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Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment

Part of: Markov processes Special processes Stochastic processes

Published online by Cambridge University Press:  02 December 2024

Wenming Hong  and
Shengli Liang
Wenming Hong*
Affiliation:
Beijing Normal University
Shengli Liang*
Affiliation:
Southern University of Science and Technology
*
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, China. Email address: wmhong@bnu.edu.cn
**Postal address: Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, Guangdong, China. Email address: liangsl@sustech.edu.cn
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Abstract

Consider a branching random walk on the real line with a random environment in time (BRWRE). A necessary and sufficient condition for the non-triviality of the limit of the derivative martingale is formulated. To this end, we investigate the random walk in a time-inhomogeneous random environment (RWRE), which is related to the BRWRE by the many-to-one formula. The key step is to figure out Tanaka’s decomposition for the RWRE conditioned to stay non-negative (or above a line), which is interesting in itself.

Keywords

Branching random walkrandom environmentderivative martingalequenched harmonic functionrandom walk conditioned to stay non-negativeTanaka’s decomposition

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments
Secondary: 60G42: Martingales with discrete parameter

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 57 , Issue 2 , June 2025 , pp. 642 - 676
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Afanasyev, V. I., Geiger, J., Kersting, G. and Vatutin, V. A. (2005). Criticality for branching processes in random environment. Ann. Prob. 33, 645–673.CrossRefGoogle Scholar
Athreya, K. B. (2000). Change of measures for Markov chains and the $L\log L$ theorem for branching processes. Bernoulli 6, 323–338.CrossRefGoogle Scholar
Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Prob. 41, 1362–1426.CrossRefGoogle Scholar
Aïdékon, E. and Shi, Z. (2014). The Seneta–Heyde scaling for the branching random walk. Ann. Prob. 42, 959–993.CrossRefGoogle Scholar
Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.CrossRefGoogle Scholar
Biggins, J. D. (2003). Random walk conditioned to stay positive. J. London Math. Soc. 67, 259–272.CrossRefGoogle Scholar
Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. Appl. Prob. 36, 544–581.CrossRefGoogle Scholar
Biggins, J. D. and Kyprianou, A. E. (2005). Fixed points of the smoothing transform: the boundary case. Electron. J. Prob. 10, 609–631.CrossRefGoogle Scholar
Chen, X. (2015). A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk. Adv. Appl. Prob. 47, 741–760.CrossRefGoogle Scholar
Durrett, R. and Liggett, M. (1983). Fixed points of the smoothing transform. Z. Wahrscheinlichkeitsth. 64, 275–301.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Gao, Z., Liu, Q. and Wang, H. (2014). Central limit theorems for a branching random walk with a random environment in time. Acta Math. Sci. 34, 501–512.CrossRefGoogle Scholar
Greven, A. and den Hollander, F. (1992). Branching random walk in random environment: phase transition for local and global growth rates. Prob. Theory Relat. Fields 91, 195–249.CrossRefGoogle Scholar
Harris, S. C. (1999). Travelling-waves for the FKPP equation via probabilistic arguments. Proc. R. Soc. Edinburgh A 129, 503–517.CrossRefGoogle Scholar
Hong, W. and Liang, S. (2022). Random walks in time-inhomogeneous random environment conditioned to stay positive. Preprint. Available at https://arxiv.org/abs/2211.15017.Google Scholar
Hu, Y. and Yoshida, N. (2009). Localization for branching random walks in random environment. Stoch. Process. Appl. 119, 1632–1651.CrossRefGoogle Scholar
Huang, C. and Liu, Q. (2014). Branching random walk with a random environment in time. Preprint. Available at https://arxiv.org/abs/1407.7623.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Kyprianou, A. E. (1998). Slow variation and uniqueness of solutions to the functional equation in the branching random walk. J. Appl. Prob. 35, 795–802.CrossRefGoogle Scholar
Lalley, S. P. and Sellke, T. (1987). A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Prob. 15, 1052–1061.CrossRefGoogle Scholar
Liu, Q. (1998). Fixed points of a generalized smoothing transform and applications to the branching processes. Adv. Appl. Prob. 30, 85–112.CrossRefGoogle Scholar
Liu, Q. (2000). On generalized multiplicative cascades. Stoch. Process. Appl. 86, 263–286.CrossRefGoogle Scholar
Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes, Springer, New York, pp. 217–221.CrossRefGoogle Scholar
Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.Google Scholar
Mallein, B. (2015). Maximal displacement in a branching random walk through interfaces. Electron. J. Prob. 20, 1–40.CrossRefGoogle Scholar
Mallein, B. and Miłoś, P. (2019). Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment. Stoch. Process. Appl. 129, 3239–3260.CrossRefGoogle Scholar
Mallein, B. and Shi, Q. (2023). A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process. Bernoulli 29, 597–624.CrossRefGoogle Scholar
McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Commun. Pure Appl. Math. 28, 323–331.CrossRefGoogle Scholar
Shi, Z. (2015). Branching Random Walks: École d’Été de Probabilités de Saint-Flour XLII—2012. Springer, Cham.CrossRefGoogle Scholar
Tanaka, H. (1989). Time reversal of random walks in one dimension. Tokyo J. Math. 12, 159–174.CrossRefGoogle Scholar
Wang, X. and Huang, C. (2017). Convergence of martingale and moderate deviations for a branching random walk with a random environment in time. J. Theoret. Prob. 30, 961–995.CrossRefGoogle Scholar
Yang, T. and Ren, Y.-X. (2011). Limit theorem for derivative martingale at criticality w.r.t. branching Brownian motion. Statist. Prob. Lett. 81, 195–200.CrossRefGoogle Scholar
Yoshida, N. (2008). Central limit theorem for branching random walks in random environment. Ann. Appl. Prob. 18, 1619–1635.CrossRefGoogle Scholar