Abstract
A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a Lévy process with no jump less than \(-1\). We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another Lévy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey’s condition. In that case, a characterization of the extinction probability is given using a random differential equation with blowup terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcritical CBRE-process with immigration.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Afanasyev, V.I., Böinghoff, C., Kersting, G., Vatutin, V.A.: Limit theorems for a weakly subcritical branching process in a random environment. J. Theor. Probab. 25, 703–732 (2012)
Afanasy’ev, V.I., Geiger, J., Kersting, G., Vatutin, V.A.: Criticality for branching processes in random environment. Ann. Probab. 33, 645–673 (2005)
Aliev, S.A.: A limit theorem for the Galton–Watson branching processes with immigration. Ukrainian Math. J. 37, 535–538 (1985)
Bansaye, V., Millan, J.C.P., Smadi, C.: On the extinction of continuous state branching processes with catastrophes. Electron. J. Probab. 18(106), 1–31 (2013)
Böinghoff, C., Hutzenthaler, M.: Branching diffusions in random environment. Markov Process. Relat. Fields 18, 269–310 (2012)
Dawson, D.A., Li, Z.: Skew convolution semigroups and affine Markov processes. Ann. Probab. 34, 1103–1142 (2006)
Dawson, D.A., Li, Z.: Stochastic equations, flows and measure-valued processes. Ann. Probab. 40, 813–857 (2012)
El Karoui, N., Méléard, S.: Martingale measures and stochastic calculus. Probab. Theory Relat. Fields 84, 83–101 (1990)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)
Feller, W.: Diffusion processes in genetics. In: Proceedings 2nd Berkeley Symp. Math. Statist. Probab., 1950, pp. 227–246. Univ. California Press, Berkeley and Los Angeles (1951)
Fu, Z., Li, Z.: Stochastic equations of non-negative processes with jumps. Stoch. Process. Appl. 120, 306–330 (2010)
Guivarch, Y., Liu, Q.S.: Propriétés asymptotiques des processus de branchement en environnement aléatoire. C.R. Acad. Sci. Paris Sér. I Math. 332, 339–344 (2001)
Hairer, M., Mattingly, J.: The strong Feller property for singular stochastic PDEs. arXiv:1610.03415 (2016)
Helland, I.S.: Minimal conditions for weak convergence to a diffusion process on the line. Ann. Probab. 9, 429–452 (1981)
Hutzenthaler, M.: Supercritical branching diffusions in random environment. Electron. Commun. Probab. 16, 781–791 (2011)
Jiřina, M.: Stochastic branching processes with continuous state space. Czech. Math. J. 8, 292–313 (1958)
Kawazu, K., Watanabe, S.: Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16, 36–54 (1971)
Keiding, N.: Extinction and exponential growth in random environments. Theor. Popul. Biol. 8, 49–63 (1975)
Kurtz, T.G.: Diffusion approximations for branching processes. In: Branching processes (Conf., Saint Hippolyte, Que., 1976), Vol. 5, 269–292 (1978)
Lamperti, J.: The limit of a sequence of branching processes. Z. Wahrsch. verw. Geb. 7, 271–288 (1967a)
Lamperti, J.: Continuous state branching processes. Bull. Am. Math. Soc. 73, 382–386 (1967b)
Li, Z.: A limit theorem for discrete Galton–Watson branching processes with immigration. J. Appl. Probab. 43, 289–295 (2006)
Li, Z.: Measure-Valued Branching Markov Processes. Springer, Heidelberg (2011)
Li, Z., Ma, C.: Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model. Stoch. Process. Appl. 125, 3196–3233 (2015)
Li, Z., Pu, F.: Strong solutions of jump-type stochastic equations. Electron. Commun. Probab. 17(33), 1–13 (2012)
Li, Z., Xu, W.: Asymptotic results for exponential functionals of Lévy processes. arXiv:1601.02363v1 (2016)
Palau, S., Pardo, J.C.: Continuous state branching processes in random environment: the Brownian case. Stoch. Process. Appl. 127, 957–994 (2017)
Palau, S., Pardo, J.C.: Branching processes in a Lévy random environment. arXiv:1512.07691v1 (2015)
Smith, W.L.: Necessary conditions for almost sure extinction of a branching process with random environment. Ann. Math. Statist. 39, 2136–2140 (1968)
Smith, W.L., Wilkinson, W.E.: On branching processes in random environments. Ann. Math. Statist. 40, 814–827 (1969)
Sharpe, M.: General Theory of Markov Processes. Academic Press, New York (1988)
Vatutin, V.A.: A limit theorem for an intermediate subcritical branching process in a random environment. Theory Probab. Appl. 48, 481–492 (2004)
Wang, F.: Harnack Inequalities for Stochastic Partial Differential Equations. Springer, Heidelberg (2013)
Acknowledgements
We thank Professors Yueyun Hu and Zhan Shi for helpful discussions on branching processes in random environments. We are grateful to Sandra Palau for pointing out a gap in the original proof of Theorem 3.1 and a referee for a list of comments which helped us in improving the presentation of the results.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (Nos. 11371061, 11531001 and 11671041).
Rights and permissions
About this article
Cite this article
He, H., Li, Z. & Xu, W. Continuous-State Branching Processes in Lévy Random Environments. J Theor Probab 31, 1952–1974 (2018). https://doi.org/10.1007/s10959-017-0765-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-017-0765-1
Keywords
- Continuous-state branching process
- Random environment
- Lévy process
- Transition semigroup
- Backward stochastic equation
- Survival probability
- Immigration
- Ergodicity