scholar
Dawson, D. A. and Li, Z. (2006). Skew convolution semigroups and affine Markov processes. Ann. Prob. 34, 1103–1142.CrossRefGoogle Scholar
Dawson, D. A. and Li, Z. (2012). Stochastic equations, flows and measure-valued processes. Ann. Prob. 40, 813–857.CrossRefGoogle Scholar
Duquesne, T. and Winkel, M. (2007). Growth of Lévy trees. Prob. Theory Relat. Fields 139, 313–371.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence . John Wiley, New York.CrossRefGoogle Scholar
Feketa, D., Fontbona, J. and Kyprianou, A. E. (2019). Skeletal stochastic differential equations for continuous-state branching processes. J. Appl. Prob. 56, 1122–1150.CrossRefGoogle Scholar
Friesen, M., Jin, P., Kremer, J. and Rüdiger, B. (2023). Exponential ergodicity for stochastic equations of nonnegative processes with jumps. ALEA, Lat. Am. J. Prob. Math. Statist. 20, 593–627.CrossRefGoogle Scholar
Fu, Z. and Li, Z. (2010). Stochastic equations of non-negative processes with jumps. Stoch. Process. Appl. 120, 306–330.CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam.Google Scholar
Ji, L. and Xiong, J. (2022). Superprocesses for the population of rabbits on grassland. Proc. Steklov Inst. Math. 316, 195–208.CrossRefGoogle Scholar
Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 36–54.CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg.CrossRefGoogle Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Li, P., Li, Z., Wang, J. and Zhou, X. (2025). Exponential ergodicity of branching processes with immigration and competition. Ann. Inst. H. Poincaré Prob. Statist. 61, 350–384.CrossRefGoogle Scholar
Li, P. and Wang, J. (2020). Exponential ergodicity for general continuous-state nonlinear branching processes. Electron. J. Prob. 25, 125.CrossRefGoogle Scholar
Li, P., Yang, X. and Zhou, X. (2019). A general continuous-state nonlinear branching process. Ann. Appl. Prob. 29, 2523–2555.CrossRefGoogle Scholar
Li, Z. (2006). A limit theorem for discrete Galton–Watson branching processes with immigration. J. Appl. Prob. 43, 289–295.CrossRefGoogle Scholar
Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer, New York.CrossRefGoogle Scholar
Li, Z. (2020). Continuous-state branching processes with immigration. In From Probability to Finance: Mathematical Lectures from Peking University, ed. Y. Jiao. Springer, Singapore.Google Scholar
Li, Z. and Ma, C. (2015). Asymptotic properties of estimators in a Cox–Ingersoll–Ross model. Stoch. Process. Appl. 125, 3196–3233.CrossRefGoogle Scholar
Liu, J. (2024). Scaling limits of controlled branching processes. Statist. Prob. Lett. 208, 110081.CrossRefGoogle Scholar
Luo, D. and Wang, J. (2019). Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises. Stoch. Process. Appl. 129, 3129–3173.CrossRefGoogle Scholar
Ma, C. (2009). A limit theorem of two-type Galton–Watson branching processes with immigration. Statist. Prob. Lett. 79, 1710–1716.CrossRefGoogle Scholar
Marguet, A. and Smadi, C. (2024). Spread of parasites affecting death and division rates in a cell population. Stoch. Process. Appl. 168, 104262.CrossRefGoogle Scholar
Osorio, L. and Winter, A. (2020). Two-level branching model for virus population under cell division. Preprint, arXiv:2004.14352.Google Scholar
Villani, C. (2009). Optimal Transport, Old and New. Springer, Berlin. CrossRefGoogle Scholar