## The Annals of Probability

### The slow regime of randomly biased walks on trees

#### Abstract

We are interested in the randomly biased random walk on the supercritical Galton–Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_{n})$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^{3}$ in the first $n$ steps. We study the localization problem of $X_{n}$ and prove that the quenched law of $X_{n}$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_{n}|}{(\log n)^{2}}$ converges in law to some non-degenerate limit on $(0,\infty)$ whose law is explicitly computed.

#### Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 3893-3933.

Dates
Revised: September 2015
First available in Project Euclid: 14 November 2016

https://projecteuclid.org/euclid.aop/1479114266

Digital Object Identifier
doi:10.1214/15-AOP1064

Mathematical Reviews number (MathSciNet)
MR3572327

Zentralblatt MATH identifier
1360.60156

#### Citation

Hu, Yueyun; Shi, Zhan. The slow regime of randomly biased walks on trees. Ann. Probab. 44 (2016), no. 6, 3893--3933. doi:10.1214/15-AOP1064. https://projecteuclid.org/euclid.aop/1479114266

#### References

• [1] Aidékon, E. (2008). Transient random walks in random environment on a Galton–Watson tree. Probab. Theory Related Fields 142 525–559.
• [2] Aidékon, E. (2010). Large deviations for transient random walks in random environment on a Galton–Watson tree. Ann. Inst. Henri Poincaré Probab. Stat. 46 159–189.
• [3] Aïdékon, E. (2010). Tail asymptotics for the total progeny of the critical killed branching random walk. Electron. Commun. Probab. 15 522–533.
• [4] Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 1362–1426.
• [5] Aidekon, E. and Shi, Z. (2014). The Seneta–Heyde scaling for the branching random walk. Ann. Probab. 42 959–993.
• [6] Andreoletti, P. and Debs, P. (2014). The number of generations entirely visited for recurrent random walks in a random environment. J. Theoret. Probab. 27 518–538.
• [7] Andreoletti, P. and Debs, P. (2014). Spread of visited sites of a random walk along the generations of a branching process. Electron. J. Probab. 19 no. 42, 22.
• [8] Biane, Ph. and Yor, M. (1987). Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. (2) 111 23–101.
• [9] Biggins, J. D. (1977). Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14 630–636.
• [10] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37.
• [11] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 544–581.
• [12] Biggins, J. D. and Kyprianou, A. E. (2005). Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 609–631.
• [13] Bingham, N. H. and Doney, R. A. (1975). Asymptotic properties of supercritical branching processes. II. Crump–Mode and Jirina processes. Adv. in Appl. Probab. 7 66–82.
• [14] Chen, X. (2015). A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk. Adv. in Appl. Probab. 47 741–760.
• [15] Faraud, G. (2011). A central limit theorem for random walk in a random environment on marked Galton–Watson trees. Electron. J. Probab. 16 174–215.
• [16] Faraud, G., Hu, Y. and Shi, Z. (2012). Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Related Fields 154 621–660.
• [17] Golosov, A. O. (1984). Localization of random walks in one-dimensional random environments. Comm. Math. Phys. 92 491–506.
• [18] Hu, Y. and Shi, Z. (2007). A subdiffusive behaviour of recurrent random walk in random environment on a regular tree. Probab. Theory Related Fields 138 521–549.
• [19] Hu, Y. and Shi, Z. (2007). Slow movement of random walk in random environment on a regular tree. Ann. Probab. 35 1978–1997.
• [20] Hu, Y. and Shi, Z. (2014). The potential energy of biased random walks on trees. Available at arXiv:1403.6799.
• [21] Hu, Y., Shi, Z. and Yor, M. (2015). The maximal drawdown of the Brownian meander. Electron. Commun. Probab. 20 no. 39, 6.
• [22] Imhof, J.-P. (1984). Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21 500–510.
• [23] Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 131–145.
• [24] Kozlov, M. V. (1976). The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment. Teor. Verojatnost. i Primenen. 21 813–825.
• [25] Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Probab. 5 601–607.
• [26] Lyons, R. and Peres, Y. (2015). Probability on Trees and Networks. Cambridge Univ. Press, Cambridge. Available at http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html.
• [27] Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931–958.
• [28] Lyons, R. (1992). Random walks, capacity and percolation on trees. Ann. Probab. 20 2043–2088.
• [29] Lyons, R. (1997). A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) (K. B. Athreya andP. Jagers, eds.). IMA Vol. Math. Appl. 84 217–221. Springer, New York.
• [30] Lyons, R. and Pemantle, R. (1992). Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 125–136.
• [31] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
• [32] Lyons, R., Pemantle, R. and Peres, Y. (1995). Ergodic theory on Galton–Watson trees: Speed of random walk and dimension of harmonic measure. Ergodic Theory Dynam. Systems 15 593–619.
• [33] Lyons, R., Pemantle, R. and Peres, Y. (1996). Biased random walks on Galton–Watson trees. Probab. Theory Related Fields 106 249–264.
• [34] Madaule, T. (2016). First order transition for the branching random walk at the critical parameter. Stochastic Process. Appl. 126 470–502.
• [35] Menshikov, M. and Petritis, D. (2002). On random walks in random environment on trees and their relationship with multiplicative chaos. In Mathematics and Computer Science, II (Versailles, 2002). Trends Math. 415–422. Birkhäuser, Basel.
• [36] Neveu, J. (1986). Arbres et processus de Galton–Watson. Ann. Inst. Henri Poincaré Probab. Stat. 22 199–207.
• [37] Peres, Y. (1999). Probability on trees: An introductory climb. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 193–280. Springer, Berlin.
• [38] Pitman, J. W. (2015). Some conditional expectations and identities for Bessel processes related to the maximal drawdown of the Brownian meander. Unpublished manuscript.
• [39] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math. 1665 301–413. Springer, Berlin.
• [40] Shi, Z. (2015). Branching Random Walks, École D’été Saint-Flour XLII (2012). Lecture Notes in Math. 2151. Springer, Berlin.
• [41] Sinaĭ, Ya. G. (1982). The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatnost. i Primenen. 27 247–258.