Scaling limit for local times and return times of a randomly biased walk on a Galton-Watson tree

2023/10/11 by Kagan, Alexis
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2310.07278

Abstract

We consider a null recurrent random walk \mathbbX on a super-critical Galton Watson marked tree \mathbbT in the (sub-)diffusive regime. We are interested in the asymptotic behaviour of the local time of its root at n, which is the total amount of time spent by the random walk \mathbbX on the root of \mathbbT up to the time n, and in its n-th return time to the root of \mathbbT. We show that properly renormalized, this local time and this n-th return time respectively converge in law to the maximum and to an hitting time of some stable Lévy process. This paper aims in particular to extent the results of Y. Hu [Hu17].

Related

• Maximal local time of randomly biased random walks on a Galton-Watson tree
• Local times in critical generations of a random walk in random environment on trees
• Scaling limit of the subdiffusive random walk on a Galton-Watson tree in random environment
• The slow regime of randomly biased walks on trees
• Scaling limit of the recurrent biased random walk on a Galton-Watson tree