## Electronic Journal of Probability

### On the speed of once-reinforced biased random walk on trees

#### Abstract

We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 86, 32 pp.

Dates
Accepted: 27 July 2018
First available in Project Euclid: 12 September 2018

https://projecteuclid.org/euclid.ejp/1536717745

Digital Object Identifier
doi:10.1214/18-EJP208

Mathematical Reviews number (MathSciNet)
MR3858914

Zentralblatt MATH identifier
06964780

#### Citation

Collevecchio, Andrea; Holmes, Mark; Kious, Daniel. On the speed of once-reinforced biased random walk on trees. Electron. J. Probab. 23 (2018), paper no. 86, 32 pp. doi:10.1214/18-EJP208. https://projecteuclid.org/euclid.ejp/1536717745

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