## The Annals of Probability

### On the transience of processes defined on Galton–Watson trees

Andrea Collevecchio

#### Abstract

We introduce a simple technique for proving the transience of certain processes defined on the random tree generated by a supercritical branching process. We prove the transience for once-reinforced random walks on , that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567–592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on . Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229–1241] and [Ann. Probab. 18 (1990) 931–958] as part of more general results. A similar technique is applied to a vertex-reinforced jump process. A by-product of our result is that this process is transient on the 3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42–62] proved that a vertex-reinforced jump process defined on the b-ary tree is transient if b≥4 and recurrent if b=1. The case b=2 is still open.

#### Article information

Source
Ann. Probab. Volume 34, Number 3 (2006), 870-878.

Dates
First available in Project Euclid: 27 June 2006

http://projecteuclid.org/euclid.aop/1151418486

Digital Object Identifier
doi:10.1214/009117905000000837

Mathematical Reviews number (MathSciNet)
MR2243872

Zentralblatt MATH identifier
1104.60048

#### Citation

Collevecchio, Andrea. On the transience of processes defined on Galton–Watson trees. The Annals of Probability 34 (2006), no. 3, 870--878. doi:10.1214/009117905000000837. http://projecteuclid.org/euclid.aop/1151418486.

#### References

• Collevecchio, A. (2005). Limit theorems for vertex-reinforced jump processes on certain trees. Unpublished manuscript.
• Collevecchio, A. (2006). Limit theorems for reinforced random walks on certain trees. Probab. Theory Related Fields. To appear.
• Coppersmith, D. and Diaconis, P. (1987). Random walks with reinforcement. Unpublished manuscript.
• Dai, J. J. (2005). A once edge-reinforced random walk on a Galton--Watson tree is transient. Statist. Probab. Lett. 73 115--124.
• Davis, B. (1990). Reinforced random walk. Probab. Theory Related Fields 84 203--229.
• Davis, B. (1999). Reinforced and perturbed random walks. In Random Walks (P. Révész and B. Tóth, eds.) 9 113--126. Bolyai Soc. Math. Studies, Budapest.
• Davis, B. and Volkov, S. (2002). Continuous time vertex-reinforced jump processes. Probab. Theory Related Fields 84 281--300.
• Davis, B. and Volkov, S. (2004). Vertex-reinforced jump process on trees and finite graphs. Probab. Theory Related Fields 128 42--62.
• Diaconis, P. and Rolles, S. W. W. (2006). Bayesian analysis for reversible Markov chains. Ann. Statist. 34. To appear.
• Durrett, R., Kesten, H. and Limic, V. (2002). Once reinforced random walk. Probab. Theory Related Fields 122 567--592.
• Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931--958.
• Lyons, R. and Pemantle, R. (1992). Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 125--136.
• Lyons, R., Pemantle, R. and Peres, Y. (1996). Biased random walks on Galton--Watson trees. Probab. Theory Related Fields 106 249--264.
• Muliere, P., Secchi, P. and Walker, S. G. (2000). Urn schemes and reinforced random walks. Stochastic Process. Appl. 88 59--78.
• Othmer, H. and Stevens, A. (1997). Aggregation, blowup, and collapse: The ABCs of taxis and reinforced random walk. SIAM J. Appl. Math. 57 1044--1081.
• Pemantle, R. (1988). Phase transition in reinforced random walks and rwre on trees. Ann. Probab. 16 1229--1241.
• Pemantle, R. (2001). Random processes with reinforcement. Preprint. Available at www.math.upenn.edu/~pemantle/papers/Papers.html.
• Pemantle, R. (1992). Vertex-reinforced random walk. Probab. Theory Related Fields 92 117--136.
• Pemantle, R. and Peres, Y. (1996). On which graphs are all random walks in random environments transient? In Random Discrete Structures (R. Pemantle and Y. Peres, eds.) 207--211. Springer, New York.
• Pemantle, R. and Stacey, A. M. (2001). The branching random walk and contact process on Galton--Watson and nonhomogeneous trees. Ann. Probab. 29 1563--1590.
• Pemantle, R. and Volkov, S. (1999). Vertex-reinforced random walks on $\mathbbZ$ have finite range. Ann. Probab. 27 1368--1388.
• Rolles, S. W. W. (2003). How edge-reinforced random walk arises naturally. Probab. Theory Related Fields 126 243--260.
• Rolles, S. W. W. (2006). On the recurrence of edge-reinforced random walks on $\mathbbZ\times G$. Probab. Theory Related Fields. To appear.
• Sellke, T. (1994). Reinforced random walk on the $d$-dimensional integer lattice. Technical Report 94--26, Dept. Statistics, Purdue Univ.
• Volkov, S. (2001). Vertex-reinforced random walk on arbitrary graphs. Ann. Probab. 29 66--91.