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Volume growth and heat kernel estimates for the continuum random tree
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Croydon, David A. (2008) Volume growth and heat kernel estimates for the continuum random tree. Probability Theory and Related Fields, Vol.140 (No.1-2). pp. 207-238. doi:10.1007/s00440-007-0063-4 ISSN 0178-8051.
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Official URL: http://dx.doi.org/10.1007/s00440-007-0063-4
Abstract
In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost-surely logarithmic global fluctuations and log-logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r similar to 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t similar to 0 almost-surely. Finally, we prove that this quenched (almost-sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Library of Congress Subject Headings (LCSH): | Trees (Graph theory), Continuum (Mathematics), Brownian motion processes, Self-similar processes, Kernel functions, Heat equation, Fractals, Fluctuations (Physics) | ||||
Journal or Publication Title: | Probability Theory and Related Fields | ||||
Publisher: | Springer | ||||
ISSN: | 0178-8051 | ||||
Official Date: | January 2008 | ||||
Dates: |
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Volume: | Vol.140 | ||||
Number: | No.1-2 | ||||
Number of Pages: | 32 | ||||
Page Range: | pp. 207-238 | ||||
DOI: | 10.1007/s00440-007-0063-4 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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