Brownian motion on stable looptrees

2019/02/05 by Eleanor Archer, Archer, Eleanor · 1 citation
Mathematics · #60K37 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60K37

paper · pdf · doi:10.48550/arxiv.1902.01713

41 pages. Some further details added to proofs in Section 5

arxiv created 2020/12/12 · arxiv updated 2020/12/15

Abstract

In this article, we introduce Brownian motion on stable looptrees using resistance techniques. We prove an invariance principle characterising it as the scaling limit of random walks on discrete looptrees, and prove precise local and global bounds on its heat kernel. We also conduct a detailed investigation of the volume growth properties of stable looptrees, and show that the random volume and heat kernel fluctuations are locally log-logarithmic, and globally logarithmic around leading terms of r^α and t^{(-α)/(α+ 1)} respectively. These volume fluctuations are the same order as for the Brownian continuum random tree, but the upper volume fluctuations (and corresponding lower heat kernel fluctuations) are different to those of stable trees.

Cited by

• Infinite stable looptrees

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