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

TL;DR

This paper investigates the long-term behavior of once-reinforced biased random walks on Galton-Watson trees, revealing conditions under which reinforcement affects recurrence, transience, and speed, with implications for understanding reinforced stochastic processes on trees.

Contribution

It demonstrates that multiplicative reinforcement can induce recurrence even in ballistic regimes and establishes monotonicity of speed on regular trees with respect to reinforcement.

Findings

Reinforcement can cause recurrence despite ballistic underlying bias.

On Galton-Watson trees without leaves, the walk has positive speed in the transient regime.

Speed decreases monotonically with reinforcement on regular trees when initial speed is high.

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.