Jump processes on the boundaries of random trees

TL;DR

This paper extends Kigami's work by analyzing trace processes on the Martin boundary of random trees, providing short-time heat kernel estimates and mean displacement bounds for these stochastic processes.

Contribution

It introduces the study of trace processes on the Martin boundary of random trees and establishes key heat kernel and displacement estimates.

Findings

Short-time on-diagonal heat kernel asymptotics

Bounds on mean displacements of the process

Extension of Kigami's deterministic tree results to random trees

Abstract

Kigami showed that a transient random walk on a deterministic infinite tree $T$ induces its trace process on the Martin boundary of $T$. In this paper, we will deal with trace processes on Martin boundaries of random trees instead of deterministic ones, and prove short time log-asymptotic of on-diagonal heat kernel estimates and estimates of mean displacements.