arXiv subjects

Dongjian Qian

Publications and source records attributed to Dongjian Qian.

4 recordsLinked to original sources

arXiv2026

Critical Self-Similar Markov Trees

Recently introduced and studied in arXiv:2407.07888, a self-similar Markov tree (ssMt) is a random decorated tree that vastly generalises the fragmentation tree. We study here the critical case that was left aside in arXiv:2407.07888. Borrowing techniques from branching random walk, in particular the recent result of Aïdékon--Hu--Shi arXiv:2409.01048, we can complete the picture by constructing critical ssMt, computing their fractal dimension and studying their associated harmonic and length measures using spinal decomposition.

math.PR↗

arXiv2024

Speed of Random Walks in Dirichlet Environment on a Galton-Watson Tree

This paper deals with a transient random walk in Dirichlet environment, or equivalently a linearly edge reinforced random walk, on a Galton-Watson tree. We compute the stationary distribution of the environment seen from the particle of an edge reinforced random walk. We obtain a formula for the speed and give a necessary and sufficient condition for the walk to have a positive speed under some moment conditions on the offspring distribution of the tree.

math.PR↗

arXiv2024

Scaling limit of the random walk on a Galton-Watson tree with regular varying offspring distribution

We consider a random walk on a Galton-Watson tree whose offspring distribution has a regular varying tail of order $κ\in (1,2)$. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable Lévy process, jointly with the convergence of the renormalised trace of the walk towards the continuum tree coded by the latter continuous-time height process.

math.PR↗

arXiv2023

Regular subspaces of symmetric stable processes

Roughly speaking, regular subspaces are regular Dirichlet forms that inherit the original forms with smaller domains. In this paper, regular subspaces of 1-dim symmetric $α$-stable processes are considered. The main result is that it admits proper regular subspaces if and only if $α\in [1,2]$. Moreover, for $α\in(1,2)$, the characterization of the regular subspaces is given. General 1-dim symmetric Lévy processes will also be investigated. It will be shown that whether it has proper regular subspaces is closely related to whether its sample paths have finite variation.

math.PR↗