On the cover time of Brownian motion on the Brownian continuum random tree

TL;DR

This paper proves a fundamental relationship between cover time and local times for Brownian motion on the Brownian CRT, and shows convergence of random walk cover times on critical Galton-Watson trees to this limit.

Contribution

It establishes the exact cover time characterization for Brownian motion on the Brownian CRT and links it to random walk cover times on Galton-Watson trees, using novel self-similarity and Ray-Knight techniques.

Findings

Cover time equals the infimum of times with positive local times everywhere.

Rescaled cover times of random walks on Galton-Watson trees converge to the CRT cover time.

Partially confirms Aldous's 1991 conjecture on cover-and-return times.

Abstract

Upon almost-every realisation of the Brownian continuum random tree (CRT), it is possible to define a canonical diffusion process or `Brownian motion'. The main result of this article establishes that the cover time of the Brownian motion on the Brownian CRT (i.e.\ the time taken by the process in question to visit the entire state space) is equal to the infimum over the times at which the associated local times are strictly positive everywhere. The proof of this result depends on the recursive self-similarity of the Brownian CRT and a novel version of the first Ray-Knight theorem for trees, which is of independent interest. As a consequence, we obtain that the suitably-rescaled cover times of simple random walks on critical, finite variance Galton-Watson trees converge in distribution with respect to their annealed laws to the cover time of Brownian motion on the Brownian CRT. Other families of graphs that have the Brownian CRT as a scaling limit are also covered. Additionally, we partially confirm a 1991 conjecture of David Aldous regarding related cover-and-return times.