Electronic Journal of Probability

Exponential ergodicity for general continuous-state nonlinear branching processes

Pei-Sen Li and Jian Wang

Full-text: Open access

Abstract

By combining the coupling by reflection for Brownian motion with the refined basic coupling for Poisson random measure, we present sufficient conditions for the exponential ergodicity of general continuous-state nonlinear branching processes in both the $L^{1}$-Wasserstein distance and the total variation norm, where the drift term is dissipative only for large distance, and either diffusion noise or jump noise is allowed to be vanished. Sufficient conditions for the corresponding strong ergodicity are also established.

Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 125, 25 pp.

Dates
Received: 13 September 2019
Accepted: 3 October 2020
First available in Project Euclid: 12 October 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1602489723

Digital Object Identifier
doi:10.1214/20-EJP528

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60G52: Stable processes 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes

Keywords
continuous-state nonlinear branching process exponential ergodicity coupling strong ergodicity

Rights
Creative Commons Attribution 4.0 International License.

Citation

Li, Pei-Sen; Wang, Jian. Exponential ergodicity for general continuous-state nonlinear branching processes. Electron. J. Probab. 25 (2020), paper no. 125, 25 pp. doi:10.1214/20-EJP528. https://projecteuclid.org/euclid.ejp/1602489723


Export citation

References

  • [1] Bertoin, J. and Le Gall, J.-F.: The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes, Probab. Theory Related Fields 117 (2000), 249–266.
  • [2] Chen, M.-F.: Eigenvalues, Inequalities, and Ergodic Theory, Springer-Verlag, London, 2005.
  • [3] Chen, M.-F. and Li. S.-F.: Coupling methods for multidimensional diffusion processes, Ann. Probab. 17 (1989), 151–177.
  • [4] Duhalde, X., Foucart, C. and Ma, C.: On the hitting times of continuous-state branching processes with immigration, Stochastic Process. Appl. 124 (2014), 4182–4201.
  • [5] Etheridge, A. M.: Survival and extinction in a locally regulated population, Ann. Appl. Probab. 14 (2004), 188–214.
  • [6] Friesen, M., Jin, P., Kremer, J. and Rüdiger, B.: Exponential ergodicity for stochastic equations of nonnegative processes with jumps, arXiv:1902.02833.
  • [7] Fu, Z. and Li, Z.: Stochastic equations of non-negative processes with jumps, Stochastic Process. Appl. 120 (2010), 306–330.
  • [8] Grey, D.R.: Asymptotic behaviour of continuous time, continuous state-space branching processes, J. Appl. Probab. 11 (1974), 669–677.
  • [9] Kyprianou, A.: Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer-Verlag, Berlin, 2006.
  • [10] Lambert, A.: The branching process with logistic growth, Ann. Appl. Probab. 15 (2005), 1506–1535.
  • [11] Lamperti, J.: Continuous-state branching processes, Bull. Am. Math. Soc. 73 (1967), 382–386.
  • [12] Li, P.-S., Yang, X. and Zhou, X.: A general continuous-state nonlinear branching process, Ann. Appl. Probab. 29 (2019), 2523–2555.
  • [13] Li, Z.: Measure-Valued Branching Markov Processes, Springer, Heidelberg, 2011.
  • [14] Li, Z. and Ma, C.: Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model, Stochastic Process. Appl. 125 (2015), 3196–3233.
  • [15] Liang, M. and Wang, J.: Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling, Stochastic Process. Appl. 130 (2020), 3053–3094.
  • [16] Liang, M., Schilling, R.L. and Wang, J.: A unified approach to coupling SDEs driven by Lévy noise and some applications, Bernoulli 26 (2020), 664–693.
  • [17] Lindvall, T., Rogers, L.C.G.: Coupling of multidimensional diffusions by reflection, Ann. Probab. 14 (1986), 860–872.
  • [18] Luo, D. and Wang, J.: Exponential convergence in $L^{p}$-Wasserstein distance for diffusion processes without uniformly dissipative drift, Math. Nachr. 289 (2016), 1909–1926.
  • [19] Luo, D. and Wang, J.: Coupling by reflection and Hölder regularity for non-local operators of variable order, Transactions of the American Mathematical Society 371 (2019), 431–459.
  • [20] Luo, D. and Wang, J.: Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises, Stochastic Process. Appl. 129 (2019), 3129–3173.
  • [21] Mao, Y.-H.: Strong ergodicity for Markov processes by coupling methods, J. Appl. Probab. 39 (2002), 839–852.
  • [22] Pardoux, É.: Probabilistic Models of Population Evolution: Scaling Limits, Genealogies and Interactions, Springer, Heidelberg, 2016.
  • [23] Wang, J.: $L_{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes, Bernoulli 22 (2016), 1598–1616.